Documentation of 'org.netlib.lapack.LAPACK' Java class
LAPACK
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org.netlib.lapack

Class LAPACK


  • public abstract class LAPACK
extends java.lang.Object
    LAPACK provider which will attempt to access a native implementation and falling back to use F2J if none is available.
    See Also:
    http://sourceforge.net/projects/f2j, http://www.netlib.org/lapack/

    • Constructor Summary

      Constructors 
      Constructor and Description
      LAPACK() 

    • Method Summary

      All Methods Static Methods Instance Methods Abstract Methods Concrete Methods 
      Modifier and Type Method and Description
      abstract void dbdsdc(java.lang.String uplo, java.lang.String compq, int n, double[] d, double[] e, double[] u, int ldu, double[] vt, int ldvt, double[] q, int[] iq, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dbdsqr(java.lang.String uplo, int n, int ncvt, int nru, int ncc, double[] d, double[] e, double[] vt, int ldvt, double[] u, int ldu, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void ddisna(java.lang.String job, int m, int n, double[] d, double[] sep, org.netlib.util.intW info)
      ..
      abstract void dgbbrd(java.lang.String vect, int m, int n, int ncc, int kl, int ku, double[] ab, int ldab, double[] d, double[] e, double[] q, int ldq, double[] pt, int ldpt, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgbcon(java.lang.String norm, int n, int kl, int ku, double[] ab, int ldab, int[] ipiv, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgbequ(int m, int n, int kl, int ku, double[] ab, int ldab, double[] r, double[] c, org.netlib.util.doubleW rowcnd, org.netlib.util.doubleW colcnd, org.netlib.util.doubleW amax, org.netlib.util.intW info)
      ..
      abstract void dgbrfs(java.lang.String trans, int n, int kl, int ku, int nrhs, double[] ab, int ldab, double[] afb, int ldafb, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgbsv(int n, int kl, int ku, int nrhs, double[] ab, int ldab, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dgbsvx(java.lang.String fact, java.lang.String trans, int n, int kl, int ku, int nrhs, double[] ab, int ldab, double[] afb, int ldafb, int[] ipiv, org.netlib.util.StringW equed, double[] r, double[] c, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgbtf2(int m, int n, int kl, int ku, double[] ab, int ldab, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dgbtrf(int m, int n, int kl, int ku, double[] ab, int ldab, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dgbtrs(java.lang.String trans, int n, int kl, int ku, int nrhs, double[] ab, int ldab, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dgebak(java.lang.String job, java.lang.String side, int n, int ilo, int ihi, double[] scale, int m, double[] v, int ldv, org.netlib.util.intW info)
      ..
      abstract void dgebal(java.lang.String job, int n, double[] a, int lda, org.netlib.util.intW ilo, org.netlib.util.intW ihi, double[] scale, org.netlib.util.intW info)
      ..
      abstract void dgebd2(int m, int n, double[] a, int lda, double[] d, double[] e, double[] tauq, double[] taup, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgebrd(int m, int n, double[] a, int lda, double[] d, double[] e, double[] tauq, double[] taup, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgecon(java.lang.String norm, int n, double[] a, int lda, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgeequ(int m, int n, double[] a, int lda, double[] r, double[] c, org.netlib.util.doubleW rowcnd, org.netlib.util.doubleW colcnd, org.netlib.util.doubleW amax, org.netlib.util.intW info)
      ..
      void dgees(java.lang.String jobvs, java.lang.String sort, java.lang.Object select, int n, double[] a, int lda, org.netlib.util.intW sdim, double[] wr, double[] wi, double[] vs, int ldvs, double[] work, int lwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      void dgeesx(java.lang.String jobvs, java.lang.String sort, java.lang.Object select, java.lang.String sense, int n, double[] a, int lda, org.netlib.util.intW sdim, double[] wr, double[] wi, double[] vs, int ldvs, org.netlib.util.doubleW rconde, org.netlib.util.doubleW rcondv, double[] work, int lwork, int[] iwork, int liwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void dgeev(java.lang.String jobvl, java.lang.String jobvr, int n, double[] a, int lda, double[] wr, double[] wi, double[] vl, int ldvl, double[] vr, int ldvr, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgeevx(java.lang.String balanc, java.lang.String jobvl, java.lang.String jobvr, java.lang.String sense, int n, double[] a, int lda, double[] wr, double[] wi, double[] vl, int ldvl, double[] vr, int ldvr, org.netlib.util.intW ilo, org.netlib.util.intW ihi, double[] scale, org.netlib.util.doubleW abnrm, double[] rconde, double[] rcondv, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgegs(java.lang.String jobvsl, java.lang.String jobvsr, int n, double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, double[] vsl, int ldvsl, double[] vsr, int ldvsr, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgegv(java.lang.String jobvl, java.lang.String jobvr, int n, double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, double[] vl, int ldvl, double[] vr, int ldvr, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgehd2(int n, int ilo, int ihi, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgehrd(int n, int ilo, int ihi, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgelq2(int m, int n, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgelqf(int m, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgels(java.lang.String trans, int m, int n, int nrhs, double[] a, int lda, double[] b, int ldb, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgelsd(int m, int n, int nrhs, double[] a, int lda, double[] b, int ldb, double[] s, double rcond, org.netlib.util.intW rank, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgelss(int m, int n, int nrhs, double[] a, int lda, double[] b, int ldb, double[] s, double rcond, org.netlib.util.intW rank, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgelsx(int m, int n, int nrhs, double[] a, int lda, double[] b, int ldb, int[] jpvt, double rcond, org.netlib.util.intW rank, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgelsy(int m, int n, int nrhs, double[] a, int lda, double[] b, int ldb, int[] jpvt, double rcond, org.netlib.util.intW rank, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgeql2(int m, int n, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgeqlf(int m, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgeqp3(int m, int n, double[] a, int lda, int[] jpvt, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgeqpf(int m, int n, double[] a, int lda, int[] jpvt, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgeqr2(int m, int n, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgeqrf(int m, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgerfs(java.lang.String trans, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgerq2(int m, int n, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgerqf(int m, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgesc2(int n, double[] a, int lda, double[] rhs, int[] ipiv, int[] jpiv, org.netlib.util.doubleW scale)
      ..
      abstract void dgesdd(java.lang.String jobz, int m, int n, double[] a, int lda, double[] s, double[] u, int ldu, double[] vt, int ldvt, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgesv(int n, int nrhs, double[] a, int lda, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dgesvd(java.lang.String jobu, java.lang.String jobvt, int m, int n, double[] a, int lda, double[] s, double[] u, int ldu, double[] vt, int ldvt, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgesvx(java.lang.String fact, java.lang.String trans, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, int[] ipiv, org.netlib.util.StringW equed, double[] r, double[] c, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgetc2(int n, double[] a, int lda, int[] ipiv, int[] jpiv, org.netlib.util.intW info)
      ..
      abstract void dgetf2(int m, int n, double[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dgetrf(int m, int n, double[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dgetri(int n, double[] a, int lda, int[] ipiv, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgetrs(java.lang.String trans, int n, int nrhs, double[] a, int lda, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dggbak(java.lang.String job, java.lang.String side, int n, int ilo, int ihi, double[] lscale, double[] rscale, int m, double[] v, int ldv, org.netlib.util.intW info)
      ..
      abstract void dggbal(java.lang.String job, int n, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW ilo, org.netlib.util.intW ihi, double[] lscale, double[] rscale, double[] work, org.netlib.util.intW info)
      ..
      void dgges(java.lang.String jobvsl, java.lang.String jobvsr, java.lang.String sort, java.lang.Object selctg, int n, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW sdim, double[] alphar, double[] alphai, double[] beta, double[] vsl, int ldvsl, double[] vsr, int ldvsr, double[] work, int lwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      void dggesx(java.lang.String jobvsl, java.lang.String jobvsr, java.lang.String sort, java.lang.Object selctg, java.lang.String sense, int n, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW sdim, double[] alphar, double[] alphai, double[] beta, double[] vsl, int ldvsl, double[] vsr, int ldvsr, double[] rconde, double[] rcondv, double[] work, int lwork, int[] iwork, int liwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void dggev(java.lang.String jobvl, java.lang.String jobvr, int n, double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, double[] vl, int ldvl, double[] vr, int ldvr, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dggevx(java.lang.String balanc, java.lang.String jobvl, java.lang.String jobvr, java.lang.String sense, int n, double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, double[] vl, int ldvl, double[] vr, int ldvr, org.netlib.util.intW ilo, org.netlib.util.intW ihi, double[] lscale, double[] rscale, org.netlib.util.doubleW abnrm, org.netlib.util.doubleW bbnrm, double[] rconde, double[] rcondv, double[] work, int lwork, int[] iwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void dggglm(int n, int m, int p, double[] a, int lda, double[] b, int ldb, double[] d, double[] x, double[] y, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dgghrd(java.lang.String compq, java.lang.String compz, int n, int ilo, int ihi, double[] a, int lda, double[] b, int ldb, double[] q, int ldq, double[] z, int ldz, org.netlib.util.intW info)
      ..
      abstract void dgglse(int m, int n, int p, double[] a, int lda, double[] b, int ldb, double[] c, double[] d, double[] x, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dggqrf(int n, int m, int p, double[] a, int lda, double[] taua, double[] b, int ldb, double[] taub, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dggrqf(int m, int p, int n, double[] a, int lda, double[] taua, double[] b, int ldb, double[] taub, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dggsvd(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int n, int p, org.netlib.util.intW k, org.netlib.util.intW l, double[] a, int lda, double[] b, int ldb, double[] alpha, double[] beta, double[] u, int ldu, double[] v, int ldv, double[] q, int ldq, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dggsvp(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int p, int n, double[] a, int lda, double[] b, int ldb, double tola, double tolb, org.netlib.util.intW k, org.netlib.util.intW l, double[] u, int ldu, double[] v, int ldv, double[] q, int ldq, int[] iwork, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dgtcon(java.lang.String norm, int n, double[] dl, double[] d, double[] du, double[] du2, int[] ipiv, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgtrfs(java.lang.String trans, int n, int nrhs, double[] dl, double[] d, double[] du, double[] dlf, double[] df, double[] duf, double[] du2, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgtsv(int n, int nrhs, double[] dl, double[] d, double[] du, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dgtsvx(java.lang.String fact, java.lang.String trans, int n, int nrhs, double[] dl, double[] d, double[] du, double[] dlf, double[] df, double[] duf, double[] du2, int[] ipiv, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dgttrf(int n, double[] dl, double[] d, double[] du, double[] du2, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dgttrs(java.lang.String trans, int n, int nrhs, double[] dl, double[] d, double[] du, double[] du2, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dgtts2(int itrans, int n, int nrhs, double[] dl, double[] d, double[] du, double[] du2, int[] ipiv, double[] b, int ldb)
      ..
      abstract void dhgeqz(java.lang.String job, java.lang.String compq, java.lang.String compz, int n, int ilo, int ihi, double[] h, int ldh, double[] t, int ldt, double[] alphar, double[] alphai, double[] beta, double[] q, int ldq, double[] z, int ldz, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dhsein(java.lang.String side, java.lang.String eigsrc, java.lang.String initv, boolean[] select, int n, double[] h, int ldh, double[] wr, double[] wi, double[] vl, int ldvl, double[] vr, int ldvr, int mm, org.netlib.util.intW m, double[] work, int[] ifaill, int[] ifailr, org.netlib.util.intW info)
      ..
      abstract void dhseqr(java.lang.String job, java.lang.String compz, int n, int ilo, int ihi, double[] h, int ldh, double[] wr, double[] wi, double[] z, int ldz, double[] work, int lwork, org.netlib.util.intW info)
      ..
      boolean disnan(double din)
      ..
      abstract void dlabad(org.netlib.util.doubleW small, org.netlib.util.doubleW large)
      ..
      abstract void dlabrd(int m, int n, int nb, double[] a, int lda, double[] d, double[] e, double[] tauq, double[] taup, double[] x, int ldx, double[] y, int ldy)
      ..
      void dlacn2(int n, double[] v, double[] x, int[] isgn, org.netlib.util.doubleW est, org.netlib.util.intW kase, int[] isave)
      ..
      abstract void dlacon(int n, double[] v, double[] x, int[] isgn, org.netlib.util.doubleW est, org.netlib.util.intW kase)
      ..
      abstract void dlacpy(java.lang.String uplo, int m, int n, double[] a, int lda, double[] b, int ldb)
      ..
      abstract void dladiv(double a, double b, double c, double d, org.netlib.util.doubleW p, org.netlib.util.doubleW q)
      ..
      abstract void dlae2(double a, double b, double c, org.netlib.util.doubleW rt1, org.netlib.util.doubleW rt2)
      ..
      abstract void dlaebz(int ijob, int nitmax, int n, int mmax, int minp, int nbmin, double abstol, double reltol, double pivmin, double[] d, double[] e, double[] e2, int[] nval, double[] ab, double[] c, org.netlib.util.intW mout, int[] nab, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlaed0(int icompq, int qsiz, int n, double[] d, double[] e, double[] q, int ldq, double[] qstore, int ldqs, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlaed1(int n, double[] d, double[] q, int ldq, int[] indxq, org.netlib.util.doubleW rho, int cutpnt, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlaed2(org.netlib.util.intW k, int n, int n1, double[] d, double[] q, int ldq, int[] indxq, org.netlib.util.doubleW rho, double[] z, double[] dlamda, double[] w, double[] q2, int[] indx, int[] indxc, int[] indxp, int[] coltyp, org.netlib.util.intW info)
      ..
      abstract void dlaed3(int k, int n, int n1, double[] d, double[] q, int ldq, double rho, double[] dlamda, double[] q2, int[] indx, int[] ctot, double[] w, double[] s, org.netlib.util.intW info)
      ..
      abstract void dlaed4(int n, int i, double[] d, double[] z, double[] delta, double rho, org.netlib.util.doubleW dlam, org.netlib.util.intW info)
      ..
      abstract void dlaed5(int i, double[] d, double[] z, double[] delta, double rho, org.netlib.util.doubleW dlam)
      ..
      abstract void dlaed6(int kniter, boolean orgati, double rho, double[] d, double[] z, double finit, org.netlib.util.doubleW tau, org.netlib.util.intW info)
      ..
      abstract void dlaed7(int icompq, int n, int qsiz, int tlvls, int curlvl, int curpbm, double[] d, double[] q, int ldq, int[] indxq, org.netlib.util.doubleW rho, int cutpnt, double[] qstore, int[] qptr, int[] prmptr, int[] perm, int[] givptr, int[] givcol, double[] givnum, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlaed8(int icompq, org.netlib.util.intW k, int n, int qsiz, double[] d, double[] q, int ldq, int[] indxq, org.netlib.util.doubleW rho, int cutpnt, double[] z, double[] dlamda, double[] q2, int ldq2, double[] w, int[] perm, org.netlib.util.intW givptr, int[] givcol, double[] givnum, int[] indxp, int[] indx, org.netlib.util.intW info)
      ..
      abstract void dlaed9(int k, int kstart, int kstop, int n, double[] d, double[] q, int ldq, double rho, double[] dlamda, double[] w, double[] s, int lds, org.netlib.util.intW info)
      ..
      abstract void dlaeda(int n, int tlvls, int curlvl, int curpbm, int[] prmptr, int[] perm, int[] givptr, int[] givcol, double[] givnum, double[] q, int[] qptr, double[] z, double[] ztemp, org.netlib.util.intW info)
      ..
      abstract void dlaein(boolean rightv, boolean noinit, int n, double[] h, int ldh, double wr, double wi, double[] vr, double[] vi, double[] b, int ldb, double[] work, double eps3, double smlnum, double bignum, org.netlib.util.intW info)
      ..
      abstract void dlaev2(double a, double b, double c, org.netlib.util.doubleW rt1, org.netlib.util.doubleW rt2, org.netlib.util.doubleW cs1, org.netlib.util.doubleW sn1)
      ..
      abstract void dlaexc(boolean wantq, int n, double[] t, int ldt, double[] q, int ldq, int j1, int n1, int n2, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlag2(double[] a, int lda, double[] b, int ldb, double safmin, org.netlib.util.doubleW scale1, org.netlib.util.doubleW scale2, org.netlib.util.doubleW wr1, org.netlib.util.doubleW wr2, org.netlib.util.doubleW wi)
      ..
      void dlag2s(int m, int n, double[] a, int lda, float[] sa, int ldsa, org.netlib.util.intW info)
      ..
      abstract void dlags2(boolean upper, double a1, double a2, double a3, double b1, double b2, double b3, org.netlib.util.doubleW csu, org.netlib.util.doubleW snu, org.netlib.util.doubleW csv, org.netlib.util.doubleW snv, org.netlib.util.doubleW csq, org.netlib.util.doubleW snq)
      ..
      abstract void dlagtf(int n, double[] a, double lambda, double[] b, double[] c, double tol, double[] d, int[] in, org.netlib.util.intW info)
      ..
      abstract void dlagtm(java.lang.String trans, int n, int nrhs, double alpha, double[] dl, double[] d, double[] du, double[] x, int ldx, double beta, double[] b, int ldb)
      ..
      abstract void dlagts(int job, int n, double[] a, double[] b, double[] c, double[] d, int[] in, double[] y, org.netlib.util.doubleW tol, org.netlib.util.intW info)
      ..
      abstract void dlagv2(double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, org.netlib.util.doubleW csl, org.netlib.util.doubleW snl, org.netlib.util.doubleW csr, org.netlib.util.doubleW snr)
      ..
      abstract void dlahqr(boolean wantt, boolean wantz, int n, int ilo, int ihi, double[] h, int ldh, double[] wr, double[] wi, int iloz, int ihiz, double[] z, int ldz, org.netlib.util.intW info)
      ..
      void dlahr2(int n, int k, int nb, double[] a, int lda, double[] tau, double[] t, int ldt, double[] y, int ldy)
      ..
      abstract void dlahrd(int n, int k, int nb, double[] a, int lda, double[] tau, double[] t, int ldt, double[] y, int ldy)
      ..
      abstract void dlaic1(int job, int j, double[] x, double sest, double[] w, double gamma, org.netlib.util.doubleW sestpr, org.netlib.util.doubleW s, org.netlib.util.doubleW c)
      ..
      boolean dlaisnan(double din1, double din2)
      ..
      abstract void dlaln2(boolean ltrans, int na, int nw, double smin, double ca, double[] a, int lda, double d1, double d2, double[] b, int ldb, double wr, double wi, double[] x, int ldx, org.netlib.util.doubleW scale, org.netlib.util.doubleW xnorm, org.netlib.util.intW info)
      ..
      abstract void dlals0(int icompq, int nl, int nr, int sqre, int nrhs, double[] b, int ldb, double[] bx, int ldbx, int[] perm, int givptr, int[] givcol, int ldgcol, double[] givnum, int ldgnum, double[] poles, double[] difl, double[] difr, double[] z, int k, double c, double s, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlalsa(int icompq, int smlsiz, int n, int nrhs, double[] b, int ldb, double[] bx, int ldbx, double[] u, int ldu, double[] vt, int[] k, double[] difl, double[] difr, double[] z, double[] poles, int[] givptr, int[] givcol, int ldgcol, int[] perm, double[] givnum, double[] c, double[] s, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlalsd(java.lang.String uplo, int smlsiz, int n, int nrhs, double[] d, double[] e, double[] b, int ldb, double rcond, org.netlib.util.intW rank, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlamc1(org.netlib.util.intW beta, org.netlib.util.intW t, org.netlib.util.booleanW rnd, org.netlib.util.booleanW ieee1)
      ..
      abstract void dlamc2(org.netlib.util.intW beta, org.netlib.util.intW t, org.netlib.util.booleanW rnd, org.netlib.util.doubleW eps, org.netlib.util.intW emin, org.netlib.util.doubleW rmin, org.netlib.util.intW emax, org.netlib.util.doubleW rmax)
      ..
      double dlamc3(double a, double b)
      ..
      abstract void dlamc4(org.netlib.util.intW emin, double start, int base)
      ..
      abstract void dlamc5(int beta, int p, int emin, boolean ieee, org.netlib.util.intW emax, org.netlib.util.doubleW rmax)
      ..
      abstract double dlamch(java.lang.String cmach)
      ..
      abstract void dlamrg(int n1, int n2, double[] a, int dtrd1, int dtrd2, int[] index)
      ..
      int dlaneg(int n, double[] d, double[] lld, double sigma, double pivmin, int r)
      ..
      double dlangb(java.lang.String norm, int n, int kl, int ku, double[] ab, int ldab, double[] work)
      ..
      double dlange(java.lang.String norm, int m, int n, double[] a, int lda, double[] work)
      ..
      double dlangt(java.lang.String norm, int n, double[] dl, double[] d, double[] du)
      ..
      double dlanhs(java.lang.String norm, int n, double[] a, int lda, double[] work)
      ..
      double dlansb(java.lang.String norm, java.lang.String uplo, int n, int k, double[] ab, int ldab, double[] work)
      ..
      double dlansp(java.lang.String norm, java.lang.String uplo, int n, double[] ap, double[] work)
      ..
      double dlanst(java.lang.String norm, int n, double[] d, double[] e)
      ..
      double dlansy(java.lang.String norm, java.lang.String uplo, int n, double[] a, int lda, double[] work)
      ..
      double dlantb(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, int k, double[] ab, int ldab, double[] work)
      ..
      double dlantp(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, double[] ap, double[] work)
      ..
      double dlantr(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int m, int n, double[] a, int lda, double[] work)
      ..
      abstract void dlanv2(org.netlib.util.doubleW a, org.netlib.util.doubleW b, org.netlib.util.doubleW c, org.netlib.util.doubleW d, org.netlib.util.doubleW rt1r, org.netlib.util.doubleW rt1i, org.netlib.util.doubleW rt2r, org.netlib.util.doubleW rt2i, org.netlib.util.doubleW cs, org.netlib.util.doubleW sn)
      ..
      abstract void dlapll(int n, double[] x, int incx, double[] y, int incy, org.netlib.util.doubleW ssmin)
      ..
      abstract void dlapmt(boolean forwrd, int m, int n, double[] x, int ldx, int[] k)
      ..
      double dlapy2(double x, double y)
      ..
      double dlapy3(double x, double y, double z)
      ..
      abstract void dlaqgb(int m, int n, int kl, int ku, double[] ab, int ldab, double[] r, double[] c, double rowcnd, double colcnd, double amax, org.netlib.util.StringW equed)
      ..
      abstract void dlaqge(int m, int n, double[] a, int lda, double[] r, double[] c, double rowcnd, double colcnd, double amax, org.netlib.util.StringW equed)
      ..
      abstract void dlaqp2(int m, int n, int offset, double[] a, int lda, int[] jpvt, double[] tau, double[] vn1, double[] vn2, double[] work)
      ..
      abstract void dlaqps(int m, int n, int offset, int nb, org.netlib.util.intW kb, double[] a, int lda, int[] jpvt, double[] tau, double[] vn1, double[] vn2, double[] auxv, double[] f, int ldf)
      ..
      void dlaqr0(boolean wantt, boolean wantz, int n, int ilo, int ihi, double[] h, int ldh, double[] wr, double[] wi, int iloz, int ihiz, double[] z, int ldz, double[] work, int lwork, org.netlib.util.intW info)
      ..
      void dlaqr1(int n, double[] h, int ldh, double sr1, double si1, double sr2, double si2, double[] v)
      ..
      void dlaqr2(boolean wantt, boolean wantz, int n, int ktop, int kbot, int nw, double[] h, int ldh, int iloz, int ihiz, double[] z, int ldz, org.netlib.util.intW ns, org.netlib.util.intW nd, double[] sr, double[] si, double[] v, int ldv, int nh, double[] t, int ldt, int nv, double[] wv, int ldwv, double[] work, int lwork)
      ..
      void dlaqr3(boolean wantt, boolean wantz, int n, int ktop, int kbot, int nw, double[] h, int ldh, int iloz, int ihiz, double[] z, int ldz, org.netlib.util.intW ns, org.netlib.util.intW nd, double[] sr, double[] si, double[] v, int ldv, int nh, double[] t, int ldt, int nv, double[] wv, int ldwv, double[] work, int lwork)
      ..
      void dlaqr4(boolean wantt, boolean wantz, int n, int ilo, int ihi, double[] h, int ldh, double[] wr, double[] wi, int iloz, int ihiz, double[] z, int ldz, double[] work, int lwork, org.netlib.util.intW info)
      ..
      void dlaqr5(boolean wantt, boolean wantz, int kacc22, int n, int ktop, int kbot, int nshfts, double[] sr, double[] si, double[] h, int ldh, int iloz, int ihiz, double[] z, int ldz, double[] v, int ldv, double[] u, int ldu, int nv, double[] wv, int ldwv, int nh, double[] wh, int ldwh)
      ..
      abstract void dlaqsb(java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] s, double scond, double amax, org.netlib.util.StringW equed)
      ..
      abstract void dlaqsp(java.lang.String uplo, int n, double[] ap, double[] s, double scond, double amax, org.netlib.util.StringW equed)
      ..
      abstract void dlaqsy(java.lang.String uplo, int n, double[] a, int lda, double[] s, double scond, double amax, org.netlib.util.StringW equed)
      ..
      abstract void dlaqtr(boolean ltran, boolean lreal, int n, double[] t, int ldt, double[] b, double w, org.netlib.util.doubleW scale, double[] x, double[] work, org.netlib.util.intW info)
      ..
      void dlar1v(int n, int b1, int bn, double lambda, double[] d, double[] l, double[] ld, double[] lld, double pivmin, double gaptol, double[] z, boolean wantnc, org.netlib.util.intW negcnt, org.netlib.util.doubleW ztz, org.netlib.util.doubleW mingma, org.netlib.util.intW r, int[] isuppz, org.netlib.util.doubleW nrminv, org.netlib.util.doubleW resid, org.netlib.util.doubleW rqcorr, double[] work)
      ..
      abstract void dlar2v(int n, double[] x, double[] y, double[] z, int incx, double[] c, double[] s, int incc)
      ..
      abstract void dlarf(java.lang.String side, int m, int n, double[] v, int incv, double tau, double[] c, int Ldc, double[] work)
      ..
      abstract void dlarfb(java.lang.String side, java.lang.String trans, java.lang.String direct, java.lang.String storev, int m, int n, int k, double[] v, int ldv, double[] t, int ldt, double[] c, int Ldc, double[] work, int ldwork)
      ..
      abstract void dlarfg(int n, org.netlib.util.doubleW alpha, double[] x, int incx, org.netlib.util.doubleW tau)
      ..
      abstract void dlarft(java.lang.String direct, java.lang.String storev, int n, int k, double[] v, int ldv, double[] tau, double[] t, int ldt)
      ..
      abstract void dlarfx(java.lang.String side, int m, int n, double[] v, double tau, double[] c, int Ldc, double[] work)
      ..
      abstract void dlargv(int n, double[] x, int incx, double[] y, int incy, double[] c, int incc)
      ..
      abstract void dlarnv(int idist, int[] iseed, int n, double[] x)
      ..
      void dlarra(int n, double[] d, double[] e, double[] e2, double spltol, double tnrm, org.netlib.util.intW nsplit, int[] isplit, org.netlib.util.intW info)
      ..
      void dlarrb(int n, double[] d, double[] lld, int ifirst, int ilast, double rtol1, double rtol2, int offset, double[] w, double[] wgap, double[] werr, double[] work, int[] iwork, double pivmin, double spdiam, int twist, org.netlib.util.intW info)
      ..
      void dlarrc(java.lang.String jobt, int n, double vl, double vu, double[] d, double[] e, double pivmin, org.netlib.util.intW eigcnt, org.netlib.util.intW lcnt, org.netlib.util.intW rcnt, org.netlib.util.intW info)
      ..
      void dlarrd(java.lang.String range, java.lang.String order, int n, double vl, double vu, int il, int iu, double[] gers, double reltol, double[] d, double[] e, double[] e2, double pivmin, int nsplit, int[] isplit, org.netlib.util.intW m, double[] w, double[] werr, org.netlib.util.doubleW wl, org.netlib.util.doubleW wu, int[] iblock, int[] indexw, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      void dlarre(java.lang.String range, int n, org.netlib.util.doubleW vl, org.netlib.util.doubleW vu, int il, int iu, double[] d, double[] e, double[] e2, double rtol1, double rtol2, double spltol, org.netlib.util.intW nsplit, int[] isplit, org.netlib.util.intW m, double[] w, double[] werr, double[] wgap, int[] iblock, int[] indexw, double[] gers, org.netlib.util.doubleW pivmin, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      void dlarrf(int n, double[] d, double[] l, double[] ld, int clstrt, int clend, double[] w, double[] wgap, double[] werr, double spdiam, double clgapl, double clgapr, double pivmin, org.netlib.util.doubleW sigma, double[] dplus, double[] lplus, double[] work, org.netlib.util.intW info)
      ..
      void dlarrj(int n, double[] d, double[] e2, int ifirst, int ilast, double rtol, int offset, double[] w, double[] werr, double[] work, int[] iwork, double pivmin, double spdiam, org.netlib.util.intW info)
      ..
      void dlarrk(int n, int iw, double gl, double gu, double[] d, double[] e2, double pivmin, double reltol, org.netlib.util.doubleW w, org.netlib.util.doubleW werr, org.netlib.util.intW info)
      ..
      void dlarrr(int n, double[] d, double[] e, org.netlib.util.intW info)
      ..
      void dlarrv(int n, double vl, double vu, double[] d, double[] l, double pivmin, int[] isplit, int m, int dol, int dou, double minrgp, org.netlib.util.doubleW rtol1, org.netlib.util.doubleW rtol2, double[] w, double[] werr, double[] wgap, int[] iblock, int[] indexw, double[] gers, double[] z, int ldz, int[] isuppz, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlartg(double f, double g, org.netlib.util.doubleW cs, org.netlib.util.doubleW sn, org.netlib.util.doubleW r)
      ..
      abstract void dlartv(int n, double[] x, int incx, double[] y, int incy, double[] c, double[] s, int incc)
      ..
      abstract void dlaruv(int[] iseed, int n, double[] x)
      ..
      abstract void dlarz(java.lang.String side, int m, int n, int l, double[] v, int incv, double tau, double[] c, int Ldc, double[] work)
      ..
      abstract void dlarzb(java.lang.String side, java.lang.String trans, java.lang.String direct, java.lang.String storev, int m, int n, int k, int l, double[] v, int ldv, double[] t, int ldt, double[] c, int Ldc, double[] work, int ldwork)
      ..
      abstract void dlarzt(java.lang.String direct, java.lang.String storev, int n, int k, double[] v, int ldv, double[] tau, double[] t, int ldt)
      ..
      abstract void dlas2(double f, double g, double h, org.netlib.util.doubleW ssmin, org.netlib.util.doubleW ssmax)
      ..
      abstract void dlascl(java.lang.String type, int kl, int ku, double cfrom, double cto, int m, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dlasd0(int n, int sqre, double[] d, double[] e, double[] u, int ldu, double[] vt, int ldvt, int smlsiz, int[] iwork, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasd1(int nl, int nr, int sqre, double[] d, org.netlib.util.doubleW alpha, org.netlib.util.doubleW beta, double[] u, int ldu, double[] vt, int ldvt, int[] idxq, int[] iwork, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasd2(int nl, int nr, int sqre, org.netlib.util.intW k, double[] d, double[] z, double alpha, double beta, double[] u, int ldu, double[] vt, int ldvt, double[] dsigma, double[] u2, int ldu2, double[] vt2, int ldvt2, int[] idxp, int[] idx, int[] idxc, int[] idxq, int[] coltyp, org.netlib.util.intW info)
      ..
      abstract void dlasd3(int nl, int nr, int sqre, int k, double[] d, double[] q, int ldq, double[] dsigma, double[] u, int ldu, double[] u2, int ldu2, double[] vt, int ldvt, double[] vt2, int ldvt2, int[] idxc, int[] ctot, double[] z, org.netlib.util.intW info)
      ..
      abstract void dlasd4(int n, int i, double[] d, double[] z, double[] delta, double rho, org.netlib.util.doubleW sigma, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasd5(int i, double[] d, double[] z, double[] delta, double rho, org.netlib.util.doubleW dsigma, double[] work)
      ..
      abstract void dlasd6(int icompq, int nl, int nr, int sqre, double[] d, double[] vf, double[] vl, org.netlib.util.doubleW alpha, org.netlib.util.doubleW beta, int[] idxq, int[] perm, org.netlib.util.intW givptr, int[] givcol, int ldgcol, double[] givnum, int ldgnum, double[] poles, double[] difl, double[] difr, double[] z, org.netlib.util.intW k, org.netlib.util.doubleW c, org.netlib.util.doubleW s, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlasd7(int icompq, int nl, int nr, int sqre, org.netlib.util.intW k, double[] d, double[] z, double[] zw, double[] vf, double[] vfw, double[] vl, double[] vlw, double alpha, double beta, double[] dsigma, int[] idx, int[] idxp, int[] idxq, int[] perm, org.netlib.util.intW givptr, int[] givcol, int ldgcol, double[] givnum, int ldgnum, org.netlib.util.doubleW c, org.netlib.util.doubleW s, org.netlib.util.intW info)
      ..
      abstract void dlasd8(int icompq, int k, double[] d, double[] z, double[] vf, double[] vl, double[] difl, double[] difr, int lddifr, double[] dsigma, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasda(int icompq, int smlsiz, int n, int sqre, double[] d, double[] e, double[] u, int ldu, double[] vt, int[] k, double[] difl, double[] difr, double[] z, double[] poles, int[] givptr, int[] givcol, int ldgcol, int[] perm, double[] givnum, double[] c, double[] s, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dlasdq(java.lang.String uplo, int sqre, int n, int ncvt, int nru, int ncc, double[] d, double[] e, double[] vt, int ldvt, double[] u, int ldu, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasdt(int n, org.netlib.util.intW lvl, org.netlib.util.intW nd, int[] inode, int[] ndiml, int[] ndimr, int msub)
      ..
      abstract void dlaset(java.lang.String uplo, int m, int n, double alpha, double beta, double[] a, int lda)
      ..
      abstract void dlasq1(int n, double[] d, double[] e, double[] work, org.netlib.util.intW info)
      ..
      abstract void dlasq2(int n, double[] z, org.netlib.util.intW info)
      ..
      abstract void dlasq3(int i0, org.netlib.util.intW n0, double[] z, int pp, org.netlib.util.doubleW dmin, org.netlib.util.doubleW sigma, org.netlib.util.doubleW desig, org.netlib.util.doubleW qmax, org.netlib.util.intW nfail, org.netlib.util.intW iter, org.netlib.util.intW ndiv, boolean ieee)
      ..
      abstract void dlasq4(int i0, int n0, double[] z, int pp, int n0in, double dmin, double dmin1, double dmin2, double dn, double dn1, double dn2, org.netlib.util.doubleW tau, org.netlib.util.intW ttype)
      ..
      abstract void dlasq5(int i0, int n0, double[] z, int pp, double tau, org.netlib.util.doubleW dmin, org.netlib.util.doubleW dmin1, org.netlib.util.doubleW dmin2, org.netlib.util.doubleW dn, org.netlib.util.doubleW dnm1, org.netlib.util.doubleW dnm2, boolean ieee)
      ..
      abstract void dlasq6(int i0, int n0, double[] z, int pp, org.netlib.util.doubleW dmin, org.netlib.util.doubleW dmin1, org.netlib.util.doubleW dmin2, org.netlib.util.doubleW dn, org.netlib.util.doubleW dnm1, org.netlib.util.doubleW dnm2)
      ..
      abstract void dlasr(java.lang.String side, java.lang.String pivot, java.lang.String direct, int m, int n, double[] c, double[] s, double[] a, int lda)
      ..
      abstract void dlasrt(java.lang.String id, int n, double[] d, org.netlib.util.intW info)
      ..
      abstract void dlassq(int n, double[] x, int incx, org.netlib.util.doubleW scale, org.netlib.util.doubleW sumsq)
      ..
      abstract void dlasv2(double f, double g, double h, org.netlib.util.doubleW ssmin, org.netlib.util.doubleW ssmax, org.netlib.util.doubleW snr, org.netlib.util.doubleW csr, org.netlib.util.doubleW snl, org.netlib.util.doubleW csl)
      ..
      abstract void dlaswp(int n, double[] a, int lda, int k1, int k2, int[] ipiv, int incx)
      ..
      abstract void dlasy2(boolean ltranl, boolean ltranr, int isgn, int n1, int n2, double[] tl, int ldtl, double[] tr, int ldtr, double[] b, int ldb, org.netlib.util.doubleW scale, double[] x, int ldx, org.netlib.util.doubleW xnorm, org.netlib.util.intW info)
      ..
      abstract void dlasyf(java.lang.String uplo, int n, int nb, org.netlib.util.intW kb, double[] a, int lda, int[] ipiv, double[] w, int ldw, org.netlib.util.intW info)
      ..
      abstract void dlatbs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, int kd, double[] ab, int ldab, double[] x, org.netlib.util.doubleW scale, double[] cnorm, org.netlib.util.intW info)
      ..
      abstract void dlatdf(int ijob, int n, double[] z, int ldz, double[] rhs, org.netlib.util.doubleW rdsum, org.netlib.util.doubleW rdscal, int[] ipiv, int[] jpiv)
      ..
      abstract void dlatps(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, double[] ap, double[] x, org.netlib.util.doubleW scale, double[] cnorm, org.netlib.util.intW info)
      ..
      abstract void dlatrd(java.lang.String uplo, int n, int nb, double[] a, int lda, double[] e, double[] tau, double[] w, int ldw)
      ..
      abstract void dlatrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, double[] a, int lda, double[] x, org.netlib.util.doubleW scale, double[] cnorm, org.netlib.util.intW info)
      ..
      abstract void dlatrz(int m, int n, int l, double[] a, int lda, double[] tau, double[] work)
      ..
      abstract void dlatzm(java.lang.String side, int m, int n, double[] v, int incv, double tau, double[] c1, double[] c2, int Ldc, double[] work)
      ..
      abstract void dlauu2(java.lang.String uplo, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dlauum(java.lang.String uplo, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      void dlazq3(int i0, org.netlib.util.intW n0, double[] z, int pp, org.netlib.util.doubleW dmin, org.netlib.util.doubleW sigma, org.netlib.util.doubleW desig, org.netlib.util.doubleW qmax, org.netlib.util.intW nfail, org.netlib.util.intW iter, org.netlib.util.intW ndiv, boolean ieee, org.netlib.util.intW ttype, org.netlib.util.doubleW dmin1, org.netlib.util.doubleW dmin2, org.netlib.util.doubleW dn, org.netlib.util.doubleW dn1, org.netlib.util.doubleW dn2, org.netlib.util.doubleW tau)
      ..
      void dlazq4(int i0, int n0, double[] z, int pp, int n0in, double dmin, double dmin1, double dmin2, double dn, double dn1, double dn2, org.netlib.util.doubleW tau, org.netlib.util.intW ttype, org.netlib.util.doubleW g)
      ..
      abstract void dopgtr(java.lang.String uplo, int n, double[] ap, double[] tau, double[] q, int ldq, double[] work, org.netlib.util.intW info)
      ..
      abstract void dopmtr(java.lang.String side, java.lang.String uplo, java.lang.String trans, int m, int n, double[] ap, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorg2l(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorg2r(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorgbr(java.lang.String vect, int m, int n, int k, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorghr(int n, int ilo, int ihi, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorgl2(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorglq(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorgql(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorgqr(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorgr2(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorgrq(int m, int n, int k, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorgtr(java.lang.String uplo, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorm2l(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dorm2r(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dormbr(java.lang.String vect, java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormhr(java.lang.String side, java.lang.String trans, int m, int n, int ilo, int ihi, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dorml2(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dormlq(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormql(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormqr(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormr2(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dormr3(java.lang.String side, java.lang.String trans, int m, int n, int k, int l, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, org.netlib.util.intW info)
      ..
      abstract void dormrq(java.lang.String side, java.lang.String trans, int m, int n, int k, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormrz(java.lang.String side, java.lang.String trans, int m, int n, int k, int l, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dormtr(java.lang.String side, java.lang.String uplo, java.lang.String trans, int m, int n, double[] a, int lda, double[] tau, double[] c, int Ldc, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dpbcon(java.lang.String uplo, int n, int kd, double[] ab, int ldab, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpbequ(java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] s, org.netlib.util.doubleW scond, org.netlib.util.doubleW amax, org.netlib.util.intW info)
      ..
      abstract void dpbrfs(java.lang.String uplo, int n, int kd, int nrhs, double[] ab, int ldab, double[] afb, int ldafb, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpbstf(java.lang.String uplo, int n, int kd, double[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void dpbsv(java.lang.String uplo, int n, int kd, int nrhs, double[] ab, int ldab, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dpbsvx(java.lang.String fact, java.lang.String uplo, int n, int kd, int nrhs, double[] ab, int ldab, double[] afb, int ldafb, org.netlib.util.StringW equed, double[] s, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpbtf2(java.lang.String uplo, int n, int kd, double[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void dpbtrf(java.lang.String uplo, int n, int kd, double[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void dpbtrs(java.lang.String uplo, int n, int kd, int nrhs, double[] ab, int ldab, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dpocon(java.lang.String uplo, int n, double[] a, int lda, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpoequ(int n, double[] a, int lda, double[] s, org.netlib.util.doubleW scond, org.netlib.util.doubleW amax, org.netlib.util.intW info)
      ..
      abstract void dporfs(java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dposv(java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dposvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, org.netlib.util.StringW equed, double[] s, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpotf2(java.lang.String uplo, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dpotrf(java.lang.String uplo, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dpotri(java.lang.String uplo, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dpotrs(java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dppcon(java.lang.String uplo, int n, double[] ap, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dppequ(java.lang.String uplo, int n, double[] ap, double[] s, org.netlib.util.doubleW scond, org.netlib.util.doubleW amax, org.netlib.util.intW info)
      ..
      abstract void dpprfs(java.lang.String uplo, int n, int nrhs, double[] ap, double[] afp, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dppsv(java.lang.String uplo, int n, int nrhs, double[] ap, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dppsvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, double[] ap, double[] afp, org.netlib.util.StringW equed, double[] s, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dpptrf(java.lang.String uplo, int n, double[] ap, org.netlib.util.intW info)
      ..
      abstract void dpptri(java.lang.String uplo, int n, double[] ap, org.netlib.util.intW info)
      ..
      abstract void dpptrs(java.lang.String uplo, int n, int nrhs, double[] ap, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dptcon(int n, double[] d, double[] e, double anorm, org.netlib.util.doubleW rcond, double[] work, org.netlib.util.intW info)
      ..
      abstract void dpteqr(java.lang.String compz, int n, double[] d, double[] e, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dptrfs(int n, int nrhs, double[] d, double[] e, double[] df, double[] ef, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, org.netlib.util.intW info)
      ..
      abstract void dptsv(int n, int nrhs, double[] d, double[] e, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dptsvx(java.lang.String fact, int n, int nrhs, double[] d, double[] e, double[] df, double[] ef, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, org.netlib.util.intW info)
      ..
      abstract void dpttrf(int n, double[] d, double[] e, org.netlib.util.intW info)
      ..
      abstract void dpttrs(int n, int nrhs, double[] d, double[] e, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dptts2(int n, int nrhs, double[] d, double[] e, double[] b, int ldb)
      ..
      abstract void drscl(int n, double sa, double[] sx, int incx)
      ..
      abstract void dsbev(java.lang.String jobz, java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] w, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsbevd(java.lang.String jobz, java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsbevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] q, int ldq, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsbgst(java.lang.String vect, java.lang.String uplo, int n, int ka, int kb, double[] ab, int ldab, double[] bb, int ldbb, double[] x, int ldx, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsbgv(java.lang.String jobz, java.lang.String uplo, int n, int ka, int kb, double[] ab, int ldab, double[] bb, int ldbb, double[] w, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsbgvd(java.lang.String jobz, java.lang.String uplo, int n, int ka, int kb, double[] ab, int ldab, double[] bb, int ldbb, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsbgvx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, int ka, int kb, double[] ab, int ldab, double[] bb, int ldbb, double[] q, int ldq, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsbtrd(java.lang.String vect, java.lang.String uplo, int n, int kd, double[] ab, int ldab, double[] d, double[] e, double[] q, int ldq, double[] work, org.netlib.util.intW info)
      ..
      double dsecnd()
      -- LAPACK auxiliary routine (version 3.1.1) -- Univ.
      void dsgesv(int n, int nrhs, double[] a, int lda, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] work, float[] swork, org.netlib.util.intW iter, org.netlib.util.intW info)
      ..
      abstract void dspcon(java.lang.String uplo, int n, double[] ap, int[] ipiv, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dspev(java.lang.String jobz, java.lang.String uplo, int n, double[] ap, double[] w, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dspevd(java.lang.String jobz, java.lang.String uplo, int n, double[] ap, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dspevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, double[] ap, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dspgst(int itype, java.lang.String uplo, int n, double[] ap, double[] bp, org.netlib.util.intW info)
      ..
      abstract void dspgv(int itype, java.lang.String jobz, java.lang.String uplo, int n, double[] ap, double[] bp, double[] w, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dspgvd(int itype, java.lang.String jobz, java.lang.String uplo, int n, double[] ap, double[] bp, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dspgvx(int itype, java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, double[] ap, double[] bp, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsprfs(java.lang.String uplo, int n, int nrhs, double[] ap, double[] afp, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dspsv(java.lang.String uplo, int n, int nrhs, double[] ap, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dspsvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, double[] ap, double[] afp, int[] ipiv, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dsptrd(java.lang.String uplo, int n, double[] ap, double[] d, double[] e, double[] tau, org.netlib.util.intW info)
      ..
      abstract void dsptrf(java.lang.String uplo, int n, double[] ap, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dsptri(java.lang.String uplo, int n, double[] ap, int[] ipiv, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsptrs(java.lang.String uplo, int n, int nrhs, double[] ap, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dstebz(java.lang.String range, java.lang.String order, int n, double vl, double vu, int il, int iu, double abstol, double[] d, double[] e, org.netlib.util.intW m, org.netlib.util.intW nsplit, double[] w, int[] iblock, int[] isplit, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dstedc(java.lang.String compz, int n, double[] d, double[] e, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dstegr(java.lang.String jobz, java.lang.String range, int n, double[] d, double[] e, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, int[] isuppz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dstein(int n, double[] d, double[] e, int m, double[] w, int[] iblock, int[] isplit, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      void dstemr(java.lang.String jobz, java.lang.String range, int n, double[] d, double[] e, double vl, double vu, int il, int iu, org.netlib.util.intW m, double[] w, double[] z, int ldz, int nzc, int[] isuppz, org.netlib.util.booleanW tryrac, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsteqr(java.lang.String compz, int n, double[] d, double[] e, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsterf(int n, double[] d, double[] e, org.netlib.util.intW info)
      ..
      abstract void dstev(java.lang.String jobz, int n, double[] d, double[] e, double[] z, int ldz, double[] work, org.netlib.util.intW info)
      ..
      abstract void dstevd(java.lang.String jobz, int n, double[] d, double[] e, double[] z, int ldz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dstevr(java.lang.String jobz, java.lang.String range, int n, double[] d, double[] e, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, int[] isuppz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dstevx(java.lang.String jobz, java.lang.String range, int n, double[] d, double[] e, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsycon(java.lang.String uplo, int n, double[] a, int lda, int[] ipiv, double anorm, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dsyev(java.lang.String jobz, java.lang.String uplo, int n, double[] a, int lda, double[] w, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dsyevd(java.lang.String jobz, java.lang.String uplo, int n, double[] a, int lda, double[] w, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsyevr(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, double[] a, int lda, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, int[] isuppz, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsyevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, double[] a, int lda, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsygs2(int itype, java.lang.String uplo, int n, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dsygst(int itype, java.lang.String uplo, int n, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dsygv(int itype, java.lang.String jobz, java.lang.String uplo, int n, double[] a, int lda, double[] b, int ldb, double[] w, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dsygvd(int itype, java.lang.String jobz, java.lang.String uplo, int n, double[] a, int lda, double[] b, int ldb, double[] w, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dsygvx(int itype, java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, double[] a, int lda, double[] b, int ldb, double vl, double vu, int il, int iu, double abstol, org.netlib.util.intW m, double[] w, double[] z, int ldz, double[] work, int lwork, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void dsyrfs(java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, int[] ipiv, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dsysv(java.lang.String uplo, int n, int nrhs, double[] a, int lda, int[] ipiv, double[] b, int ldb, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dsysvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, double[] a, int lda, double[] af, int ldaf, int[] ipiv, double[] b, int ldb, double[] x, int ldx, org.netlib.util.doubleW rcond, double[] ferr, double[] berr, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dsytd2(java.lang.String uplo, int n, double[] a, int lda, double[] d, double[] e, double[] tau, org.netlib.util.intW info)
      ..
      abstract void dsytf2(java.lang.String uplo, int n, double[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void dsytrd(java.lang.String uplo, int n, double[] a, int lda, double[] d, double[] e, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dsytrf(java.lang.String uplo, int n, double[] a, int lda, int[] ipiv, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dsytri(java.lang.String uplo, int n, double[] a, int lda, int[] ipiv, double[] work, org.netlib.util.intW info)
      ..
      abstract void dsytrs(java.lang.String uplo, int n, int nrhs, double[] a, int lda, int[] ipiv, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dtbcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, int kd, double[] ab, int ldab, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtbrfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int kd, int nrhs, double[] ab, int ldab, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtbtrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int kd, int nrhs, double[] ab, int ldab, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dtgevc(java.lang.String side, java.lang.String howmny, boolean[] select, int n, double[] s, int lds, double[] p, int ldp, double[] vl, int ldvl, double[] vr, int ldvr, int mm, org.netlib.util.intW m, double[] work, org.netlib.util.intW info)
      ..
      abstract void dtgex2(boolean wantq, boolean wantz, int n, double[] a, int lda, double[] b, int ldb, double[] q, int ldq, double[] z, int ldz, int j1, int n1, int n2, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dtgexc(boolean wantq, boolean wantz, int n, double[] a, int lda, double[] b, int ldb, double[] q, int ldq, double[] z, int ldz, org.netlib.util.intW ifst, org.netlib.util.intW ilst, double[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void dtgsen(int ijob, boolean wantq, boolean wantz, boolean[] select, int n, double[] a, int lda, double[] b, int ldb, double[] alphar, double[] alphai, double[] beta, double[] q, int ldq, double[] z, int ldz, org.netlib.util.intW m, org.netlib.util.doubleW pl, org.netlib.util.doubleW pr, double[] dif, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dtgsja(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int p, int n, int k, int l, double[] a, int lda, double[] b, int ldb, double tola, double tolb, double[] alpha, double[] beta, double[] u, int ldu, double[] v, int ldv, double[] q, int ldq, double[] work, org.netlib.util.intW ncycle, org.netlib.util.intW info)
      ..
      abstract void dtgsna(java.lang.String job, java.lang.String howmny, boolean[] select, int n, double[] a, int lda, double[] b, int ldb, double[] vl, int ldvl, double[] vr, int ldvr, double[] s, double[] dif, int mm, org.netlib.util.intW m, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtgsy2(java.lang.String trans, int ijob, int m, int n, double[] a, int lda, double[] b, int ldb, double[] c, int Ldc, double[] d, int ldd, double[] e, int lde, double[] f, int ldf, org.netlib.util.doubleW scale, org.netlib.util.doubleW rdsum, org.netlib.util.doubleW rdscal, int[] iwork, org.netlib.util.intW pq, org.netlib.util.intW info)
      ..
      abstract void dtgsyl(java.lang.String trans, int ijob, int m, int n, double[] a, int lda, double[] b, int ldb, double[] c, int Ldc, double[] d, int ldd, double[] e, int lde, double[] f, int ldf, org.netlib.util.doubleW scale, org.netlib.util.doubleW dif, double[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtpcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, double[] ap, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtprfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, double[] ap, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtptri(java.lang.String uplo, java.lang.String diag, int n, double[] ap, org.netlib.util.intW info)
      ..
      abstract void dtptrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, double[] ap, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dtrcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, double[] a, int lda, org.netlib.util.doubleW rcond, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtrevc(java.lang.String side, java.lang.String howmny, boolean[] select, int n, double[] t, int ldt, double[] vl, int ldvl, double[] vr, int ldvr, int mm, org.netlib.util.intW m, double[] work, org.netlib.util.intW info)
      ..
      abstract void dtrexc(java.lang.String compq, int n, double[] t, int ldt, double[] q, int ldq, org.netlib.util.intW ifst, org.netlib.util.intW ilst, double[] work, org.netlib.util.intW info)
      ..
      abstract void dtrrfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, double[] a, int lda, double[] b, int ldb, double[] x, int ldx, double[] ferr, double[] berr, double[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtrsen(java.lang.String job, java.lang.String compq, boolean[] select, int n, double[] t, int ldt, double[] q, int ldq, double[] wr, double[] wi, org.netlib.util.intW m, org.netlib.util.doubleW s, org.netlib.util.doubleW sep, double[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void dtrsna(java.lang.String job, java.lang.String howmny, boolean[] select, int n, double[] t, int ldt, double[] vl, int ldvl, double[] vr, int ldvr, double[] s, double[] sep, int mm, org.netlib.util.intW m, double[] work, int ldwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void dtrsyl(java.lang.String trana, java.lang.String tranb, int isgn, int m, int n, double[] a, int lda, double[] b, int ldb, double[] c, int Ldc, org.netlib.util.doubleW scale, org.netlib.util.intW info)
      ..
      abstract void dtrti2(java.lang.String uplo, java.lang.String diag, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dtrtri(java.lang.String uplo, java.lang.String diag, int n, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void dtrtrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, double[] a, int lda, double[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void dtzrqf(int m, int n, double[] a, int lda, double[] tau, org.netlib.util.intW info)
      ..
      abstract void dtzrzf(int m, int n, double[] a, int lda, double[] tau, double[] work, int lwork, org.netlib.util.intW info)
      ..
      static LAPACK getInstance() 
      abstract int ieeeck(int ispec, float zero, float one)
      ..
      abstract int ilaenv(int ispec, java.lang.String name, java.lang.String opts, int n1, int n2, int n3, int n4)
      ..
      void ilaver(org.netlib.util.intW vers_major, org.netlib.util.intW vers_minor, org.netlib.util.intW vers_patch)
      -- LAPACK routine (version 3.1.1) -- Univ.
      int iparmq(int ispec, java.lang.String name, java.lang.String opts, int n, int ilo, int ihi, int lwork)
      Purpose ======= This program sets problem and machine dependent parameters useful for xHSEQR and its subroutines.
      boolean lsame(java.lang.String ca, java.lang.String cb)
      ..
      boolean lsamen(int n, java.lang.String ca, java.lang.String cb)
      ..
      abstract void sbdsdc(java.lang.String uplo, java.lang.String compq, int n, float[] d, float[] e, float[] u, int ldu, float[] vt, int ldvt, float[] q, int[] iq, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sbdsqr(java.lang.String uplo, int n, int ncvt, int nru, int ncc, float[] d, float[] e, float[] vt, int ldvt, float[] u, int ldu, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sdisna(java.lang.String job, int m, int n, float[] d, float[] sep, org.netlib.util.intW info)
      ..
      float second()
      -- LAPACK auxiliary routine (version 3.1.1) -- Univ.
      abstract void sgbbrd(java.lang.String vect, int m, int n, int ncc, int kl, int ku, float[] ab, int ldab, float[] d, float[] e, float[] q, int ldq, float[] pt, int ldpt, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgbcon(java.lang.String norm, int n, int kl, int ku, float[] ab, int ldab, int[] ipiv, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgbequ(int m, int n, int kl, int ku, float[] ab, int ldab, float[] r, float[] c, org.netlib.util.floatW rowcnd, org.netlib.util.floatW colcnd, org.netlib.util.floatW amax, org.netlib.util.intW info)
      ..
      abstract void sgbrfs(java.lang.String trans, int n, int kl, int ku, int nrhs, float[] ab, int ldab, float[] afb, int ldafb, int[] ipiv, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgbsv(int n, int kl, int ku, int nrhs, float[] ab, int ldab, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sgbsvx(java.lang.String fact, java.lang.String trans, int n, int kl, int ku, int nrhs, float[] ab, int ldab, float[] afb, int ldafb, int[] ipiv, org.netlib.util.StringW equed, float[] r, float[] c, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgbtf2(int m, int n, int kl, int ku, float[] ab, int ldab, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void sgbtrf(int m, int n, int kl, int ku, float[] ab, int ldab, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void sgbtrs(java.lang.String trans, int n, int kl, int ku, int nrhs, float[] ab, int ldab, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sgebak(java.lang.String job, java.lang.String side, int n, int ilo, int ihi, float[] scale, int m, float[] v, int ldv, org.netlib.util.intW info)
      ..
      abstract void sgebal(java.lang.String job, int n, float[] a, int lda, org.netlib.util.intW ilo, org.netlib.util.intW ihi, float[] scale, org.netlib.util.intW info)
      ..
      abstract void sgebd2(int m, int n, float[] a, int lda, float[] d, float[] e, float[] tauq, float[] taup, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgebrd(int m, int n, float[] a, int lda, float[] d, float[] e, float[] tauq, float[] taup, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgecon(java.lang.String norm, int n, float[] a, int lda, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgeequ(int m, int n, float[] a, int lda, float[] r, float[] c, org.netlib.util.floatW rowcnd, org.netlib.util.floatW colcnd, org.netlib.util.floatW amax, org.netlib.util.intW info)
      ..
      void sgees(java.lang.String jobvs, java.lang.String sort, java.lang.Object select, int n, float[] a, int lda, org.netlib.util.intW sdim, float[] wr, float[] wi, float[] vs, int ldvs, float[] work, int lwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      void sgeesx(java.lang.String jobvs, java.lang.String sort, java.lang.Object select, java.lang.String sense, int n, float[] a, int lda, org.netlib.util.intW sdim, float[] wr, float[] wi, float[] vs, int ldvs, org.netlib.util.floatW rconde, org.netlib.util.floatW rcondv, float[] work, int lwork, int[] iwork, int liwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void sgeev(java.lang.String jobvl, java.lang.String jobvr, int n, float[] a, int lda, float[] wr, float[] wi, float[] vl, int ldvl, float[] vr, int ldvr, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgeevx(java.lang.String balanc, java.lang.String jobvl, java.lang.String jobvr, java.lang.String sense, int n, float[] a, int lda, float[] wr, float[] wi, float[] vl, int ldvl, float[] vr, int ldvr, org.netlib.util.intW ilo, org.netlib.util.intW ihi, float[] scale, org.netlib.util.floatW abnrm, float[] rconde, float[] rcondv, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgegs(java.lang.String jobvsl, java.lang.String jobvsr, int n, float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, float[] vsl, int ldvsl, float[] vsr, int ldvsr, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgegv(java.lang.String jobvl, java.lang.String jobvr, int n, float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, float[] vl, int ldvl, float[] vr, int ldvr, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgehd2(int n, int ilo, int ihi, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgehrd(int n, int ilo, int ihi, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgelq2(int m, int n, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgelqf(int m, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgels(java.lang.String trans, int m, int n, int nrhs, float[] a, int lda, float[] b, int ldb, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgelsd(int m, int n, int nrhs, float[] a, int lda, float[] b, int ldb, float[] s, float rcond, org.netlib.util.intW rank, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgelss(int m, int n, int nrhs, float[] a, int lda, float[] b, int ldb, float[] s, float rcond, org.netlib.util.intW rank, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgelsx(int m, int n, int nrhs, float[] a, int lda, float[] b, int ldb, int[] jpvt, float rcond, org.netlib.util.intW rank, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgelsy(int m, int n, int nrhs, float[] a, int lda, float[] b, int ldb, int[] jpvt, float rcond, org.netlib.util.intW rank, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgeql2(int m, int n, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgeqlf(int m, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgeqp3(int m, int n, float[] a, int lda, int[] jpvt, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgeqpf(int m, int n, float[] a, int lda, int[] jpvt, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgeqr2(int m, int n, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgeqrf(int m, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgerfs(java.lang.String trans, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, int[] ipiv, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgerq2(int m, int n, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgerqf(int m, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgesc2(int n, float[] a, int lda, float[] rhs, int[] ipiv, int[] jpiv, org.netlib.util.floatW scale)
      ..
      abstract void sgesdd(java.lang.String jobz, int m, int n, float[] a, int lda, float[] s, float[] u, int ldu, float[] vt, int ldvt, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgesv(int n, int nrhs, float[] a, int lda, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sgesvd(java.lang.String jobu, java.lang.String jobvt, int m, int n, float[] a, int lda, float[] s, float[] u, int ldu, float[] vt, int ldvt, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgesvx(java.lang.String fact, java.lang.String trans, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, int[] ipiv, org.netlib.util.StringW equed, float[] r, float[] c, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgetc2(int n, float[] a, int lda, int[] ipiv, int[] jpiv, org.netlib.util.intW info)
      ..
      abstract void sgetf2(int m, int n, float[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void sgetrf(int m, int n, float[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void sgetri(int n, float[] a, int lda, int[] ipiv, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgetrs(java.lang.String trans, int n, int nrhs, float[] a, int lda, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sggbak(java.lang.String job, java.lang.String side, int n, int ilo, int ihi, float[] lscale, float[] rscale, int m, float[] v, int ldv, org.netlib.util.intW info)
      ..
      abstract void sggbal(java.lang.String job, int n, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW ilo, org.netlib.util.intW ihi, float[] lscale, float[] rscale, float[] work, org.netlib.util.intW info)
      ..
      void sgges(java.lang.String jobvsl, java.lang.String jobvsr, java.lang.String sort, java.lang.Object selctg, int n, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW sdim, float[] alphar, float[] alphai, float[] beta, float[] vsl, int ldvsl, float[] vsr, int ldvsr, float[] work, int lwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      void sggesx(java.lang.String jobvsl, java.lang.String jobvsr, java.lang.String sort, java.lang.Object selctg, java.lang.String sense, int n, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW sdim, float[] alphar, float[] alphai, float[] beta, float[] vsl, int ldvsl, float[] vsr, int ldvsr, float[] rconde, float[] rcondv, float[] work, int lwork, int[] iwork, int liwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void sggev(java.lang.String jobvl, java.lang.String jobvr, int n, float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, float[] vl, int ldvl, float[] vr, int ldvr, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sggevx(java.lang.String balanc, java.lang.String jobvl, java.lang.String jobvr, java.lang.String sense, int n, float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, float[] vl, int ldvl, float[] vr, int ldvr, org.netlib.util.intW ilo, org.netlib.util.intW ihi, float[] lscale, float[] rscale, org.netlib.util.floatW abnrm, org.netlib.util.floatW bbnrm, float[] rconde, float[] rcondv, float[] work, int lwork, int[] iwork, boolean[] bwork, org.netlib.util.intW info)
      ..
      abstract void sggglm(int n, int m, int p, float[] a, int lda, float[] b, int ldb, float[] d, float[] x, float[] y, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sgghrd(java.lang.String compq, java.lang.String compz, int n, int ilo, int ihi, float[] a, int lda, float[] b, int ldb, float[] q, int ldq, float[] z, int ldz, org.netlib.util.intW info)
      ..
      abstract void sgglse(int m, int n, int p, float[] a, int lda, float[] b, int ldb, float[] c, float[] d, float[] x, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sggqrf(int n, int m, int p, float[] a, int lda, float[] taua, float[] b, int ldb, float[] taub, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sggrqf(int m, int p, int n, float[] a, int lda, float[] taua, float[] b, int ldb, float[] taub, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sggsvd(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int n, int p, org.netlib.util.intW k, org.netlib.util.intW l, float[] a, int lda, float[] b, int ldb, float[] alpha, float[] beta, float[] u, int ldu, float[] v, int ldv, float[] q, int ldq, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sggsvp(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int p, int n, float[] a, int lda, float[] b, int ldb, float tola, float tolb, org.netlib.util.intW k, org.netlib.util.intW l, float[] u, int ldu, float[] v, int ldv, float[] q, int ldq, int[] iwork, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sgtcon(java.lang.String norm, int n, float[] dl, float[] d, float[] du, float[] du2, int[] ipiv, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgtrfs(java.lang.String trans, int n, int nrhs, float[] dl, float[] d, float[] du, float[] dlf, float[] df, float[] duf, float[] du2, int[] ipiv, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgtsv(int n, int nrhs, float[] dl, float[] d, float[] du, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sgtsvx(java.lang.String fact, java.lang.String trans, int n, int nrhs, float[] dl, float[] d, float[] du, float[] dlf, float[] df, float[] duf, float[] du2, int[] ipiv, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sgttrf(int n, float[] dl, float[] d, float[] du, float[] du2, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void sgttrs(java.lang.String trans, int n, int nrhs, float[] dl, float[] d, float[] du, float[] du2, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sgtts2(int itrans, int n, int nrhs, float[] dl, float[] d, float[] du, float[] du2, int[] ipiv, float[] b, int ldb)
      ..
      abstract void shgeqz(java.lang.String job, java.lang.String compq, java.lang.String compz, int n, int ilo, int ihi, float[] h, int ldh, float[] t, int ldt, float[] alphar, float[] alphai, float[] beta, float[] q, int ldq, float[] z, int ldz, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void shsein(java.lang.String side, java.lang.String eigsrc, java.lang.String initv, boolean[] select, int n, float[] h, int ldh, float[] wr, float[] wi, float[] vl, int ldvl, float[] vr, int ldvr, int mm, org.netlib.util.intW m, float[] work, int[] ifaill, int[] ifailr, org.netlib.util.intW info)
      ..
      abstract void shseqr(java.lang.String job, java.lang.String compz, int n, int ilo, int ihi, float[] h, int ldh, float[] wr, float[] wi, float[] z, int ldz, float[] work, int lwork, org.netlib.util.intW info)
      ..
      boolean sisnan(float sin)
      ..
      abstract void slabad(org.netlib.util.floatW small, org.netlib.util.floatW large)
      ..
      abstract void slabrd(int m, int n, int nb, float[] a, int lda, float[] d, float[] e, float[] tauq, float[] taup, float[] x, int ldx, float[] y, int ldy)
      ..
      void slacn2(int n, float[] v, float[] x, int[] isgn, org.netlib.util.floatW est, org.netlib.util.intW kase, int[] isave)
      ..
      abstract void slacon(int n, float[] v, float[] x, int[] isgn, org.netlib.util.floatW est, org.netlib.util.intW kase)
      ..
      abstract void slacpy(java.lang.String uplo, int m, int n, float[] a, int lda, float[] b, int ldb)
      ..
      abstract void sladiv(float a, float b, float c, float d, org.netlib.util.floatW p, org.netlib.util.floatW q)
      ..
      abstract void slae2(float a, float b, float c, org.netlib.util.floatW rt1, org.netlib.util.floatW rt2)
      ..
      abstract void slaebz(int ijob, int nitmax, int n, int mmax, int minp, int nbmin, float abstol, float reltol, float pivmin, float[] d, float[] e, float[] e2, int[] nval, float[] ab, float[] c, org.netlib.util.intW mout, int[] nab, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slaed0(int icompq, int qsiz, int n, float[] d, float[] e, float[] q, int ldq, float[] qstore, int ldqs, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slaed1(int n, float[] d, float[] q, int ldq, int[] indxq, org.netlib.util.floatW rho, int cutpnt, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slaed2(org.netlib.util.intW k, int n, int n1, float[] d, float[] q, int ldq, int[] indxq, org.netlib.util.floatW rho, float[] z, float[] dlamda, float[] w, float[] q2, int[] indx, int[] indxc, int[] indxp, int[] coltyp, org.netlib.util.intW info)
      ..
      abstract void slaed3(int k, int n, int n1, float[] d, float[] q, int ldq, float rho, float[] dlamda, float[] q2, int[] indx, int[] ctot, float[] w, float[] s, org.netlib.util.intW info)
      ..
      abstract void slaed4(int n, int i, float[] d, float[] z, float[] delta, float rho, org.netlib.util.floatW dlam, org.netlib.util.intW info)
      ..
      abstract void slaed5(int i, float[] d, float[] z, float[] delta, float rho, org.netlib.util.floatW dlam)
      ..
      abstract void slaed6(int kniter, boolean orgati, float rho, float[] d, float[] z, float finit, org.netlib.util.floatW tau, org.netlib.util.intW info)
      ..
      abstract void slaed7(int icompq, int n, int qsiz, int tlvls, int curlvl, int curpbm, float[] d, float[] q, int ldq, int[] indxq, org.netlib.util.floatW rho, int cutpnt, float[] qstore, int[] qptr, int[] prmptr, int[] perm, int[] givptr, int[] givcol, float[] givnum, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slaed8(int icompq, org.netlib.util.intW k, int n, int qsiz, float[] d, float[] q, int ldq, int[] indxq, org.netlib.util.floatW rho, int cutpnt, float[] z, float[] dlamda, float[] q2, int ldq2, float[] w, int[] perm, org.netlib.util.intW givptr, int[] givcol, float[] givnum, int[] indxp, int[] indx, org.netlib.util.intW info)
      ..
      abstract void slaed9(int k, int kstart, int kstop, int n, float[] d, float[] q, int ldq, float rho, float[] dlamda, float[] w, float[] s, int lds, org.netlib.util.intW info)
      ..
      abstract void slaeda(int n, int tlvls, int curlvl, int curpbm, int[] prmptr, int[] perm, int[] givptr, int[] givcol, float[] givnum, float[] q, int[] qptr, float[] z, float[] ztemp, org.netlib.util.intW info)
      ..
      abstract void slaein(boolean rightv, boolean noinit, int n, float[] h, int ldh, float wr, float wi, float[] vr, float[] vi, float[] b, int ldb, float[] work, float eps3, float smlnum, float bignum, org.netlib.util.intW info)
      ..
      abstract void slaev2(float a, float b, float c, org.netlib.util.floatW rt1, org.netlib.util.floatW rt2, org.netlib.util.floatW cs1, org.netlib.util.floatW sn1)
      ..
      abstract void slaexc(boolean wantq, int n, float[] t, int ldt, float[] q, int ldq, int j1, int n1, int n2, float[] work, org.netlib.util.intW info)
      ..
      abstract void slag2(float[] a, int lda, float[] b, int ldb, float safmin, org.netlib.util.floatW scale1, org.netlib.util.floatW scale2, org.netlib.util.floatW wr1, org.netlib.util.floatW wr2, org.netlib.util.floatW wi)
      ..
      void slag2d(int m, int n, float[] sa, int ldsa, double[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void slags2(boolean upper, float a1, float a2, float a3, float b1, float b2, float b3, org.netlib.util.floatW csu, org.netlib.util.floatW snu, org.netlib.util.floatW csv, org.netlib.util.floatW snv, org.netlib.util.floatW csq, org.netlib.util.floatW snq)
      ..
      abstract void slagtf(int n, float[] a, float lambda, float[] b, float[] c, float tol, float[] d, int[] in, org.netlib.util.intW info)
      ..
      abstract void slagtm(java.lang.String trans, int n, int nrhs, float alpha, float[] dl, float[] d, float[] du, float[] x, int ldx, float beta, float[] b, int ldb)
      ..
      abstract void slagts(int job, int n, float[] a, float[] b, float[] c, float[] d, int[] in, float[] y, org.netlib.util.floatW tol, org.netlib.util.intW info)
      ..
      abstract void slagv2(float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, org.netlib.util.floatW csl, org.netlib.util.floatW snl, org.netlib.util.floatW csr, org.netlib.util.floatW snr)
      ..
      abstract void slahqr(boolean wantt, boolean wantz, int n, int ilo, int ihi, float[] h, int ldh, float[] wr, float[] wi, int iloz, int ihiz, float[] z, int ldz, org.netlib.util.intW info)
      ..
      void slahr2(int n, int k, int nb, float[] a, int lda, float[] tau, float[] t, int ldt, float[] y, int ldy)
      ..
      abstract void slahrd(int n, int k, int nb, float[] a, int lda, float[] tau, float[] t, int ldt, float[] y, int ldy)
      ..
      abstract void slaic1(int job, int j, float[] x, float sest, float[] w, float gamma, org.netlib.util.floatW sestpr, org.netlib.util.floatW s, org.netlib.util.floatW c)
      ..
      boolean slaisnan(float sin1, float sin2)
      ..
      abstract void slaln2(boolean ltrans, int na, int nw, float smin, float ca, float[] a, int lda, float d1, float d2, float[] b, int ldb, float wr, float wi, float[] x, int ldx, org.netlib.util.floatW scale, org.netlib.util.floatW xnorm, org.netlib.util.intW info)
      ..
      abstract void slals0(int icompq, int nl, int nr, int sqre, int nrhs, float[] b, int ldb, float[] bx, int ldbx, int[] perm, int givptr, int[] givcol, int ldgcol, float[] givnum, int ldgnum, float[] poles, float[] difl, float[] difr, float[] z, int k, float c, float s, float[] work, org.netlib.util.intW info)
      ..
      abstract void slalsa(int icompq, int smlsiz, int n, int nrhs, float[] b, int ldb, float[] bx, int ldbx, float[] u, int ldu, float[] vt, int[] k, float[] difl, float[] difr, float[] z, float[] poles, int[] givptr, int[] givcol, int ldgcol, int[] perm, float[] givnum, float[] c, float[] s, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slalsd(java.lang.String uplo, int smlsiz, int n, int nrhs, float[] d, float[] e, float[] b, int ldb, float rcond, org.netlib.util.intW rank, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slamc1(org.netlib.util.intW beta, org.netlib.util.intW t, org.netlib.util.booleanW rnd, org.netlib.util.booleanW ieee1)
      ..
      abstract void slamc2(org.netlib.util.intW beta, org.netlib.util.intW t, org.netlib.util.booleanW rnd, org.netlib.util.floatW eps, org.netlib.util.intW emin, org.netlib.util.floatW rmin, org.netlib.util.intW emax, org.netlib.util.floatW rmax)
      ..
      float slamc3(float a, float b)
      ..
      abstract void slamc4(org.netlib.util.intW emin, float start, int base)
      ..
      abstract void slamc5(int beta, int p, int emin, boolean ieee, org.netlib.util.intW emax, org.netlib.util.floatW rmax)
      ..
      float slamch(java.lang.String cmach)
      ..
      abstract void slamrg(int n1, int n2, float[] a, int strd1, int strd2, int[] index)
      ..
      int slaneg(int n, float[] d, float[] lld, float sigma, float pivmin, int r)
      ..
      float slangb(java.lang.String norm, int n, int kl, int ku, float[] ab, int ldab, float[] work)
      ..
      float slange(java.lang.String norm, int m, int n, float[] a, int lda, float[] work)
      ..
      float slangt(java.lang.String norm, int n, float[] dl, float[] d, float[] du)
      ..
      float slanhs(java.lang.String norm, int n, float[] a, int lda, float[] work)
      ..
      float slansb(java.lang.String norm, java.lang.String uplo, int n, int k, float[] ab, int ldab, float[] work)
      ..
      float slansp(java.lang.String norm, java.lang.String uplo, int n, float[] ap, float[] work)
      ..
      float slanst(java.lang.String norm, int n, float[] d, float[] e)
      ..
      float slansy(java.lang.String norm, java.lang.String uplo, int n, float[] a, int lda, float[] work)
      ..
      float slantb(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, int k, float[] ab, int ldab, float[] work)
      ..
      float slantp(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, float[] ap, float[] work)
      ..
      float slantr(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int m, int n, float[] a, int lda, float[] work)
      ..
      abstract void slanv2(org.netlib.util.floatW a, org.netlib.util.floatW b, org.netlib.util.floatW c, org.netlib.util.floatW d, org.netlib.util.floatW rt1r, org.netlib.util.floatW rt1i, org.netlib.util.floatW rt2r, org.netlib.util.floatW rt2i, org.netlib.util.floatW cs, org.netlib.util.floatW sn)
      ..
      abstract void slapll(int n, float[] x, int incx, float[] y, int incy, org.netlib.util.floatW ssmin)
      ..
      abstract void slapmt(boolean forwrd, int m, int n, float[] x, int ldx, int[] k)
      ..
      float slapy2(float x, float y)
      ..
      float slapy3(float x, float y, float z)
      ..
      abstract void slaqgb(int m, int n, int kl, int ku, float[] ab, int ldab, float[] r, float[] c, float rowcnd, float colcnd, float amax, org.netlib.util.StringW equed)
      ..
      abstract void slaqge(int m, int n, float[] a, int lda, float[] r, float[] c, float rowcnd, float colcnd, float amax, org.netlib.util.StringW equed)
      ..
      abstract void slaqp2(int m, int n, int offset, float[] a, int lda, int[] jpvt, float[] tau, float[] vn1, float[] vn2, float[] work)
      ..
      abstract void slaqps(int m, int n, int offset, int nb, org.netlib.util.intW kb, float[] a, int lda, int[] jpvt, float[] tau, float[] vn1, float[] vn2, float[] auxv, float[] f, int ldf)
      ..
      void slaqr0(boolean wantt, boolean wantz, int n, int ilo, int ihi, float[] h, int ldh, float[] wr, float[] wi, int iloz, int ihiz, float[] z, int ldz, float[] work, int lwork, org.netlib.util.intW info)
      ..
      void slaqr1(int n, float[] h, int ldh, float sr1, float si1, float sr2, float si2, float[] v)
      ..
      void slaqr2(boolean wantt, boolean wantz, int n, int ktop, int kbot, int nw, float[] h, int ldh, int iloz, int ihiz, float[] z, int ldz, org.netlib.util.intW ns, org.netlib.util.intW nd, float[] sr, float[] si, float[] v, int ldv, int nh, float[] t, int ldt, int nv, float[] wv, int ldwv, float[] work, int lwork)
      ..
      void slaqr3(boolean wantt, boolean wantz, int n, int ktop, int kbot, int nw, float[] h, int ldh, int iloz, int ihiz, float[] z, int ldz, org.netlib.util.intW ns, org.netlib.util.intW nd, float[] sr, float[] si, float[] v, int ldv, int nh, float[] t, int ldt, int nv, float[] wv, int ldwv, float[] work, int lwork)
      ..
      void slaqr4(boolean wantt, boolean wantz, int n, int ilo, int ihi, float[] h, int ldh, float[] wr, float[] wi, int iloz, int ihiz, float[] z, int ldz, float[] work, int lwork, org.netlib.util.intW info)
      ..
      void slaqr5(boolean wantt, boolean wantz, int kacc22, int n, int ktop, int kbot, int nshfts, float[] sr, float[] si, float[] h, int ldh, int iloz, int ihiz, float[] z, int ldz, float[] v, int ldv, float[] u, int ldu, int nv, float[] wv, int ldwv, int nh, float[] wh, int ldwh)
      ..
      abstract void slaqsb(java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] s, float scond, float amax, org.netlib.util.StringW equed)
      ..
      abstract void slaqsp(java.lang.String uplo, int n, float[] ap, float[] s, float scond, float amax, org.netlib.util.StringW equed)
      ..
      abstract void slaqsy(java.lang.String uplo, int n, float[] a, int lda, float[] s, float scond, float amax, org.netlib.util.StringW equed)
      ..
      abstract void slaqtr(boolean ltran, boolean lreal, int n, float[] t, int ldt, float[] b, float w, org.netlib.util.floatW scale, float[] x, float[] work, org.netlib.util.intW info)
      ..
      void slar1v(int n, int b1, int bn, float lambda, float[] d, float[] l, float[] ld, float[] lld, float pivmin, float gaptol, float[] z, boolean wantnc, org.netlib.util.intW negcnt, org.netlib.util.floatW ztz, org.netlib.util.floatW mingma, org.netlib.util.intW r, int[] isuppz, org.netlib.util.floatW nrminv, org.netlib.util.floatW resid, org.netlib.util.floatW rqcorr, float[] work)
      ..
      abstract void slar2v(int n, float[] x, float[] y, float[] z, int incx, float[] c, float[] s, int incc)
      ..
      abstract void slarf(java.lang.String side, int m, int n, float[] v, int incv, float tau, float[] c, int Ldc, float[] work)
      ..
      abstract void slarfb(java.lang.String side, java.lang.String trans, java.lang.String direct, java.lang.String storev, int m, int n, int k, float[] v, int ldv, float[] t, int ldt, float[] c, int Ldc, float[] work, int ldwork)
      ..
      abstract void slarfg(int n, org.netlib.util.floatW alpha, float[] x, int incx, org.netlib.util.floatW tau)
      ..
      abstract void slarft(java.lang.String direct, java.lang.String storev, int n, int k, float[] v, int ldv, float[] tau, float[] t, int ldt)
      ..
      abstract void slarfx(java.lang.String side, int m, int n, float[] v, float tau, float[] c, int Ldc, float[] work)
      ..
      abstract void slargv(int n, float[] x, int incx, float[] y, int incy, float[] c, int incc)
      ..
      abstract void slarnv(int idist, int[] iseed, int n, float[] x)
      ..
      void slarra(int n, float[] d, float[] e, float[] e2, float spltol, float tnrm, org.netlib.util.intW nsplit, int[] isplit, org.netlib.util.intW info)
      ..
      void slarrb(int n, float[] d, float[] lld, int ifirst, int ilast, float rtol1, float rtol2, int offset, float[] w, float[] wgap, float[] werr, float[] work, int[] iwork, float pivmin, float spdiam, int twist, org.netlib.util.intW info)
      ..
      void slarrc(java.lang.String jobt, int n, float vl, float vu, float[] d, float[] e, float pivmin, org.netlib.util.intW eigcnt, org.netlib.util.intW lcnt, org.netlib.util.intW rcnt, org.netlib.util.intW info)
      ..
      void slarrd(java.lang.String range, java.lang.String order, int n, float vl, float vu, int il, int iu, float[] gers, float reltol, float[] d, float[] e, float[] e2, float pivmin, int nsplit, int[] isplit, org.netlib.util.intW m, float[] w, float[] werr, org.netlib.util.floatW wl, org.netlib.util.floatW wu, int[] iblock, int[] indexw, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      void slarre(java.lang.String range, int n, org.netlib.util.floatW vl, org.netlib.util.floatW vu, int il, int iu, float[] d, float[] e, float[] e2, float rtol1, float rtol2, float spltol, org.netlib.util.intW nsplit, int[] isplit, org.netlib.util.intW m, float[] w, float[] werr, float[] wgap, int[] iblock, int[] indexw, float[] gers, org.netlib.util.floatW pivmin, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      void slarrf(int n, float[] d, float[] l, float[] ld, int clstrt, int clend, float[] w, float[] wgap, float[] werr, float spdiam, float clgapl, float clgapr, float pivmin, org.netlib.util.floatW sigma, float[] dplus, float[] lplus, float[] work, org.netlib.util.intW info)
      ..
      void slarrj(int n, float[] d, float[] e2, int ifirst, int ilast, float rtol, int offset, float[] w, float[] werr, float[] work, int[] iwork, float pivmin, float spdiam, org.netlib.util.intW info)
      ..
      void slarrk(int n, int iw, float gl, float gu, float[] d, float[] e2, float pivmin, float reltol, org.netlib.util.floatW w, org.netlib.util.floatW werr, org.netlib.util.intW info)
      ..
      void slarrr(int n, float[] d, float[] e, org.netlib.util.intW info)
      ..
      void slarrv(int n, float vl, float vu, float[] d, float[] l, float pivmin, int[] isplit, int m, int dol, int dou, float minrgp, org.netlib.util.floatW rtol1, org.netlib.util.floatW rtol2, float[] w, float[] werr, float[] wgap, int[] iblock, int[] indexw, float[] gers, float[] z, int ldz, int[] isuppz, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slartg(float f, float g, org.netlib.util.floatW cs, org.netlib.util.floatW sn, org.netlib.util.floatW r)
      ..
      abstract void slartv(int n, float[] x, int incx, float[] y, int incy, float[] c, float[] s, int incc)
      ..
      abstract void slaruv(int[] iseed, int n, float[] x)
      ..
      abstract void slarz(java.lang.String side, int m, int n, int l, float[] v, int incv, float tau, float[] c, int Ldc, float[] work)
      ..
      abstract void slarzb(java.lang.String side, java.lang.String trans, java.lang.String direct, java.lang.String storev, int m, int n, int k, int l, float[] v, int ldv, float[] t, int ldt, float[] c, int Ldc, float[] work, int ldwork)
      ..
      abstract void slarzt(java.lang.String direct, java.lang.String storev, int n, int k, float[] v, int ldv, float[] tau, float[] t, int ldt)
      ..
      abstract void slas2(float f, float g, float h, org.netlib.util.floatW ssmin, org.netlib.util.floatW ssmax)
      ..
      abstract void slascl(java.lang.String type, int kl, int ku, float cfrom, float cto, int m, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void slasd0(int n, int sqre, float[] d, float[] e, float[] u, int ldu, float[] vt, int ldvt, int smlsiz, int[] iwork, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasd1(int nl, int nr, int sqre, float[] d, org.netlib.util.floatW alpha, org.netlib.util.floatW beta, float[] u, int ldu, float[] vt, int ldvt, int[] idxq, int[] iwork, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasd2(int nl, int nr, int sqre, org.netlib.util.intW k, float[] d, float[] z, float alpha, float beta, float[] u, int ldu, float[] vt, int ldvt, float[] dsigma, float[] u2, int ldu2, float[] vt2, int ldvt2, int[] idxp, int[] idx, int[] idxc, int[] idxq, int[] coltyp, org.netlib.util.intW info)
      ..
      abstract void slasd3(int nl, int nr, int sqre, int k, float[] d, float[] q, int ldq, float[] dsigma, float[] u, int ldu, float[] u2, int ldu2, float[] vt, int ldvt, float[] vt2, int ldvt2, int[] idxc, int[] ctot, float[] z, org.netlib.util.intW info)
      ..
      abstract void slasd4(int n, int i, float[] d, float[] z, float[] delta, float rho, org.netlib.util.floatW sigma, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasd5(int i, float[] d, float[] z, float[] delta, float rho, org.netlib.util.floatW dsigma, float[] work)
      ..
      abstract void slasd6(int icompq, int nl, int nr, int sqre, float[] d, float[] vf, float[] vl, org.netlib.util.floatW alpha, org.netlib.util.floatW beta, int[] idxq, int[] perm, org.netlib.util.intW givptr, int[] givcol, int ldgcol, float[] givnum, int ldgnum, float[] poles, float[] difl, float[] difr, float[] z, org.netlib.util.intW k, org.netlib.util.floatW c, org.netlib.util.floatW s, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slasd7(int icompq, int nl, int nr, int sqre, org.netlib.util.intW k, float[] d, float[] z, float[] zw, float[] vf, float[] vfw, float[] vl, float[] vlw, float alpha, float beta, float[] dsigma, int[] idx, int[] idxp, int[] idxq, int[] perm, org.netlib.util.intW givptr, int[] givcol, int ldgcol, float[] givnum, int ldgnum, org.netlib.util.floatW c, org.netlib.util.floatW s, org.netlib.util.intW info)
      ..
      abstract void slasd8(int icompq, int k, float[] d, float[] z, float[] vf, float[] vl, float[] difl, float[] difr, int lddifr, float[] dsigma, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasda(int icompq, int smlsiz, int n, int sqre, float[] d, float[] e, float[] u, int ldu, float[] vt, int[] k, float[] difl, float[] difr, float[] z, float[] poles, int[] givptr, int[] givcol, int ldgcol, int[] perm, float[] givnum, float[] c, float[] s, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void slasdq(java.lang.String uplo, int sqre, int n, int ncvt, int nru, int ncc, float[] d, float[] e, float[] vt, int ldvt, float[] u, int ldu, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasdt(int n, org.netlib.util.intW lvl, org.netlib.util.intW nd, int[] inode, int[] ndiml, int[] ndimr, int msub)
      ..
      abstract void slaset(java.lang.String uplo, int m, int n, float alpha, float beta, float[] a, int lda)
      ..
      abstract void slasq1(int n, float[] d, float[] e, float[] work, org.netlib.util.intW info)
      ..
      abstract void slasq2(int n, float[] z, org.netlib.util.intW info)
      ..
      abstract void slasq3(int i0, org.netlib.util.intW n0, float[] z, int pp, org.netlib.util.floatW dmin, org.netlib.util.floatW sigma, org.netlib.util.floatW desig, org.netlib.util.floatW qmax, org.netlib.util.intW nfail, org.netlib.util.intW iter, org.netlib.util.intW ndiv, boolean ieee)
      ..
      abstract void slasq4(int i0, int n0, float[] z, int pp, int n0in, float dmin, float dmin1, float dmin2, float dn, float dn1, float dn2, org.netlib.util.floatW tau, org.netlib.util.intW ttype)
      ..
      abstract void slasq5(int i0, int n0, float[] z, int pp, float tau, org.netlib.util.floatW dmin, org.netlib.util.floatW dmin1, org.netlib.util.floatW dmin2, org.netlib.util.floatW dn, org.netlib.util.floatW dnm1, org.netlib.util.floatW dnm2, boolean ieee)
      ..
      abstract void slasq6(int i0, int n0, float[] z, int pp, org.netlib.util.floatW dmin, org.netlib.util.floatW dmin1, org.netlib.util.floatW dmin2, org.netlib.util.floatW dn, org.netlib.util.floatW dnm1, org.netlib.util.floatW dnm2)
      ..
      abstract void slasr(java.lang.String side, java.lang.String pivot, java.lang.String direct, int m, int n, float[] c, float[] s, float[] a, int lda)
      ..
      abstract void slasrt(java.lang.String id, int n, float[] d, org.netlib.util.intW info)
      ..
      abstract void slassq(int n, float[] x, int incx, org.netlib.util.floatW scale, org.netlib.util.floatW sumsq)
      ..
      abstract void slasv2(float f, float g, float h, org.netlib.util.floatW ssmin, org.netlib.util.floatW ssmax, org.netlib.util.floatW snr, org.netlib.util.floatW csr, org.netlib.util.floatW snl, org.netlib.util.floatW csl)
      ..
      abstract void slaswp(int n, float[] a, int lda, int k1, int k2, int[] ipiv, int incx)
      ..
      abstract void slasy2(boolean ltranl, boolean ltranr, int isgn, int n1, int n2, float[] tl, int ldtl, float[] tr, int ldtr, float[] b, int ldb, org.netlib.util.floatW scale, float[] x, int ldx, org.netlib.util.floatW xnorm, org.netlib.util.intW info)
      ..
      abstract void slasyf(java.lang.String uplo, int n, int nb, org.netlib.util.intW kb, float[] a, int lda, int[] ipiv, float[] w, int ldw, org.netlib.util.intW info)
      ..
      abstract void slatbs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, int kd, float[] ab, int ldab, float[] x, org.netlib.util.floatW scale, float[] cnorm, org.netlib.util.intW info)
      ..
      abstract void slatdf(int ijob, int n, float[] z, int ldz, float[] rhs, org.netlib.util.floatW rdsum, org.netlib.util.floatW rdscal, int[] ipiv, int[] jpiv)
      ..
      abstract void slatps(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, float[] ap, float[] x, org.netlib.util.floatW scale, float[] cnorm, org.netlib.util.intW info)
      ..
      abstract void slatrd(java.lang.String uplo, int n, int nb, float[] a, int lda, float[] e, float[] tau, float[] w, int ldw)
      ..
      abstract void slatrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, java.lang.String normin, int n, float[] a, int lda, float[] x, org.netlib.util.floatW scale, float[] cnorm, org.netlib.util.intW info)
      ..
      abstract void slatrz(int m, int n, int l, float[] a, int lda, float[] tau, float[] work)
      ..
      abstract void slatzm(java.lang.String side, int m, int n, float[] v, int incv, float tau, float[] c1, float[] c2, int Ldc, float[] work)
      ..
      abstract void slauu2(java.lang.String uplo, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void slauum(java.lang.String uplo, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      void slazq3(int i0, org.netlib.util.intW n0, float[] z, int pp, org.netlib.util.floatW dmin, org.netlib.util.floatW sigma, org.netlib.util.floatW desig, org.netlib.util.floatW qmax, org.netlib.util.intW nfail, org.netlib.util.intW iter, org.netlib.util.intW ndiv, boolean ieee, org.netlib.util.intW ttype, org.netlib.util.floatW dmin1, org.netlib.util.floatW dmin2, org.netlib.util.floatW dn, org.netlib.util.floatW dn1, org.netlib.util.floatW dn2, org.netlib.util.floatW tau)
      ..
      void slazq4(int i0, int n0, float[] z, int pp, int n0in, float dmin, float dmin1, float dmin2, float dn, float dn1, float dn2, org.netlib.util.floatW tau, org.netlib.util.intW ttype, org.netlib.util.floatW g)
      ..
      abstract void sopgtr(java.lang.String uplo, int n, float[] ap, float[] tau, float[] q, int ldq, float[] work, org.netlib.util.intW info)
      ..
      abstract void sopmtr(java.lang.String side, java.lang.String uplo, java.lang.String trans, int m, int n, float[] ap, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorg2l(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorg2r(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorgbr(java.lang.String vect, int m, int n, int k, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorghr(int n, int ilo, int ihi, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorgl2(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorglq(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorgql(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorgqr(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorgr2(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorgrq(int m, int n, int k, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorgtr(java.lang.String uplo, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorm2l(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sorm2r(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sormbr(java.lang.String vect, java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormhr(java.lang.String side, java.lang.String trans, int m, int n, int ilo, int ihi, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sorml2(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sormlq(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormql(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormqr(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormr2(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sormr3(java.lang.String side, java.lang.String trans, int m, int n, int k, int l, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, org.netlib.util.intW info)
      ..
      abstract void sormrq(java.lang.String side, java.lang.String trans, int m, int n, int k, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormrz(java.lang.String side, java.lang.String trans, int m, int n, int k, int l, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void sormtr(java.lang.String side, java.lang.String uplo, java.lang.String trans, int m, int n, float[] a, int lda, float[] tau, float[] c, int Ldc, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void spbcon(java.lang.String uplo, int n, int kd, float[] ab, int ldab, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spbequ(java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] s, org.netlib.util.floatW scond, org.netlib.util.floatW amax, org.netlib.util.intW info)
      ..
      abstract void spbrfs(java.lang.String uplo, int n, int kd, int nrhs, float[] ab, int ldab, float[] afb, int ldafb, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spbstf(java.lang.String uplo, int n, int kd, float[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void spbsv(java.lang.String uplo, int n, int kd, int nrhs, float[] ab, int ldab, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void spbsvx(java.lang.String fact, java.lang.String uplo, int n, int kd, int nrhs, float[] ab, int ldab, float[] afb, int ldafb, org.netlib.util.StringW equed, float[] s, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spbtf2(java.lang.String uplo, int n, int kd, float[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void spbtrf(java.lang.String uplo, int n, int kd, float[] ab, int ldab, org.netlib.util.intW info)
      ..
      abstract void spbtrs(java.lang.String uplo, int n, int kd, int nrhs, float[] ab, int ldab, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void spocon(java.lang.String uplo, int n, float[] a, int lda, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spoequ(int n, float[] a, int lda, float[] s, org.netlib.util.floatW scond, org.netlib.util.floatW amax, org.netlib.util.intW info)
      ..
      abstract void sporfs(java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sposv(java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sposvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, org.netlib.util.StringW equed, float[] s, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spotf2(java.lang.String uplo, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void spotrf(java.lang.String uplo, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void spotri(java.lang.String uplo, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void spotrs(java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sppcon(java.lang.String uplo, int n, float[] ap, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sppequ(java.lang.String uplo, int n, float[] ap, float[] s, org.netlib.util.floatW scond, org.netlib.util.floatW amax, org.netlib.util.intW info)
      ..
      abstract void spprfs(java.lang.String uplo, int n, int nrhs, float[] ap, float[] afp, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sppsv(java.lang.String uplo, int n, int nrhs, float[] ap, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sppsvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, float[] ap, float[] afp, org.netlib.util.StringW equed, float[] s, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void spptrf(java.lang.String uplo, int n, float[] ap, org.netlib.util.intW info)
      ..
      abstract void spptri(java.lang.String uplo, int n, float[] ap, org.netlib.util.intW info)
      ..
      abstract void spptrs(java.lang.String uplo, int n, int nrhs, float[] ap, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sptcon(int n, float[] d, float[] e, float anorm, org.netlib.util.floatW rcond, float[] work, org.netlib.util.intW info)
      ..
      abstract void spteqr(java.lang.String compz, int n, float[] d, float[] e, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void sptrfs(int n, int nrhs, float[] d, float[] e, float[] df, float[] ef, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, org.netlib.util.intW info)
      ..
      abstract void sptsv(int n, int nrhs, float[] d, float[] e, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sptsvx(java.lang.String fact, int n, int nrhs, float[] d, float[] e, float[] df, float[] ef, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, org.netlib.util.intW info)
      ..
      abstract void spttrf(int n, float[] d, float[] e, org.netlib.util.intW info)
      ..
      abstract void spttrs(int n, int nrhs, float[] d, float[] e, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sptts2(int n, int nrhs, float[] d, float[] e, float[] b, int ldb)
      ..
      abstract void srscl(int n, float sa, float[] sx, int incx)
      ..
      abstract void ssbev(java.lang.String jobz, java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] w, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssbevd(java.lang.String jobz, java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssbevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] q, int ldq, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssbgst(java.lang.String vect, java.lang.String uplo, int n, int ka, int kb, float[] ab, int ldab, float[] bb, int ldbb, float[] x, int ldx, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssbgv(java.lang.String jobz, java.lang.String uplo, int n, int ka, int kb, float[] ab, int ldab, float[] bb, int ldbb, float[] w, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssbgvd(java.lang.String jobz, java.lang.String uplo, int n, int ka, int kb, float[] ab, int ldab, float[] bb, int ldbb, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssbgvx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, int ka, int kb, float[] ab, int ldab, float[] bb, int ldbb, float[] q, int ldq, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssbtrd(java.lang.String vect, java.lang.String uplo, int n, int kd, float[] ab, int ldab, float[] d, float[] e, float[] q, int ldq, float[] work, org.netlib.util.intW info)
      ..
      abstract void sspcon(java.lang.String uplo, int n, float[] ap, int[] ipiv, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sspev(java.lang.String jobz, java.lang.String uplo, int n, float[] ap, float[] w, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void sspevd(java.lang.String jobz, java.lang.String uplo, int n, float[] ap, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sspevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, float[] ap, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void sspgst(int itype, java.lang.String uplo, int n, float[] ap, float[] bp, org.netlib.util.intW info)
      ..
      abstract void sspgv(int itype, java.lang.String jobz, java.lang.String uplo, int n, float[] ap, float[] bp, float[] w, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void sspgvd(int itype, java.lang.String jobz, java.lang.String uplo, int n, float[] ap, float[] bp, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sspgvx(int itype, java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, float[] ap, float[] bp, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssprfs(java.lang.String uplo, int n, int nrhs, float[] ap, float[] afp, int[] ipiv, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sspsv(java.lang.String uplo, int n, int nrhs, float[] ap, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sspsvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, float[] ap, float[] afp, int[] ipiv, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void ssptrd(java.lang.String uplo, int n, float[] ap, float[] d, float[] e, float[] tau, org.netlib.util.intW info)
      ..
      abstract void ssptrf(java.lang.String uplo, int n, float[] ap, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void ssptri(java.lang.String uplo, int n, float[] ap, int[] ipiv, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssptrs(java.lang.String uplo, int n, int nrhs, float[] ap, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void sstebz(java.lang.String range, java.lang.String order, int n, float vl, float vu, int il, int iu, float abstol, float[] d, float[] e, org.netlib.util.intW m, org.netlib.util.intW nsplit, float[] w, int[] iblock, int[] isplit, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void sstedc(java.lang.String compz, int n, float[] d, float[] e, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sstegr(java.lang.String jobz, java.lang.String range, int n, float[] d, float[] e, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, int[] isuppz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sstein(int n, float[] d, float[] e, int m, float[] w, int[] iblock, int[] isplit, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      void sstemr(java.lang.String jobz, java.lang.String range, int n, float[] d, float[] e, float vl, float vu, int il, int iu, org.netlib.util.intW m, float[] w, float[] z, int ldz, int nzc, int[] isuppz, org.netlib.util.booleanW tryrac, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssteqr(java.lang.String compz, int n, float[] d, float[] e, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssterf(int n, float[] d, float[] e, org.netlib.util.intW info)
      ..
      abstract void sstev(java.lang.String jobz, int n, float[] d, float[] e, float[] z, int ldz, float[] work, org.netlib.util.intW info)
      ..
      abstract void sstevd(java.lang.String jobz, int n, float[] d, float[] e, float[] z, int ldz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sstevr(java.lang.String jobz, java.lang.String range, int n, float[] d, float[] e, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, int[] isuppz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void sstevx(java.lang.String jobz, java.lang.String range, int n, float[] d, float[] e, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssycon(java.lang.String uplo, int n, float[] a, int lda, int[] ipiv, float anorm, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void ssyev(java.lang.String jobz, java.lang.String uplo, int n, float[] a, int lda, float[] w, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void ssyevd(java.lang.String jobz, java.lang.String uplo, int n, float[] a, int lda, float[] w, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssyevr(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, float[] a, int lda, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, int[] isuppz, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssyevx(java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, float[] a, int lda, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssygs2(int itype, java.lang.String uplo, int n, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void ssygst(int itype, java.lang.String uplo, int n, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void ssygv(int itype, java.lang.String jobz, java.lang.String uplo, int n, float[] a, int lda, float[] b, int ldb, float[] w, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void ssygvd(int itype, java.lang.String jobz, java.lang.String uplo, int n, float[] a, int lda, float[] b, int ldb, float[] w, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void ssygvx(int itype, java.lang.String jobz, java.lang.String range, java.lang.String uplo, int n, float[] a, int lda, float[] b, int ldb, float vl, float vu, int il, int iu, float abstol, org.netlib.util.intW m, float[] w, float[] z, int ldz, float[] work, int lwork, int[] iwork, int[] ifail, org.netlib.util.intW info)
      ..
      abstract void ssyrfs(java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, int[] ipiv, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void ssysv(java.lang.String uplo, int n, int nrhs, float[] a, int lda, int[] ipiv, float[] b, int ldb, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void ssysvx(java.lang.String fact, java.lang.String uplo, int n, int nrhs, float[] a, int lda, float[] af, int ldaf, int[] ipiv, float[] b, int ldb, float[] x, int ldx, org.netlib.util.floatW rcond, float[] ferr, float[] berr, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void ssytd2(java.lang.String uplo, int n, float[] a, int lda, float[] d, float[] e, float[] tau, org.netlib.util.intW info)
      ..
      abstract void ssytf2(java.lang.String uplo, int n, float[] a, int lda, int[] ipiv, org.netlib.util.intW info)
      ..
      abstract void ssytrd(java.lang.String uplo, int n, float[] a, int lda, float[] d, float[] e, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void ssytrf(java.lang.String uplo, int n, float[] a, int lda, int[] ipiv, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void ssytri(java.lang.String uplo, int n, float[] a, int lda, int[] ipiv, float[] work, org.netlib.util.intW info)
      ..
      abstract void ssytrs(java.lang.String uplo, int n, int nrhs, float[] a, int lda, int[] ipiv, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void stbcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, int kd, float[] ab, int ldab, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stbrfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int kd, int nrhs, float[] ab, int ldab, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stbtrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int kd, int nrhs, float[] ab, int ldab, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void stgevc(java.lang.String side, java.lang.String howmny, boolean[] select, int n, float[] s, int lds, float[] p, int ldp, float[] vl, int ldvl, float[] vr, int ldvr, int mm, org.netlib.util.intW m, float[] work, org.netlib.util.intW info)
      ..
      abstract void stgex2(boolean wantq, boolean wantz, int n, float[] a, int lda, float[] b, int ldb, float[] q, int ldq, float[] z, int ldz, int j1, int n1, int n2, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void stgexc(boolean wantq, boolean wantz, int n, float[] a, int lda, float[] b, int ldb, float[] q, int ldq, float[] z, int ldz, org.netlib.util.intW ifst, org.netlib.util.intW ilst, float[] work, int lwork, org.netlib.util.intW info)
      ..
      abstract void stgsen(int ijob, boolean wantq, boolean wantz, boolean[] select, int n, float[] a, int lda, float[] b, int ldb, float[] alphar, float[] alphai, float[] beta, float[] q, int ldq, float[] z, int ldz, org.netlib.util.intW m, org.netlib.util.floatW pl, org.netlib.util.floatW pr, float[] dif, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void stgsja(java.lang.String jobu, java.lang.String jobv, java.lang.String jobq, int m, int p, int n, int k, int l, float[] a, int lda, float[] b, int ldb, float tola, float tolb, float[] alpha, float[] beta, float[] u, int ldu, float[] v, int ldv, float[] q, int ldq, float[] work, org.netlib.util.intW ncycle, org.netlib.util.intW info)
      ..
      abstract void stgsna(java.lang.String job, java.lang.String howmny, boolean[] select, int n, float[] a, int lda, float[] b, int ldb, float[] vl, int ldvl, float[] vr, int ldvr, float[] s, float[] dif, int mm, org.netlib.util.intW m, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stgsy2(java.lang.String trans, int ijob, int m, int n, float[] a, int lda, float[] b, int ldb, float[] c, int Ldc, float[] d, int ldd, float[] e, int lde, float[] f, int ldf, org.netlib.util.floatW scale, org.netlib.util.floatW rdsum, org.netlib.util.floatW rdscal, int[] iwork, org.netlib.util.intW pq, org.netlib.util.intW info)
      ..
      abstract void stgsyl(java.lang.String trans, int ijob, int m, int n, float[] a, int lda, float[] b, int ldb, float[] c, int Ldc, float[] d, int ldd, float[] e, int lde, float[] f, int ldf, org.netlib.util.floatW scale, org.netlib.util.floatW dif, float[] work, int lwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stpcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, float[] ap, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stprfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, float[] ap, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void stptri(java.lang.String uplo, java.lang.String diag, int n, float[] ap, org.netlib.util.intW info)
      ..
      abstract void stptrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, float[] ap, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void strcon(java.lang.String norm, java.lang.String uplo, java.lang.String diag, int n, float[] a, int lda, org.netlib.util.floatW rcond, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void strevc(java.lang.String side, java.lang.String howmny, boolean[] select, int n, float[] t, int ldt, float[] vl, int ldvl, float[] vr, int ldvr, int mm, org.netlib.util.intW m, float[] work, org.netlib.util.intW info)
      ..
      abstract void strexc(java.lang.String compq, int n, float[] t, int ldt, float[] q, int ldq, org.netlib.util.intW ifst, org.netlib.util.intW ilst, float[] work, org.netlib.util.intW info)
      ..
      abstract void strrfs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, float[] a, int lda, float[] b, int ldb, float[] x, int ldx, float[] ferr, float[] berr, float[] work, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void strsen(java.lang.String job, java.lang.String compq, boolean[] select, int n, float[] t, int ldt, float[] q, int ldq, float[] wr, float[] wi, org.netlib.util.intW m, org.netlib.util.floatW s, org.netlib.util.floatW sep, float[] work, int lwork, int[] iwork, int liwork, org.netlib.util.intW info)
      ..
      abstract void strsna(java.lang.String job, java.lang.String howmny, boolean[] select, int n, float[] t, int ldt, float[] vl, int ldvl, float[] vr, int ldvr, float[] s, float[] sep, int mm, org.netlib.util.intW m, float[] work, int ldwork, int[] iwork, org.netlib.util.intW info)
      ..
      abstract void strsyl(java.lang.String trana, java.lang.String tranb, int isgn, int m, int n, float[] a, int lda, float[] b, int ldb, float[] c, int Ldc, org.netlib.util.floatW scale, org.netlib.util.intW info)
      ..
      abstract void strti2(java.lang.String uplo, java.lang.String diag, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void strtri(java.lang.String uplo, java.lang.String diag, int n, float[] a, int lda, org.netlib.util.intW info)
      ..
      abstract void strtrs(java.lang.String uplo, java.lang.String trans, java.lang.String diag, int n, int nrhs, float[] a, int lda, float[] b, int ldb, org.netlib.util.intW info)
      ..
      abstract void stzrqf(int m, int n, float[] a, int lda, float[] tau, org.netlib.util.intW info)
      ..
      abstract void stzrzf(int m, int n, float[] a, int lda, float[] tau, float[] work, int lwork, org.netlib.util.intW info)
      ..

      • Methods inherited from class java.lang.Object

        equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

    • Constructor Detail

      • LAPACK

        public LAPACK()

    • Method Detail

      • getInstance

        public static LAPACK getInstance()

      • dbdsdc

        public abstract void dbdsdc(java.lang.String uplo,
                            java.lang.String compq,
                            int n,
                            double[] d,
                            double[] e,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            double[] q,
                            int[] iq,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DBDSDC computes the singular value decomposition (SVD) of a real
  N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
  using a divide and conquer method, where S is a diagonal matrix
  with non-negative diagonal elements (the singular values of B), and
  U and VT are orthogonal matrices of left and right singular vectors,

  respectively. DBDSDC can be used to compute all singular values,
  and optionally, singular vectors or singular vectors in compact form.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.  See DLASD3 for details.

  The code currently calls DLASDQ if singular values only are desired.

  However, it can be slightly modified to compute singular values
  using the divide and conquer method.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  B is upper bidiagonal.
          = 'L':  B is lower bidiagonal.

  COMPQ   (input) CHARACTER*1
          Specifies whether singular vectors are to be computed
          as follows:
          = 'N':  Compute singular values only;
          = 'P':  Compute singular values and compute singular
                  vectors in compact form;
          = 'I':  Compute singular values and singular vectors.

  N       (input) INTEGER
          The order of the matrix B.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the elements of E contain the offdiagonal
          elements of the bidiagonal matrix whose SVD is desired.
          On exit, E has been destroyed.

  U       (output) DOUBLE PRECISION array, dimension (LDU,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, U contains the left singular vectors
             of the bidiagonal matrix.
          For other values of COMPQ, U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1.
          If singular vectors are desired, then LDU >= max( 1, N ).

  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, VT' contains the right singular
             vectors of the bidiagonal matrix.
          For other values of COMPQ, VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1.
          If singular vectors are desired, then LDVT >= max( 1, N ).

  Q       (output) DOUBLE PRECISION array, dimension (LDQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, Q contains all the DOUBLE PRECISION data in
             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, Q is not referenced.

  IQ      (output) INTEGER array, dimension (LDIQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, IQ contains all INTEGER data in
             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, IQ is not referenced.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

          If COMPQ = 'N' then LWORK >= (4 * N).
          If COMPQ = 'P' then LWORK >= (6 * N).
          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

  IWORK   (workspace) INTEGER array, dimension (8*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an singular value.
                The update process of divide and conquer failed.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================
  Changed dimension statement in comment describing E from (N) to
  (N-1).  Sven, 17 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        compq -
        n -
        d -
        e -
        u -
        ldu -
        vt -
        ldvt -
        q -
        iq -
        work -
        iwork -
        info -

      • dbdsqr

        public abstract void dbdsqr(java.lang.String uplo,
                            int n,
                            int ncvt,
                            int nru,
                            int ncc,
                            double[] d,
                            double[] e,
                            double[] vt,
                            int ldvt,
                            double[] u,
                            int ldu,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DBDSQR computes the singular values and, optionally, the right and/or
  left singular vectors from the singular value decomposition (SVD) of

  a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
  zero-shift QR algorithm.  The SVD of B has the form
 
     B = Q * S * P**T
 
  where S is the diagonal matrix of singular values, Q is an orthogonal
  matrix of left singular vectors, and P is an orthogonal matrix of
  right singular vectors.  If left singular vectors are requested, this
  subroutine actually returns U*Q instead of Q, and, if right singular

  vectors are requested, this subroutine returns P**T*VT instead of
  P**T, for given real input matrices U and VT.  When U and VT are the

  orthogonal matrices that reduce a general matrix A to bidiagonal
  form:  A = U*B*VT, as computed by DGEBRD, then

     A = (U*Q) * S * (P**T*VT)

  is the SVD of A.  Optionally, the subroutine may also compute Q**T*C

  for a given real input matrix C.

  See "Computing  Small Singular Values of Bidiagonal Matrices With
  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
  LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
  no. 5, pp. 873-912, Sept 1990) and
  "Accurate singular values and differential qd algorithms," by
  B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
  Department, University of California at Berkeley, July 1992
  for a detailed description of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

  N       (input) INTEGER
          The order of the matrix B.  N >= 0.

  NCVT    (input) INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

  NRU     (input) INTEGER
          The number of rows of the matrix U. NRU >= 0.

  NCC     (input) INTEGER
          The number of columns of the matrix C. NCC >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B. 
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

  VT      (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * VT.
          Not referenced if NCVT = 0.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

  U       (input/output) DOUBLE PRECISION array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.
          On exit, U is overwritten by U * Q.
          Not referenced if NRU = 0.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * C.
          Not referenced if NCC = 0.

  LDC     (input) INTEGER
          The leading dimension of the array C.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
          if NCVT = NRU = NCC = 0, (max(1, 4*N)) otherwise

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm did not converge; D and E contain the
                elements of a bidiagonal matrix which is orthogonally
                similar to the input matrix B;  if INFO = i, i
                elements of E have not converged to zero.

  Internal Parameters
  ===================

  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          If it is positive, TOLMUL*EPS is the desired relative
             precision in the computed singular values.
          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
             desired absolute accuracy in the computed singular
             values (corresponds to relative accuracy
             abs(TOLMUL*EPS) in the largest singular value.
          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
             between 10 (for fast convergence) and .1/EPS
             (for there to be some accuracy in the results).
          Default is to lose at either one eighth or 2 of the
             available decimal digits in each computed singular value
             (whichever is smaller).

  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the
          algorithm through its inner loop. The algorithms stops
          (and so fails to converge) if the number of passes
          through the inner loop exceeds MAXITR*N**2.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ncvt -
        nru -
        ncc -
        d -
        e -
        vt -
        ldvt -
        u -
        ldu -
        c -
        Ldc -
        work -
        info -

      • ddisna

        public abstract void ddisna(java.lang.String job,
                            int m,
                            int n,
                            double[] d,
                            double[] sep,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DDISNA computes the reciprocal condition numbers for the eigenvectors
  of a real symmetric or complex Hermitian matrix or for the left or
  right singular vectors of a general m-by-n matrix. The reciprocal
  condition number is the 'gap' between the corresponding eigenvalue or
  singular value and the nearest other one.

  The bound on the error, measured by angle in radians, in the I-th
  computed vector is given by

         DLAMCH( 'E' ) * ( ANORM / SEP( I ) )

  where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed

  to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of

  the error bound.

  DDISNA may also be used to compute error bounds for eigenvectors of
  the generalized symmetric definite eigenproblem.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies for which problem the reciprocal condition numbers

          should be computed:
          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
          = 'L':  the left singular vectors of a general matrix;
          = 'R':  the right singular vectors of a general matrix.

  M       (input) INTEGER
          The number of rows of the matrix. M >= 0.

  N       (input) INTEGER
          If JOB = 'L' or 'R', the number of columns of the matrix,
          in which case N >= 0. Ignored if JOB = 'E'.

  D       (input) DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                              dimension (min(M,N)) if JOB = 'L' or 'R'

          The eigenvalues (if JOB = 'E') or singular values (if JOB =
          'L' or 'R') of the matrix, in either increasing or decreasing
          order. If singular values, they must be non-negative.

  SEP     (output) DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                               dimension (min(M,N)) if JOB = 'L' or 'R'
          The reciprocal condition numbers of the vectors.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        m -
        n -
        d -
        sep -
        info -

      • dgbbrd

        public abstract void dgbbrd(java.lang.String vect,
                            int m,
                            int n,
                            int ncc,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            double[] d,
                            double[] e,
                            double[] q,
                            int ldq,
                            double[] pt,
                            int ldpt,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBBRD reduces a real general m-by-n band matrix A to upper
  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.

  The routine computes B, and optionally forms Q or P', or computes
  Q'*C for a given matrix C.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          Specifies whether or not the matrices Q and P' are to be
          formed.
          = 'N': do not form Q or P';
          = 'Q': form Q only;
          = 'P': form P' only;
          = 'B': form both.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NCC     (input) INTEGER
          The number of columns of the matrix C.  NCC >= 0.

  KL      (input) INTEGER
          The number of subdiagonals of the matrix A. KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals of the matrix A. KU >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the m-by-n band matrix A, stored in rows 1 to
          KL+KU+1. The j-th column of A is stored in the j-th column of
          the array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
          On exit, A is overwritten by values generated during the
          reduction.

  LDAB    (input) INTEGER
          The leading dimension of the array A. LDAB >= KL+KU+1.

  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B.

  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
          The superdiagonal elements of the bidiagonal matrix B.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,M)
          If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
          If VECT = 'N' or 'P', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.

  PT      (output) DOUBLE PRECISION array, dimension (LDPT,N)
          If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
          If VECT = 'N' or 'Q', the array PT is not referenced.

  LDPT    (input) INTEGER
          The leading dimension of the array PT.
          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,NCC)
          On entry, an m-by-ncc matrix C.
          On exit, C is overwritten by Q'*C.
          C is not referenced if NCC = 0.

  LDC     (input) INTEGER
          The leading dimension of the array C.
          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*max(M,N))

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        m -
        n -
        ncc -
        kl -
        ku -
        ab -
        ldab -
        d -
        e -
        q -
        ldq -
        pt -
        ldpt -
        c -
        Ldc -
        work -
        info -

      • dgbcon

        public abstract void dgbcon(java.lang.String norm,
                            int n,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            int[] ipiv,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBCON estimates the reciprocal of the condition number of a real
  general band matrix A, in either the 1-norm or the infinity-norm,
  using the LU factorization computed by DGBTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).

  ANORM   (input) DOUBLE PRECISION
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dgbequ

        public abstract void dgbequ(int m,
                            int n,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            double[] r,
                            double[] c,
                            org.netlib.util.doubleW rowcnd,
                            org.netlib.util.doubleW colcnd,
                            org.netlib.util.doubleW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBEQU computes row and column scalings intended to equilibrate an
  M-by-N band matrix A and reduce its condition number.  R returns the

  row scale factors and C the column scale factors, chosen to try to
  make the largest element in each row and column of the matrix B with

  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
  number and BIGNUM = largest safe number.  Use of these scaling
  factors is not guaranteed to reduce the condition number of A but
  works well in practice.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  R       (output) DOUBLE PRECISION array, dimension (M)
          If INFO = 0, or INFO > M, R contains the row scale factors
          for A.

  C       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, C contains the column scale factors for A.

  ROWCND  (output) DOUBLE PRECISION
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.

  COLCND  (output) DOUBLE PRECISION
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.

  AMAX    (output) DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        info -

      • dgbrfs

        public abstract void dgbrfs(java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] afb,
                            int ldafb,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is banded, and provides
  error bounds and backward error estimates for the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The original band matrix A, stored in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.


  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from DGBTRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DGBTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgbsv

        public abstract void dgbsv(int n,
                           int kl,
                           int ku,
                           int nrhs,
                           double[] ab,
                           int ldab,
                           int[] ipiv,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBSV computes the solution to a real system of linear equations
  A * X = B, where A is a band matrix of order N with KL subdiagonals
  and KU superdiagonals, and X and B are N-by-NRHS matrices.

  The LU decomposition with partial pivoting and row interchanges is
  used to factor A as A = L * U, where L is a product of permutation
  and unit lower triangular matrices with KL subdiagonals, and U is
  upper triangular with KL+KU superdiagonals.  The factored form of A
  is then used to solve the system of equations A * X = B.

  Arguments
  =========

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                has been completed, but the factor U is exactly
                singular, and the solution has not been computed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U because of fill-in resulting from the row interchanges.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        ipiv -
        b -
        ldb -
        info -

      • dgbsvx

        public abstract void dgbsvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] afb,
                            int ldafb,
                            int[] ipiv,
                            org.netlib.util.StringW equed,
                            double[] r,
                            double[] c,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBSVX uses the LU factorization to compute the solution to a real
  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

  where A is a band matrix of order N with KL subdiagonals and KU
  superdiagonals, and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed by this subroutine:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
     or diag(C)*B (if TRANS = 'T' or 'C').

  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
     matrix A (after equilibration if FACT = 'E') as
        A = L * U,
     where L is a product of permutation and unit lower triangular
     matrices with KL subdiagonals, and U is upper triangular with
     KL+KU superdiagonals.

  3. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
     that it solves the original system before equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFB and IPIV contain the factored form of
                  A.  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  AB, AFB, and IPIV are not modified.
          = 'N':  The matrix A will be copied to AFB and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFB and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

          If FACT = 'F' and EQUED is not 'N', then A must have been
          equilibrated by the scaling factors in R and/or C.  AB is not
          modified if FACT = 'F' or 'N', or if FACT = 'E' and
          EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry
          contains details of the LU factorization of the band matrix
          A, as computed by DGBTRF.  U is stored as an upper triangular
          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
          and the multipliers used during the factorization are stored

          in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
          the factored form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit
          returns details of the LU factorization of A.

          If FACT = 'E', then AFB is an output argument and on exit
          returns details of the LU factorization of the equilibrated
          matrix A (see the description of AB for the form of the
          equilibrated matrix).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.


  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = L*U
          as computed by DGBTRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the equilibrated matrix A.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  R       (input or output) DOUBLE PRECISION array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.

  C       (input or output) DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is

          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C must be positive.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit,
          if EQUED = 'N', B is not modified;
          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
          to the original system of equations.  Note that A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (3*N)
          On exit, WORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If WORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has been completed, but the factor U is exactly

                       singular, so the solution and error bounds
                       could not be computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        ipiv -
        equed -
        r -
        c -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgbtf2

        public abstract void dgbtf2(int m,
                            int n,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBTF2 computes an LU factorization of a real m-by-n band matrix A
  using partial pivoting with row interchanges.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization

               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U, because of fill-in resulting from the row
  interchanges.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        info -

      • dgbtrf

        public abstract void dgbtrf(int m,
                            int n,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBTRF computes an LU factorization of a real m-by-n band matrix A
  using partial pivoting with row interchanges.

  This is the blocked version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization

               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U because of fill-in resulting from the row interchanges.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        info -

      • dgbtrs

        public abstract void dgbtrs(java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGBTRS solves a system of linear equations
     A * X = B  or  A' * X = B
  with a general band matrix A using the LU factorization computed
  by DGBTRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        ipiv -
        b -
        ldb -
        info -

      • dgebak

        public abstract void dgebak(java.lang.String job,
                            java.lang.String side,
                            int n,
                            int ilo,
                            int ihi,
                            double[] scale,
                            int m,
                            double[] v,
                            int ldv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEBAK forms the right or left eigenvectors of a real general matrix

  by backward transformation on the computed eigenvectors of the
  balanced matrix output by DGEBAL.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the type of backward transformation required:
          = 'N', do nothing, return immediately;
          = 'P', do backward transformation for permutation only;
          = 'S', do backward transformation for scaling only;
          = 'B', do backward transformations for both permutation and
                 scaling.
          JOB must be the same as the argument JOB supplied to DGEBAL.


  SIDE    (input) CHARACTER*1
          = 'R':  V contains right eigenvectors;
          = 'L':  V contains left eigenvectors.

  N       (input) INTEGER
          The number of rows of the matrix V.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          The integers ILO and IHI determined by DGEBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  SCALE   (input) DOUBLE PRECISION array, dimension (N)
          Details of the permutation and scaling factors, as returned
          by DGEBAL.

  M       (input) INTEGER
          The number of columns of the matrix V.  M >= 0.

  V       (input/output) DOUBLE PRECISION array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by DHSEIN or DTREVC.
          On exit, V is overwritten by the transformed eigenvectors.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        side -
        n -
        ilo -
        ihi -
        scale -
        m -
        v -
        ldv -
        info -

      • dgebal

        public abstract void dgebal(java.lang.String job,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            double[] scale,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEBAL balances a general real matrix A.  This involves, first,
  permuting A by a similarity transformation to isolate eigenvalues
  in the first 1 to ILO-1 and last IHI+1 to N elements on the
  diagonal; and second, applying a diagonal similarity transformation
  to rows and columns ILO to IHI to make the rows and columns as
  close in norm as possible.  Both steps are optional.

  Balancing may reduce the 1-norm of the matrix, and improve the
  accuracy of the computed eigenvalues and/or eigenvectors.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the operations to be performed on A:
          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0
                  for i = 1,...,N;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the input matrix A.
          On exit,  A is overwritten by the balanced matrix.
          If JOB = 'N', A is not referenced.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are set to integers such that on exit
          A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
          If JOB = 'N' or 'S', ILO = 1 and IHI = N.

  SCALE   (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied to
          A.  If P(j) is the index of the row and column interchanged
          with row and column j and D(j) is the scaling factor
          applied to row and column j, then
          SCALE(j) = P(j)    for j = 1,...,ILO-1
                   = D(j)    for j = ILO,...,IHI
                   = P(j)    for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The permutations consist of row and column interchanges which put
  the matrix in the form

             ( T1   X   Y  )
     P A P = (  0   B   Z  )
             (  0   0   T2 )

  where T1 and T2 are upper triangular matrices whose eigenvalues lie
  along the diagonal.  The column indices ILO and IHI mark the starting
  and ending columns of the submatrix B. Balancing consists of applying
  a diagonal similarity transformation inv(D) * B * D to make the
  1-norms of each row of B and its corresponding column nearly equal.
  The output matrix is

     ( T1     X*D          Y    )
     (  0  inv(D)*B*D  inv(D)*Z ).
     (  0      0           T2   )

  Information about the permutations P and the diagonal matrix D is
  returned in the vector SCALE.

  This subroutine is based on the EISPACK routine BALANC.

  Modified by Tzu-Yi Chen, Computer Science Division, University of
    California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        lda -
        ilo -
        ihi -
        scale -
        info -

      • dgebd2

        public abstract void dgebd2(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] d,
                            double[] e,
                            double[] tauq,
                            double[] taup,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEBD2 reduces a real general m by n matrix A to upper or lower
  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.

  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns in the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit,
          if m >= n, the diagonal and the first superdiagonal are
            overwritten with the upper bidiagonal matrix B; the
            elements below the diagonal, with the array TAUQ, represent
            the orthogonal matrix Q as a product of elementary
            reflectors, and the elements above the first superdiagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors;
          if m < n, the diagonal and the first subdiagonal are
            overwritten with the lower bidiagonal matrix B; the
            elements below the first subdiagonal, with the array TAUQ,

            represent the orthogonal matrix Q as a product of
            elementary reflectors, and the elements above the diagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
          The off-diagonal elements of the bidiagonal matrix B:
          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  WORK    (workspace) DOUBLE PRECISION array, dimension (max(M,N))

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  The contents of A on exit are illustrated by the following examples:


  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi
  denotes an element of the vector defining H(i), and ui an element of

  the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        work -
        info -

      • dgebrd

        public abstract void dgebrd(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] d,
                            double[] e,
                            double[] tauq,
                            double[] taup,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEBRD reduces a general real M-by-N matrix A to upper or lower
  bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.


  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns in the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N general matrix to be reduced.
          On exit,
          if m >= n, the diagonal and the first superdiagonal are
            overwritten with the upper bidiagonal matrix B; the
            elements below the diagonal, with the array TAUQ, represent
            the orthogonal matrix Q as a product of elementary
            reflectors, and the elements above the first superdiagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors;
          if m < n, the diagonal and the first subdiagonal are
            overwritten with the lower bidiagonal matrix B; the
            elements below the first subdiagonal, with the array TAUQ,

            represent the orthogonal matrix Q as a product of
            elementary reflectors, and the elements above the diagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
          The off-diagonal elements of the bidiagonal matrix B:
          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,M,N).
          For optimum performance LWORK >= (M+N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  The contents of A on exit are illustrated by the following examples:


  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi
  denotes an element of the vector defining H(i), and ui an element of

  the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        work -
        lwork -
        info -

      • dgecon

        public abstract void dgecon(java.lang.String norm,
                            int n,
                            double[] a,
                            int lda,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGECON estimates the reciprocal of the condition number of a general

  real matrix A, in either the 1-norm or the infinity-norm, using
  the LU factorization computed by DGETRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The factors L and U from the factorization A = P*L*U
          as computed by DGETRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ANORM   (input) DOUBLE PRECISION
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        a -
        lda -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dgeequ

        public abstract void dgeequ(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] r,
                            double[] c,
                            org.netlib.util.doubleW rowcnd,
                            org.netlib.util.doubleW colcnd,
                            org.netlib.util.doubleW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEEQU computes row and column scalings intended to equilibrate an
  M-by-N matrix A and reduce its condition number.  R returns the row
  scale factors and C the column scale factors, chosen to try to make
  the largest element in each row and column of the matrix B with
  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
  number and BIGNUM = largest safe number.  Use of these scaling
  factors is not guaranteed to reduce the condition number of A but
  works well in practice.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The M-by-N matrix whose equilibration factors are
          to be computed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  R       (output) DOUBLE PRECISION array, dimension (M)
          If INFO = 0 or INFO > M, R contains the row scale factors
          for A.

  C       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0,  C contains the column scale factors for A.

  ROWCND  (output) DOUBLE PRECISION
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.

  COLCND  (output) DOUBLE PRECISION
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.

  AMAX    (output) DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i,  and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        info -

      • dgees

        public void dgees(java.lang.String jobvs,
                  java.lang.String sort,
                  java.lang.Object select,
                  int n,
                  double[] a,
                  int lda,
                  org.netlib.util.intW sdim,
                  double[] wr,
                  double[] wi,
                  double[] vs,
                  int ldvs,
                  double[] work,
                  int lwork,
                  boolean[] bwork,
                  org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEES computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues, the real Schur form T, and, optionally, the matrix of
  Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).


  Optionally, it also orders the eigenvalues on the diagonal of the
  real Schur form so that selected eigenvalues are at the top left.
  The leading columns of Z then form an orthonormal basis for the
  invariant subspace corresponding to the selected eigenvalues.

  A matrix is in real Schur form if it is upper quasi-triangular with
  1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
  form
          [  a  b  ]
          [  c  a  ]

  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

  Arguments
  =========

  JOBVS   (input) CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

  SELECT  (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
          conjugate pair of eigenvalues is selected, then both complex

          eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO is set to N+2 (see INFO below).

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten by its real Schur form T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues in the same order
          that they appear on the diagonal of the output Schur form T.

          Complex conjugate pairs of eigenvalues will appear
          consecutively with the eigenvalue having the positive
          imaginary part first.

  VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur

          vectors.
          If JOBVS = 'N', VS is not referenced.

  LDVS    (input) INTEGER
          The leading dimension of the array VS.  LDVS >= 1; if
          JOBVS = 'V', LDVS >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the matrix which reduces A

                   to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem

                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvs -
        sort -
        select -
        n -
        a -
        lda -
        sdim -
        wr -
        wi -
        vs -
        ldvs -
        work -
        lwork -
        bwork -
        info -

      • dgeesx

        public void dgeesx(java.lang.String jobvs,
                   java.lang.String sort,
                   java.lang.Object select,
                   java.lang.String sense,
                   int n,
                   double[] a,
                   int lda,
                   org.netlib.util.intW sdim,
                   double[] wr,
                   double[] wi,
                   double[] vs,
                   int ldvs,
                   org.netlib.util.doubleW rconde,
                   org.netlib.util.doubleW rcondv,
                   double[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   boolean[] bwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEESX computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues, the real Schur form T, and, optionally, the matrix of
  Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).


  Optionally, it also orders the eigenvalues on the diagonal of the
  real Schur form so that selected eigenvalues are at the top left;
  computes a reciprocal condition number for the average of the
  selected eigenvalues (RCONDE); and computes a reciprocal condition
  number for the right invariant subspace corresponding to the
  selected eigenvalues (RCONDV).  The leading columns of Z form an
  orthonormal basis for this invariant subspace.

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
  these quantities are called s and sep respectively).

  A real matrix is in real Schur form if it is upper quasi-triangular
  with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in

  the form
            [  a  b  ]
            [  c  a  ]

  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

  Arguments
  =========

  JOBVS   (input) CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

  SELECT  (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a
          complex conjugate pair of eigenvalues is selected, then both

          are.  Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO may be set to N+3 (see INFO below).

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for average of selected eigenvalues only;
          = 'V': Computed for selected right invariant subspace only;
          = 'B': Computed for both.
          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N matrix A.
          On exit, A is overwritten by its real Schur form T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          WR and WI contain the real and imaginary parts, respectively,
          of the computed eigenvalues, in the same order that they
          appear on the diagonal of the output Schur form T.  Complex
          conjugate pairs of eigenvalues appear consecutively with the

          eigenvalue having the positive imaginary part first.

  VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur

          vectors.
          If JOBVS = 'N', VS is not referenced.

  LDVS    (input) INTEGER
          The leading dimension of the array VS.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N.

  RCONDE  (output) DOUBLE PRECISION
          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
          condition number for the average of the selected eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

  RCONDV  (output) DOUBLE PRECISION
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace.
          Not referenced if SENSE = 'N' or 'E'.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).
          Also, if SENSE = 'E' or 'V' or 'B',
          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
          selected eigenvalues computed by this routine.  Note that
          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or

          'B' this may not be large enough.
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates upper bounds on the optimal sizes of the
          arrays WORK and IWORK, returns these values as the first
          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).

          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is

          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
          may not be large enough.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates upper bounds on the optimal sizes of

          the arrays WORK and IWORK, returns these values as the first

          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the transformation which
                   reduces A to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem

                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvs -
        sort -
        select -
        sense -
        n -
        a -
        lda -
        sdim -
        wr -
        wi -
        vs -
        ldvs -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        liwork -
        bwork -
        info -

      • dgeev

        public abstract void dgeev(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           double[] a,
                           int lda,
                           double[] wr,
                           double[] wi,
                           double[] vl,
                           int ldvl,
                           double[] vr,
                           int ldvr,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEEV computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues and, optionally, the left and/or right eigenvectors.

  The right eigenvector v(j) of A satisfies
                   A * v(j) = lambda(j) * v(j)
  where lambda(j) is its eigenvalue.
  The left eigenvector u(j) of A satisfies
                u(j)**H * A = lambda(j) * u(j)**H
  where u(j)**H denotes the conjugate transpose of u(j).

  The computed eigenvectors are normalized to have Euclidean norm
  equal to 1 and largest component real.

  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.

  JOBVR   (input) CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1; if
          JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N), and
          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good
          performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors have been computed;
                elements i+1:N of WR and WI contain eigenvalues which
                have converged.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • dgeevx

        public abstract void dgeevx(java.lang.String balanc,
                            java.lang.String jobvl,
                            java.lang.String jobvr,
                            java.lang.String sense,
                            int n,
                            double[] a,
                            int lda,
                            double[] wr,
                            double[] wi,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            double[] scale,
                            org.netlib.util.doubleW abnrm,
                            double[] rconde,
                            double[] rcondv,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues and, optionally, the left and/or right eigenvectors.

  Optionally also, it computes a balancing transformation to improve
  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
  SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
  (RCONDE), and reciprocal condition numbers for the right
  eigenvectors (RCONDV).

  The right eigenvector v(j) of A satisfies
                   A * v(j) = lambda(j) * v(j)
  where lambda(j) is its eigenvalue.
  The left eigenvector u(j) of A satisfies
                u(j)**H * A = lambda(j) * u(j)**H
  where u(j)**H denotes the conjugate transpose of u(j).

  The computed eigenvectors are normalized to have Euclidean norm
  equal to 1 and largest component real.

  Balancing a matrix means permuting the rows and columns to make it
  more nearly upper triangular, and applying a diagonal similarity
  transformation D * A * D**(-1), where D is a diagonal matrix, to
  make its rows and columns closer in norm and the condition numbers
  of its eigenvalues and eigenvectors smaller.  The computed
  reciprocal condition numbers correspond to the balanced matrix.
  Permuting rows and columns will not change the condition numbers
  (in exact arithmetic) but diagonal scaling will.  For further
  explanation of balancing, see section 4.10.2 of the LAPACK
  Users' Guide.

  Arguments
  =========

  BALANC  (input) CHARACTER*1
          Indicates how the input matrix should be diagonally scaled
          and/or permuted to improve the conditioning of its
          eigenvalues.
          = 'N': Do not diagonally scale or permute;
          = 'P': Perform permutations to make the matrix more nearly
                 upper triangular. Do not diagonally scale;
          = 'S': Diagonally scale the matrix, i.e. replace A by
                 D*A*D**(-1), where D is a diagonal matrix chosen
                 to make the rows and columns of A more equal in
                 norm. Do not permute;
          = 'B': Both diagonally scale and permute A.

          Computed reciprocal condition numbers will be for the matrix

          after balancing and/or permuting. Permuting does not change
          condition numbers (in exact arithmetic), but balancing does.


  JOBVL   (input) CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVL must = 'V'.

  JOBVR   (input) CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVR must = 'V'.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for eigenvalues only;
          = 'V': Computed for right eigenvectors only;
          = 'B': Computed for eigenvalues and right eigenvectors.

          If SENSE = 'E' or 'B', both left and right eigenvectors
          must also be computed (JOBVL = 'V' and JOBVR = 'V').

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.  If JOBVL = 'V' or
          JOBVR = 'V', A contains the real Schur form of the balanced
          version of the input matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues will appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          JOBVR = 'V', LDVR >= N.

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are integer values determined when A was
          balanced.  The balanced A(i,j) = 0 if I > J and
          J = 1,...,ILO-1 or I = IHI+1,...,N.

  SCALE   (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          when balancing A.  If P(j) is the index of the row and column
          interchanged with row and column j, and D(j) is the scaling
          factor applied to row and column j, then
          SCALE(J) = P(J),    for J = 1,...,ILO-1
                   = D(J),    for J = ILO,...,IHI
                   = P(J)     for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  ABNRM   (output) DOUBLE PRECISION
          The one-norm of the balanced matrix (the maximum
          of the sum of absolute values of elements of any column).

  RCONDE  (output) DOUBLE PRECISION array, dimension (N)
          RCONDE(j) is the reciprocal condition number of the j-th
          eigenvalue.

  RCONDV  (output) DOUBLE PRECISION array, dimension (N)
          RCONDV(j) is the reciprocal condition number of the j-th
          right eigenvector.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.   If SENSE = 'N' or 'E',
          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (2*N-2)
          If SENSE = 'N' or 'E', not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors or condition numbers
                have been computed; elements 1:ILO-1 and i+1:N of WR
                and WI contain eigenvalues which have converged.

  =====================================================================

     .. Parameters ..
        Parameters:
        balanc -
        jobvl -
        jobvr -
        sense -
        n -
        a -
        lda -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        ilo -
        ihi -
        scale -
        abnrm -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        info -

      • dgegs

        public abstract void dgegs(java.lang.String jobvsl,
                           java.lang.String jobvsr,
                           int n,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double[] alphar,
                           double[] alphai,
                           double[] beta,
                           double[] vsl,
                           int ldvsl,
                           double[] vsr,
                           int ldvsr,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DGGES.

  DGEGS computes the eigenvalues, real Schur form, and, optionally,
  left and or/right Schur vectors of a real matrix pair (A,B).
  Given two square matrices A and B, the generalized real Schur
  factorization has the form

    A = Q*S*Z**T,  B = Q*T*Z**T

  where Q and Z are orthogonal matrices, T is upper triangular, and S
  is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
  blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
  of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
  and the columns of Z are the right Schur vectors.

  If only the eigenvalues of (A,B) are needed, the driver routine
  DGEGV should be used instead.  See DGEGV for a description of the
  eigenvalues of the generalized nonsymmetric eigenvalue problem
  (GNEP).

  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors (returned in VSL).

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors (returned in VSR).

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the matrix A.
          On exit, the upper quasi-triangular matrix S from the
          generalized real Schur factorization.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the matrix B.
          On exit, the upper triangular matrix T from the generalized
          real Schur factorization.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and (j+1)-st
          eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) DOUBLE PRECISION array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  VSL     (output) DOUBLE PRECISION array, dimension (LDVSL,N)
          If JOBVSL = 'V', the matrix of left Schur vectors Q.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) DOUBLE PRECISION array, dimension (LDVSR,N)
          If JOBVSR = 'V', the matrix of right Schur vectors Z.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,4*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
          The optimal LWORK is  2*N + N*(NB+1).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from DGGBAL
                =N+2: error return from DGEQRF
                =N+3: error return from DORMQR
                =N+4: error return from DORGQR
                =N+5: error return from DGGHRD
                =N+6: error return from DHGEQZ (other than failed
                                                iteration)
                =N+7: error return from DGGBAK (computing VSL)
                =N+8: error return from DGGBAK (computing VSR)
                =N+9: error return from DLASCL (various places)

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        work -
        lwork -
        info -

      • dgegv

        public abstract void dgegv(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double[] alphar,
                           double[] alphai,
                           double[] beta,
                           double[] vl,
                           int ldvl,
                           double[] vr,
                           int ldvr,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DGGEV.

  DGEGV computes the eigenvalues and, optionally, the left and/or right
  eigenvectors of a real matrix pair (A,B).
  Given two square matrices A and B,
  the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
  eigenvalues lambda and corresponding (non-zero) eigenvectors x such
  that

     A*x = lambda*B*x.

  An alternate form is to find the eigenvalues mu and corresponding
  eigenvectors y such that

     mu*A*y = B*y.

  These two forms are equivalent with mu = 1/lambda and x = y if
  neither lambda nor mu is zero.  In order to deal with the case that
  lambda or mu is zero or small, two values alpha and beta are returned
  for each eigenvalue, such that lambda = alpha/beta and
  mu = beta/alpha.

  The vectors x and y in the above equations are right eigenvectors of

  the matrix pair (A,B).  Vectors u and v satisfying

     u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

  are left eigenvectors of (A,B).

  Note: this routine performs "full balancing" on A and B -- see
  "Further Details", below.

  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors (returned
                  in VL).

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors (returned

                  in VR).

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the matrix A.
          If JOBVL = 'V' or JOBVR = 'V', then on exit A
          contains the real Schur form of A from the generalized Schur

          factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only the diagonal
          blocks from the Schur form will be correct.  See DGGHRD and
          DHGEQZ for details.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the matrix B.
          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
          upper triangular matrix obtained from B in the generalized
          Schur factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only those elements of
          B corresponding to the diagonal blocks from the Schur form of
          A will be correct.  See DGGHRD and DHGEQZ for details.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue of
          GNEP.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) DOUBLE PRECISION array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored
          in the columns of VL, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then u(j) = VL(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
             u(j) = VL(:,j) + i*VL(:,j+1)
          and
            u(j+1) = VL(:,j) - i*VL(:,j+1).

          Each eigenvector is scaled so that its largest component has

          abs(real part) + abs(imag. part) = 1, except for eigenvectors
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors x(j) are stored
          in the columns of VR, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then x(j) = VR(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
            x(j) = VR(:,j) + i*VR(:,j+1)
          and
            x(j+1) = VR(:,j) - i*VR(:,j+1).

          Each eigenvector is scaled so that its largest component has

          abs(real part) + abs(imag. part) = 1, except for eigenvalues

          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;

          The optimal LWORK is:
              2*N + MAX( 6*N, N*(NB+1) ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from DGGBAL
                =N+2: error return from DGEQRF
                =N+3: error return from DORMQR
                =N+4: error return from DORGQR
                =N+5: error return from DGGHRD
                =N+6: error return from DHGEQZ (other than failed
                                                iteration)
                =N+7: error return from DTGEVC
                =N+8: error return from DGGBAK (computing VL)
                =N+9: error return from DGGBAK (computing VR)
                =N+10: error return from DLASCL (various calls)

  Further Details
  ===============

  Balancing
  ---------

  This driver calls DGGBAL to both permute and scale rows and columns
  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
  and PL*B*R will be upper triangular except for the diagonal blocks
  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
  possible.  The diagonal scaling matrices DL and DR are chosen so
  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
  one (except for the elements that start out zero.)

  After the eigenvalues and eigenvectors of the balanced matrices
  have been computed, DGGBAK transforms the eigenvectors back to what
  they would have been (in perfect arithmetic) if they had not been
  balanced.

  Contents of A and B on Exit
  -------- -- - --- - -- ----

  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
  both), then on exit the arrays A and B will contain the real Schur
  form[*] of the "balanced" versions of A and B.  If no eigenvectors
  are computed, then only the diagonal blocks will be correct.

  [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
      by Golub & van Loan, pub. by Johns Hopkins U. Press.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • dgehd2

        public abstract void dgehd2(int n,
                            int ilo,
                            int ihi,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEHD2 reduces a real general matrix A to upper Hessenberg form H by

  an orthogonal similarity transformation:  Q' * A * Q = H .

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          It is assumed that A is already upper triangular in rows
          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
          set by a previous call to DGEBAL; otherwise they should be
          set to 1 and N respectively. See Further Details.
          1 <= ILO <= IHI <= max(1,N).

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the n by n general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          elements below the first subdiagonal, with the array TAU,
          represent the orthogonal matrix Q as a product of elementary

          reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of (ihi-ilo) elementary
  reflectors

     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
  exit in A(i+2:ihi,i), and tau in TAU(i).

  The contents of A are illustrated by the following example, with
  n = 7, ilo = 2 and ihi = 6:

  on entry,                        on exit,

  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
  (                         a )    (                          a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        info -

      • dgehrd

        public abstract void dgehrd(int n,
                            int ilo,
                            int ihi,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEHRD reduces a real general matrix A to upper Hessenberg form H by

  an orthogonal similarity transformation:  Q' * A * Q = H .

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          It is assumed that A is already upper triangular in rows
          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
          set by a previous call to DGEBAL; otherwise they should be
          set to 1 and N respectively. See Further Details.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          elements below the first subdiagonal, with the array TAU,
          represent the orthogonal matrix Q as a product of elementary

          reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
          zero.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of (ihi-ilo) elementary
  reflectors

     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
  exit in A(i+2:ihi,i), and tau in TAU(i).

  The contents of A are illustrated by the following example, with
  n = 7, ilo = 2 and ihi = 6:

  on entry,                        on exit,

  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
  (                         a )    (                          a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  This file is a slight modification of LAPACK-3.0's DGEHRD
  subroutine incorporating improvements proposed by Quintana-Orti and
  Van de Geijn (2005). 

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dgelq2

        public abstract void dgelq2(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELQ2 computes an LQ factorization of a real m by n matrix A:
  A = L * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, the elements on and below the diagonal of the array

          contain the m by min(m,n) lower trapezoidal matrix L (L is
          lower triangular if m <= n); the elements above the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) DOUBLE PRECISION array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

  and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • dgelqf

        public abstract void dgelqf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELQF computes an LQ factorization of a real M-by-N matrix A:
  A = L * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and below the diagonal of the array

          contain the m-by-min(m,n) lower trapezoidal matrix L (L is
          lower triangular if m <= n); the elements above the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

  and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dgels

        public abstract void dgels(java.lang.String trans,
                           int m,
                           int n,
                           int nrhs,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELS solves overdetermined or underdetermined real linear systems
  involving an M-by-N matrix A, or its transpose, using a QR or LQ
  factorization of A.  It is assumed that A has full rank.

  The following options are provided:

  1. If TRANS = 'N' and m >= n:  find the least squares solution of
     an overdetermined system, i.e., solve the least squares problem
                  minimize || B - A*X ||.

  2. If TRANS = 'N' and m < n:  find the minimum norm solution of
     an underdetermined system A * X = B.

  3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
     an undetermined system A**T * X = B.

  4. If TRANS = 'T' and m < n:  find the least squares solution of
     an overdetermined system, i.e., solve the least squares problem
                  minimize || B - A**T * X ||.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N': the linear system involves A;
          = 'T': the linear system involves A**T.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of the matrices B and X. NRHS >=0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
            if M >= N, A is overwritten by details of its QR
                       factorization as returned by DGEQRF;
            if M <  N, A is overwritten by details of its LQ
                       factorization as returned by DGELQF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the matrix B of right hand side vectors, stored
          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
          if TRANS = 'T'.
          On exit, if INFO = 0, B is overwritten by the solution
          vectors, stored columnwise:
          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
          squares solution vectors; the residual sum of squares for the
          solution in each column is given by the sum of squares of
          elements N+1 to M in that column;
          if TRANS = 'N' and m < n, rows 1 to N of B contain the
          minimum norm solution vectors;
          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
          minimum norm solution vectors;
          if TRANS = 'T' and m < n, rows 1 to M of B contain the
          least squares solution vectors; the residual sum of squares
          for the solution in each column is given by the sum of
          squares of elements M+1 to N in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= MAX(1,M,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= max( 1, MN + max( MN, NRHS ) ).
          For optimal performance,
          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
          where MN = min(M,N) and NB is the optimum block size.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO =  i, the i-th diagonal element of the
                triangular factor of A is zero, so that A does not have
                full rank; the least squares solution could not be
                computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        work -
        lwork -
        info -

      • dgelsd

        public abstract void dgelsd(int m,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] s,
                            double rcond,
                            org.netlib.util.intW rank,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELSD computes the minimum-norm solution to a real linear least
  squares problem:
      minimize 2-norm(| b - A*x |)
  using the singular value decomposition (SVD) of A. A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The problem is solved in three steps:
  (1) Reduce the coefficient matrix A to bidiagonal form with
      Householder transformations, reducing the original problem
      into a "bidiagonal least squares problem" (BLS)
  (2) Solve the BLS using a divide and conquer approach.
  (3) Apply back all the Householder tranformations to solve
      the original least squares problem.

  The effective rank of A is determined by treating as zero those
  singular values which are less than RCOND times the largest singular

  value.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of A. M >= 0.

  N       (input) INTEGER
          The number of columns of A. N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X. NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, B is overwritten by the N-by-NRHS solution
          matrix X.  If m >= n and RANK = n, the residual
          sum-of-squares for the solution in the i-th column is given
          by the sum of squares of elements n+1:m in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,max(M,N)).

  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A in decreasing order.
          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

  RCOND   (input) DOUBLE PRECISION
          RCOND is used to determine the effective rank of A.
          Singular values S(i) <= RCOND*S(1) are treated as zero.
          If RCOND < 0, machine precision is used instead.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the number of singular values

          which are greater than RCOND*S(1).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK must be at least 1.
          The exact minimum amount of workspace needed depends on M,
          N and NRHS. As long as LWORK is at least
              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
          if M is greater than or equal to N or
              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
          if M is less than N, the code will execute correctly.
          SMLSIZ is returned by ILAENV and is equal to the maximum
          size of the subproblems at the bottom of the computation
          tree (usually about 25), and
             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          LIWORK >= 3 * MINMN * NLVL + 11 * MINMN,
          where MINMN = MIN( M,N ).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the algorithm for computing the SVD failed to converge;
                if INFO = i, i off-diagonal elements of an intermediate
                bidiagonal form did not converge to zero.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        s -
        rcond -
        rank -
        work -
        lwork -
        iwork -
        info -

      • dgelss

        public abstract void dgelss(int m,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] s,
                            double rcond,
                            org.netlib.util.intW rank,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELSS computes the minimum norm solution to a real linear least
  squares problem:

  Minimize 2-norm(| b - A*x |).

  using the singular value decomposition (SVD) of A. A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix

  X.

  The effective rank of A is determined by treating as zero those
  singular values which are less than RCOND times the largest singular

  value.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X. NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the first min(m,n) rows of A are overwritten with
          its right singular vectors, stored rowwise.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, B is overwritten by the N-by-NRHS solution
          matrix X.  If m >= n and RANK = n, the residual
          sum-of-squares for the solution in the i-th column is given
          by the sum of squares of elements n+1:m in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,max(M,N)).

  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A in decreasing order.
          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

  RCOND   (input) DOUBLE PRECISION
          RCOND is used to determine the effective rank of A.
          Singular values S(i) <= RCOND*S(1) are treated as zero.
          If RCOND < 0, machine precision is used instead.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the number of singular values

          which are greater than RCOND*S(1).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 1, and also:
          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the algorithm for computing the SVD failed to converge;
                if INFO = i, i off-diagonal elements of an intermediate
                bidiagonal form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        s -
        rcond -
        rank -
        work -
        lwork -
        info -

      • dgelsx

        public abstract void dgelsx(int m,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            int[] jpvt,
                            double rcond,
                            org.netlib.util.intW rank,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DGELSY.

  DGELSX computes the minimum-norm solution to a real linear least
  squares problem:
      minimize || A * X - B ||
  using a complete orthogonal factorization of A.  A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The routine first computes a QR factorization with column pivoting:
      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]
  with R11 defined as the largest leading submatrix whose estimated
  condition number is less than 1/RCOND.  The order of R11, RANK,
  is the effective rank of A.

  Then, R22 is considered to be negligible, and R12 is annihilated
  by orthogonal transformations from the right, arriving at the
  complete orthogonal factorization:
     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]
  The minimum-norm solution is then
     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]
  where Q1 consists of the first RANK columns of Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of matrices B and X. NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been overwritten by details of its
          complete orthogonal factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, the N-by-NRHS solution matrix X.
          If m >= n and RANK = n, the residual sum-of-squares for
          the solution in the i-th column is given by the sum of
          squares of elements N+1:M in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,M,N).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
          initial column, otherwise it is a free column.  Before
          the QR factorization of A, all initial columns are
          permuted to the leading positions; only the remaining
          free columns are moved as a result of column pivoting
          during the factorization.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  RCOND   (input) DOUBLE PRECISION
          RCOND is used to determine the effective rank of A, which
          is defined as the order of the largest leading triangular
          submatrix R11 in the QR factorization with pivoting of A,
          whose estimated condition number < 1/RCOND.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the order of the submatrix
          R11.  This is the same as the order of the submatrix T11
          in the complete orthogonal factorization of A.

  WORK    (workspace) DOUBLE PRECISION array, dimension
                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        jpvt -
        rcond -
        rank -
        work -
        info -

      • dgelsy

        public abstract void dgelsy(int m,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            int[] jpvt,
                            double rcond,
                            org.netlib.util.intW rank,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGELSY computes the minimum-norm solution to a real linear least
  squares problem:
      minimize || A * X - B ||
  using a complete orthogonal factorization of A.  A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The routine first computes a QR factorization with column pivoting:
      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]
  with R11 defined as the largest leading submatrix whose estimated
  condition number is less than 1/RCOND.  The order of R11, RANK,
  is the effective rank of A.

  Then, R22 is considered to be negligible, and R12 is annihilated
  by orthogonal transformations from the right, arriving at the
  complete orthogonal factorization:
     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]
  The minimum-norm solution is then
     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]
  where Q1 consists of the first RANK columns of Q.

  This routine is basically identical to the original xGELSX except
  three differences:
    o The call to the subroutine xGEQPF has been substituted by the
      the call to the subroutine xGEQP3. This subroutine is a Blas-3
      version of the QR factorization with column pivoting.
    o Matrix B (the right hand side) is updated with Blas-3.
    o The permutation of matrix B (the right hand side) is faster and
      more simple.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of matrices B and X. NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been overwritten by details of its
          complete orthogonal factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,M,N).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of AP, otherwise column i is a free column.
          On exit, if JPVT(i) = k, then the i-th column of AP
          was the k-th column of A.

  RCOND   (input) DOUBLE PRECISION
          RCOND is used to determine the effective rank of A, which
          is defined as the order of the largest leading triangular
          submatrix R11 in the QR factorization with pivoting of A,
          whose estimated condition number < 1/RCOND.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the order of the submatrix
          R11.  This is the same as the order of the submatrix T11
          in the complete orthogonal factorization of A.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          The unblocked strategy requires that:
             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
          where MN = min( M, N ).
          The block algorithm requires that:
             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
          where NB is an upper bound on the blocksize returned
          by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
          and DORMRZ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: If INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        jpvt -
        rcond -
        rank -
        work -
        lwork -
        info -

      • dgeql2

        public abstract void dgeql2(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEQL2 computes a QL factorization of a real m by n matrix A:
  A = Q * L.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, if m >= n, the lower triangle of the subarray
          A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
          if m <= n, the elements on and below the (n-m)-th
          superdiagonal contain the m by n lower trapezoidal matrix L;

          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of elementary reflectors
          (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

  A(1:m-k+i-1,n-k+i), and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • dgeqlf

        public abstract void dgeqlf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEQLF computes a QL factorization of a real M-by-N matrix A:
  A = Q * L.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if m >= n, the lower triangle of the subarray
          A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
          if m <= n, the elements on and below the (n-m)-th
          superdiagonal contain the M-by-N lower trapezoidal matrix L;

          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of elementary reflectors
          (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

  A(1:m-k+i-1,n-k+i), and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dgeqp3

        public abstract void dgeqp3(int m,
                            int n,
                            double[] a,
                            int lda,
                            int[] jpvt,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEQP3 computes a QR factorization with column pivoting of a
  matrix A:  A*P = Q*R  using Level 3 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of the array contains the
          min(M,N)-by-N upper trapezoidal matrix R; the elements below

          the diagonal, together with the array TAU, represent the
          orthogonal matrix Q as a product of min(M,N) elementary
          reflectors.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(J).ne.0, the J-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(J)=0,
          the J-th column of A is a free column.
          On exit, if JPVT(J)=K, then the J-th column of A*P was the
          the K-th column of A.

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO=0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 3*N+1.
          For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real/complex scalar, and v is a real/complex vector
  with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
  A(i+1:m,i), and tau in TAU(i).

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        jpvt -
        tau -
        work -
        lwork -
        info -

      • dgeqpf

        public abstract void dgeqpf(int m,
                            int n,
                            double[] a,
                            int lda,
                            int[] jpvt,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DGEQP3.

  DGEQPF computes a QR factorization with column pivoting of a
  real M-by-N matrix A: A*P = Q*R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of the array contains the
          min(M,N)-by-N upper triangular matrix R; the elements
          below the diagonal, together with the array TAU,
          represent the orthogonal matrix Q as a product of
          min(m,n) elementary reflectors.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(n)

  Each H(i) has the form

     H = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).


  The matrix P is represented in jpvt as follows: If
     jpvt(j) = i
  then the jth column of P is the ith canonical unit vector.

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        jpvt -
        tau -
        work -
        info -

      • dgeqr2

        public abstract void dgeqr2(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEQR2 computes a QR factorization of a real m by n matrix A:
  A = Q * R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(m,n) by n upper trapezoidal matrix R (R is
          upper triangular if m >= n); the elements below the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

  and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • dgeqrf

        public abstract void dgeqrf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGEQRF computes a QR factorization of a real M-by-N matrix A:
  A = Q * R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
          upper triangular if m >= n); the elements below the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of min(m,n) elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

  and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dgerfs

        public abstract void dgerfs(java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGERFS improves the computed solution to a system of linear
  equations and provides error bounds and backward error estimates for

  the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The original N-by-N matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) DOUBLE PRECISION array, dimension (LDAF,N)
          The factors L and U from the factorization A = P*L*U
          as computed by DGETRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from DGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DGETRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgerq2

        public abstract void dgerq2(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGERQ2 computes an RQ factorization of a real m by n matrix A:
  A = R * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the m by n upper trapezoidal matrix R; the remaining

          elements, with the array TAU, represent the orthogonal matrix
          Q as a product of elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) DOUBLE PRECISION array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • dgerqf

        public abstract void dgerqf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGERQF computes an RQ factorization of a real M-by-N matrix A:
  A = R * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the M-by-N upper trapezoidal matrix R;
          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of min(m,n) elementary
          reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dgesc2

        public abstract void dgesc2(int n,
                            double[] a,
                            int lda,
                            double[] rhs,
                            int[] ipiv,
                            int[] jpiv,
                            org.netlib.util.doubleW scale)
        
     ..

  Purpose
  =======

  DGESC2 solves a system of linear equations

            A * X = scale* RHS

  with a general N-by-N matrix A using the LU factorization with
  complete pivoting computed by DGETC2.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the  LU part of the factorization of the n-by-n
          matrix A computed by DGETC2:  A = P * L * U * Q

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1, N).

  RHS     (input/output) DOUBLE PRECISION array, dimension (N).
          On entry, the right hand side vector b.
          On exit, the solution vector X.

  IPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  SCALE    (output) DOUBLE PRECISION
           On exit, SCALE contains the scale factor. SCALE is chosen
           0 <= SCALE <= 1 to prevent owerflow in the solution.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        rhs -
        ipiv -
        jpiv -
        scale -

      • dgesdd

        public abstract void dgesdd(java.lang.String jobz,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] s,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGESDD computes the singular value decomposition (SVD) of a real
  M-by-N matrix A, optionally computing the left and right singular
  vectors.  If singular vectors are desired, it uses a
  divide-and-conquer algorithm.

  The SVD is written

       A = U * SIGMA * transpose(V)

  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
  are the singular values of A; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.

  Note that the routine returns VT = V**T, not V.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          Specifies options for computing all or part of the matrix U:

          = 'A':  all M columns of U and all N rows of V**T are
                  returned in the arrays U and VT;
          = 'S':  the first min(M,N) columns of U and the first
                  min(M,N) rows of V**T are returned in the arrays U
                  and VT;
          = 'O':  If M >= N, the first N columns of U are overwritten
                  on the array A and all rows of V**T are returned in
                  the array VT;
                  otherwise, all columns of U are returned in the
                  array U and the first M rows of V**T are overwritten

                  in the array A;
          = 'N':  no columns of U or rows of V**T are computed.

  M       (input) INTEGER
          The number of rows of the input matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the input matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBZ = 'O',  A is overwritten with the first N columns
                          of U (the left singular vectors, stored
                          columnwise) if M >= N;
                          A is overwritten with the first M rows
                          of V**T (the right singular vectors, stored
                          rowwise) otherwise.
          if JOBZ .ne. 'O', the contents of A are destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

  U       (output) DOUBLE PRECISION array, dimension (LDU,UCOL)
          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
          UCOL = min(M,N) if JOBZ = 'S'.
          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M

          orthogonal matrix U;
          if JOBZ = 'S', U contains the first min(M,N) columns of U
          (the left singular vectors, stored columnwise);
          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
          N-by-N orthogonal matrix V**T;
          if JOBZ = 'S', VT contains the first min(M,N) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
          if JOBZ = 'S', LDVT >= min(M,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 1.
          If JOBZ = 'N',
            LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)).
          If JOBZ = 'O',
            LWORK >= 3*min(M,N)*min(M,N) + 
                     max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
          If JOBZ = 'S' or 'A'
            LWORK >= 3*min(M,N)*min(M,N) +
                     max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)).
          For good performance, LWORK should generally be larger.
          If LWORK = -1 but other input arguments are legal, WORK(1)
          returns the optimal LWORK.

  IWORK   (workspace) INTEGER array, dimension (8*min(M,N))

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  DBDSDC did not converge, updating process failed.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        m -
        n -
        a -
        lda -
        s -
        u -
        ldu -
        vt -
        ldvt -
        work -
        lwork -
        iwork -
        info -

      • dgesv

        public abstract void dgesv(int n,
                           int nrhs,
                           double[] a,
                           int lda,
                           int[] ipiv,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGESV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

  The LU decomposition with partial pivoting and row interchanges is
  used to factor A as
     A = P * L * U,
  where P is a permutation matrix, L is unit lower triangular, and U is
  upper triangular.  The factored form of A is then used to solve the
  system of equations A * X = B.

  Arguments
  =========

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N coefficient matrix A.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS matrix of right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                has been completed, but the factor U is exactly
                singular, so the solution could not be computed.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • dgesvd

        public abstract void dgesvd(java.lang.String jobu,
                            java.lang.String jobvt,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] s,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGESVD computes the singular value decomposition (SVD) of a real
  M-by-N matrix A, optionally computing the left and/or right singular

  vectors. The SVD is written

       A = U * SIGMA * transpose(V)

  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
  are the singular values of A; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.

  Note that the routine returns V**T, not V.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          Specifies options for computing all or part of the matrix U:

          = 'A':  all M columns of U are returned in array U:
          = 'S':  the first min(m,n) columns of U (the left singular
                  vectors) are returned in the array U;
          = 'O':  the first min(m,n) columns of U (the left singular
                  vectors) are overwritten on the array A;
          = 'N':  no columns of U (no left singular vectors) are
                  computed.

  JOBVT   (input) CHARACTER*1
          Specifies options for computing all or part of the matrix
          V**T:
          = 'A':  all N rows of V**T are returned in the array VT;
          = 'S':  the first min(m,n) rows of V**T (the right singular
                  vectors) are returned in the array VT;
          = 'O':  the first min(m,n) rows of V**T (the right singular
                  vectors) are overwritten on the array A;
          = 'N':  no rows of V**T (no right singular vectors) are
                  computed.

          JOBVT and JOBU cannot both be 'O'.

  M       (input) INTEGER
          The number of rows of the input matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the input matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBU = 'O',  A is overwritten with the first min(m,n)
                          columns of U (the left singular vectors,
                          stored columnwise);
          if JOBVT = 'O', A is overwritten with the first min(m,n)
                          rows of V**T (the right singular vectors,
                          stored rowwise);
          if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
                          are destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

  U       (output) DOUBLE PRECISION array, dimension (LDU,UCOL)
          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
          If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
          if JOBU = 'S', U contains the first min(m,n) columns of U
          (the left singular vectors, stored columnwise);
          if JOBU = 'N' or 'O', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBU = 'S' or 'A', LDU >= M.

  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
          If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
          V**T;
          if JOBVT = 'S', VT contains the first min(m,n) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBVT = 'N' or 'O', VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
          if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
          superdiagonal elements of an upper bidiagonal matrix B
          whose diagonal is in S (not necessarily sorted). B
          satisfies A = U * B * VT, so it has the same singular values

          as A, and singular vectors related by U and VT.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)).
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if DBDSQR did not converge, INFO specifies how many
                superdiagonals of an intermediate bidiagonal form B
                did not converge to zero. See the description of WORK
                above for details.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobvt -
        m -
        n -
        a -
        lda -
        s -
        u -
        ldu -
        vt -
        ldvt -
        work -
        lwork -
        info -

      • dgesvx

        public abstract void dgesvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            int[] ipiv,
                            org.netlib.util.StringW equed,
                            double[] r,
                            double[] c,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGESVX uses the LU factorization to compute the solution to a real
  system of linear equations
     A * X = B,
  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
     or diag(C)*B (if TRANS = 'T' or 'C').

  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
     matrix A (after equilibration if FACT = 'E') as
        A = P * L * U,
     where P is a permutation matrix, L is a unit lower triangular
     matrix, and U is upper triangular.

  3. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
     that it solves the original system before equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AF and IPIV contain the factored form of A.
                  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  A, AF, and IPIV are not modified.
          = 'N':  The matrix A will be copied to AF and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AF and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
          not 'N', then A must have been equilibrated by the scaling
          factors in R and/or C.  A is not modified if FACT = 'F' or
          'N', or if FACT = 'E' and EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)

          If FACT = 'F', then AF is an input argument and on entry
          contains the factors L and U from the factorization
          A = P*L*U as computed by DGETRF.  If EQUED .ne. 'N', then
          AF is the factored form of the equilibrated matrix A.

          If FACT = 'N', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U

          of the original matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U

          of the equilibrated matrix A (see the description of A for
          the form of the equilibrated matrix).

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = P*L*U
          as computed by DGETRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the equilibrated matrix A.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  R       (input or output) DOUBLE PRECISION array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.

  C       (input or output) DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is

          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C must be positive.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit,
          if EQUED = 'N', B is not modified;
          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
          to the original system of equations.  Note that A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (4*N)
          On exit, WORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If WORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization has
                       been completed, but the factor U is exactly
                       singular, so the solution and error bounds
                       could not be computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        equed -
        r -
        c -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgetc2

        public abstract void dgetc2(int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            int[] jpiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGETC2 computes an LU factorization with complete pivoting of the
  n-by-n matrix A. The factorization has the form A = P * L * U * Q,
  where P and Q are permutation matrices, L is lower triangular with
  unit diagonal elements and U is upper triangular.

  This is the Level 2 BLAS algorithm.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the n-by-n matrix A to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U*Q; the unit diagonal elements of L are not stored.

          If U(k, k) appears to be less than SMIN, U(k, k) is given the
          value of SMIN, i.e., giving a nonsingular perturbed system.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension(N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (output) INTEGER array, dimension(N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  INFO    (output) INTEGER
           = 0: successful exit
           > 0: if INFO = k, U(k, k) is likely to produce owerflow if
                we try to solve for x in Ax = b. So U is perturbed to
                avoid the overflow.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        ipiv -
        jpiv -
        info -

      • dgetf2

        public abstract void dgetf2(int m,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGETF2 computes an LU factorization of a general m-by-n matrix A
  using partial pivoting with row interchanges.

  The factorization has the form
     A = P * L * U
  where P is a permutation matrix, L is lower triangular with unit
  diagonal elements (lower trapezoidal if m > n), and U is upper
  triangular (upper trapezoidal if m < n).

  This is the right-looking Level 2 BLAS version of the algorithm.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n matrix to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, U(k,k) is exactly zero. The factorization
               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        ipiv -
        info -

      • dgetrf

        public abstract void dgetrf(int m,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGETRF computes an LU factorization of a general M-by-N matrix A
  using partial pivoting with row interchanges.

  The factorization has the form
     A = P * L * U
  where P is a permutation matrix, L is lower triangular with unit
  diagonal elements (lower trapezoidal if m > n), and U is upper
  triangular (upper trapezoidal if m < n).

  This is the right-looking Level 3 BLAS version of the algorithm.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization

                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        ipiv -
        info -

      • dgetri

        public abstract void dgetri(int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGETRI computes the inverse of a matrix using the LU factorization
  computed by DGETRF.

  This method inverts U and then computes inv(A) by solving the system

  inv(A)*L = inv(U) for inv(A).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the factors L and U from the factorization
          A = P*L*U as computed by DGETRF.
          On exit, if INFO = 0, the inverse of the original matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from DGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimal performance LWORK >= N*NB, where NB is
          the optimal blocksize returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
                singular and its inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        ipiv -
        work -
        lwork -
        info -

      • dgetrs

        public abstract void dgetrs(java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGETRS solves a system of linear equations
     A * X = B  or  A' * X = B
  with a general N-by-N matrix A using the LU factorization computed
  by DGETRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The factors L and U from the factorization A = P*L*U
          as computed by DGETRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from DGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • dggbak

        public abstract void dggbak(java.lang.String job,
                            java.lang.String side,
                            int n,
                            int ilo,
                            int ihi,
                            double[] lscale,
                            double[] rscale,
                            int m,
                            double[] v,
                            int ldv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGBAK forms the right or left eigenvectors of a real generalized
  eigenvalue problem A*x = lambda*B*x, by backward transformation on
  the computed eigenvectors of the balanced pair of matrices output by

  DGGBAL.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the type of backward transformation required:
          = 'N':  do nothing, return immediately;
          = 'P':  do backward transformation for permutation only;
          = 'S':  do backward transformation for scaling only;
          = 'B':  do backward transformations for both permutation and

                  scaling.
          JOB must be the same as the argument JOB supplied to DGGBAL.


  SIDE    (input) CHARACTER*1
          = 'R':  V contains right eigenvectors;
          = 'L':  V contains left eigenvectors.

  N       (input) INTEGER
          The number of rows of the matrix V.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          The integers ILO and IHI determined by DGGBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  LSCALE  (input) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the left side of A and B, as returned by DGGBAL.

  RSCALE  (input) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the right side of A and B, as returned by DGGBAL.

  M       (input) INTEGER
          The number of columns of the matrix V.  M >= 0.

  V       (input/output) DOUBLE PRECISION array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by DTGEVC.
          On exit, V is overwritten by the transformed eigenvectors.

  LDV     (input) INTEGER
          The leading dimension of the matrix V. LDV >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  See R.C. Ward, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        job -
        side -
        n -
        ilo -
        ihi -
        lscale -
        rscale -
        m -
        v -
        ldv -
        info -

      • dggbal

        public abstract void dggbal(java.lang.String job,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            double[] lscale,
                            double[] rscale,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGBAL balances a pair of general real matrices (A,B).  This
  involves, first, permuting A and B by similarity transformations to
  isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
  elements on the diagonal; and second, applying a diagonal similarity

  transformation to rows and columns ILO to IHI to make the rows
  and columns as close in norm as possible. Both steps are optional.

  Balancing may reduce the 1-norm of the matrices, and improve the
  accuracy of the computed eigenvalues and/or eigenvectors in the
  generalized eigenvalue problem A*x = lambda*B*x.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
                  and RSCALE(I) = 1.0 for i = 1,...,N.
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the input matrix A.
          On exit,  A is overwritten by the balanced matrix.
          If JOB = 'N', A is not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the input matrix B.
          On exit,  B is overwritten by the balanced matrix.
          If JOB = 'N', B is not referenced.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are set to integers such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If JOB = 'N' or 'S', ILO = 1 and IHI = N.

  LSCALE  (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If P(j) is the index of the
          row interchanged with row j, and D(j)
          is the scaling factor applied to row j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  RSCALE  (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If P(j) is the index of the
          column interchanged with column j, and D(j)
          is the scaling factor applied to column j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  WORK    (workspace) REAL array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and

          at least 1 when JOB = 'N' or 'P'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  See R.C. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        lda -
        b -
        ldb -
        ilo -
        ihi -
        lscale -
        rscale -
        work -
        info -

      • dgges

        public void dgges(java.lang.String jobvsl,
                  java.lang.String jobvsr,
                  java.lang.String sort,
                  java.lang.Object selctg,
                  int n,
                  double[] a,
                  int lda,
                  double[] b,
                  int ldb,
                  org.netlib.util.intW sdim,
                  double[] alphar,
                  double[] alphai,
                  double[] beta,
                  double[] vsl,
                  int ldvsl,
                  double[] vsr,
                  int ldvsr,
                  double[] work,
                  int lwork,
                  boolean[] bwork,
                  org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
  the generalized eigenvalues, the generalized real Schur form (S,T),
  optionally, the left and/or right matrices of Schur vectors (VSL and

  VSR). This gives the generalized Schur factorization

           (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

  Optionally, it also orders the eigenvalues so that a selected cluster
  of eigenvalues appears in the leading diagonal blocks of the upper
  quasi-triangular matrix S and the upper triangular matrix T.The
  leading columns of VSL and VSR then form an orthonormal basis for the
  corresponding left and right eigenspaces (deflating subspaces).

  (If only the generalized eigenvalues are needed, use the driver
  DGGEV instead, which is faster.)

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
  or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
  usually represented as the pair (alpha,beta), as there is a
  reasonable interpretation for beta=0 or both being zero.

  A pair of matrices (S,T) is in generalized real Schur form if T is
  upper triangular with non-negative diagonal and S is block upper
  triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
  to real generalized eigenvalues, while 2-by-2 blocks of S will be
  "standardized" by making the corresponding elements of T have the
  form:
          [  a  0  ]
          [  0  b  ]

  and the pair of corresponding 2-by-2 blocks in S and T will have a
  complex conjugate pair of generalized eigenvalues.


  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);

  SELCTG  (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISI
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),

          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

  VSL     (output) DOUBLE PRECISION array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) DOUBLE PRECISION array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N = 0, LWORK >= 1, else LWORK >= 8*N+16.
          For good performance , LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in DTGSEN.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        sort -
        selctg -
        n -
        a -
        lda -
        b -
        ldb -
        sdim -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        work -
        lwork -
        bwork -
        info -

      • dggesx

        public void dggesx(java.lang.String jobvsl,
                   java.lang.String jobvsr,
                   java.lang.String sort,
                   java.lang.Object selctg,
                   java.lang.String sense,
                   int n,
                   double[] a,
                   int lda,
                   double[] b,
                   int ldb,
                   org.netlib.util.intW sdim,
                   double[] alphar,
                   double[] alphai,
                   double[] beta,
                   double[] vsl,
                   int ldvsl,
                   double[] vsr,
                   int ldvsr,
                   double[] rconde,
                   double[] rcondv,
                   double[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   boolean[] bwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGESX computes for a pair of N-by-N real nonsymmetric matrices
  (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
  optionally, the left and/or right matrices of Schur vectors (VSL and

  VSR).  This gives the generalized Schur factorization

       (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

  Optionally, it also orders the eigenvalues so that a selected cluster
  of eigenvalues appears in the leading diagonal blocks of the upper
  quasi-triangular matrix S and the upper triangular matrix T; computes
  a reciprocal condition number for the average of the selected
  eigenvalues (RCONDE); and computes a reciprocal condition number for

  the right and left deflating subspaces corresponding to the selected

  eigenvalues (RCONDV). The leading columns of VSL and VSR then form
  an orthonormal basis for the corresponding left and right eigenspaces
  (deflating subspaces).

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
  or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
  usually represented as the pair (alpha,beta), as there is a
  reasonable interpretation for beta=0 or for both being zero.

  A pair of matrices (S,T) is in generalized real Schur form if T is
  upper triangular with non-negative diagonal and S is block upper
  triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
  to real generalized eigenvalues, while 2-by-2 blocks of S will be
  "standardized" by making the corresponding elements of T have the
  form:
          [  a  0  ]
          [  0  b  ]

  and the pair of corresponding 2-by-2 blocks in S and T will have a
  complex conjugate pair of generalized eigenvalues.


  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG).

  SELCTG  (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISI
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer satisfy
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,

          since ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned), in this
          case INFO is set to N+3.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N' : None are computed;
          = 'E' : Computed for average of selected eigenvalues only;
          = 'V' : Computed for selected deflating subspaces only;
          = 'B' : Computed for both.
          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

  VSL     (output) DOUBLE PRECISION array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) DOUBLE PRECISION array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  RCONDE  (output) DOUBLE PRECISION array, dimension ( 2 )
          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
          reciprocal condition numbers for the average of the selected

          eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

  RCONDV  (output) DOUBLE PRECISION array, dimension ( 2 )
          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
          reciprocal condition numbers for the selected deflating
          subspaces.
          Not referenced if SENSE = 'N' or 'E'.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
          LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
          LWORK >= max( 8*N, 6*N+16 ).
          Note that 2*SDIM*(N-SDIM) <= N*N/2.
          Note also that an error is only returned if
          LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
          this may not be large enough.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the bound on the optimal size of the WORK
          array and the minimum size of the IWORK array, returns these

          values as the first entries of the WORK and IWORK arrays, and
          no error message related to LWORK or LIWORK is issued by
          XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
          LIWORK >= N+6.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the bound on the optimal size of the

          WORK array and the minimum size of the IWORK array, returns
          these values as the first entries of the WORK and IWORK
          arrays, and no error message related to LWORK or LIWORK is
          issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in DHGEQZ
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in DTGSEN.

  Further details
  ===============

  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / RCONDE( 1 ).

  An approximate (asymptotic) bound on the maximum angular error in
  the computed deflating subspaces is

       EPS * norm((A, B)) / RCONDV( 2 ).

  See LAPACK User's Guide, section 4.11 for more information.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        sort -
        selctg -
        sense -
        n -
        a -
        lda -
        b -
        ldb -
        sdim -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        liwork -
        bwork -
        info -

      • dggev

        public abstract void dggev(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double[] alphar,
                           double[] alphai,
                           double[] beta,
                           double[] vl,
                           int ldvl,
                           double[] vr,
                           int ldvr,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)

  the generalized eigenvalues, and optionally, the left and/or right
  generalized eigenvectors.

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
  singular. It is usually represented as the pair (alpha,beta), as
  there is a reasonable interpretation for beta=0, and even for both
  being zero.

  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)

  of (A,B) satisfies

                   A * v(j) = lambda(j) * B * v(j).

  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
  of (A,B) satisfies

                   u(j)**H * A  = lambda(j) * u(j)**H * B .

  where u(j)**H is the conjugate-transpose of u(j).


  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio
          alpha/beta.  However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).

          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).

          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
                =N+2: error return from DTGEVC.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • dggevx

        public abstract void dggevx(java.lang.String balanc,
                            java.lang.String jobvl,
                            java.lang.String jobvr,
                            java.lang.String sense,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] alphar,
                            double[] alphai,
                            double[] beta,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            double[] lscale,
                            double[] rscale,
                            org.netlib.util.doubleW abnrm,
                            org.netlib.util.doubleW bbnrm,
                            double[] rconde,
                            double[] rcondv,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            boolean[] bwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
  the generalized eigenvalues, and optionally, the left and/or right
  generalized eigenvectors.

  Optionally also, it computes a balancing transformation to improve
  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
  LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
  the eigenvalues (RCONDE), and reciprocal condition numbers for the
  right eigenvectors (RCONDV).

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
  singular. It is usually represented as the pair (alpha,beta), as
  there is a reasonable interpretation for beta=0, and even for both
  being zero.

  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)

  of (A,B) satisfies

                   A * v(j) = lambda(j) * B * v(j) .

  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
  of (A,B) satisfies

                   u(j)**H * A  = lambda(j) * u(j)**H * B.

  where u(j)**H is the conjugate-transpose of u(j).


  Arguments
  =========

  BALANC  (input) CHARACTER*1
          Specifies the balance option to be performed.
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then A contains the first part of the real Schur
          form of the "balanced" versions of the input A and B.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then B contains the second part of the real Schur
          form of the "balanced" versions of the input A and B.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio
          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).

          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).

          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are integer values such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

  LSCALE  (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If PL(j) is the index of the
          row interchanged with row j, and DL(j) is the scaling
          factor applied to row j, then
            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                      = DL(j)  for j = ILO,...,IHI
                      = PL(j)  for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  RSCALE  (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If PR(j) is the index of the
          column interchanged with column j, and DR(j) is the scaling
          factor applied to column j, then
            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                      = DR(j)  for j = ILO,...,IHI
                      = PR(j)  for j = IHI+1,...,N
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  ABNRM   (output) DOUBLE PRECISION
          The one-norm of the balanced matrix A.

  BBNRM   (output) DOUBLE PRECISION
          The one-norm of the balanced matrix B.

  RCONDE  (output) DOUBLE PRECISION array, dimension (N)
          If SENSE = 'E' or 'B', the reciprocal condition numbers of
          the eigenvalues, stored in consecutive elements of the array.
          For a complex conjugate pair of eigenvalues two consecutive
          elements of RCONDE are set to the same value. Thus RCONDE(j),
          RCONDV(j), and the j-th columns of VL and VR all correspond
          to the j-th eigenpair.
          If SENSE = 'N or 'V', RCONDE is not referenced.

  RCONDV  (output) DOUBLE PRECISION array, dimension (N)
          If SENSE = 'V' or 'B', the estimated reciprocal condition
          numbers of the eigenvectors, stored in consecutive elements
          of the array. For a complex eigenvector two consecutive
          elements of RCONDV are set to the same value. If the
          eigenvalues cannot be reordered to compute RCONDV(j),
          RCONDV(j) is set to 0; this can only occur when the true
          value would be very small anyway.
          If SENSE = 'N' or 'E', RCONDV is not referenced.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,2*N).
          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
          LWORK >= max(1,6*N).
          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N+6)
          If SENSE = 'E', IWORK is not referenced.

  BWORK   (workspace) LOGICAL array, dimension (N)
          If SENSE = 'N', BWORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
                =N+2: error return from DTGEVC.

  Further Details
  ===============

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity

  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

  =====================================================================

     .. Parameters ..
        Parameters:
        balanc -
        jobvl -
        jobvr -
        sense -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        ilo -
        ihi -
        lscale -
        rscale -
        abnrm -
        bbnrm -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        bwork -
        info -

      • dggglm

        public abstract void dggglm(int n,
                            int m,
                            int p,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] d,
                            double[] x,
                            double[] y,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGGLM solves a general Gauss-Markov linear model (GLM) problem:

          minimize || y ||_2   subject to   d = A*x + B*y
              x

  where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
  given N-vector. It is assumed that M <= N <= M+P, and

             rank(A) = M    and    rank( A B ) = N.

  Under these assumptions, the constrained equation is always
  consistent, and there is a unique solution x and a minimal 2-norm
  solution y, which is obtained using a generalized QR factorization
  of the matrices (A, B) given by

     A = Q*(R),   B = Q*T*Z.
           (0)

  In particular, if matrix B is square nonsingular, then the problem
  GLM is equivalent to the following weighted linear least squares
  problem

               minimize || inv(B)*(d-A*x) ||_2
                   x

  where inv(B) denotes the inverse of B.

  Arguments
  =========

  N       (input) INTEGER
          The number of rows of the matrices A and B.  N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix A.  0 <= M <= N.

  P       (input) INTEGER
          The number of columns of the matrix B.  P >= N-M.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the upper triangular part of the array A contains
          the M-by-M upper triangular matrix R.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)th subdiagonal
          contain the N-by-P upper trapezoidal matrix T.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, D is the left hand side of the GLM equation.
          On exit, D is destroyed.

  X       (output) DOUBLE PRECISION array, dimension (M)
  Y       (output) DOUBLE PRECISION array, dimension (P)
          On exit, X and Y are the solutions of the GLM problem.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N+M+P).
          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
          where NB is an upper bound for the optimal blocksizes for
          DGEQRF, SGERQF, DORMQR and SORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the upper triangular factor R associated with A in the

                generalized QR factorization of the pair (A, B) is
                singular, so that rank(A) < M; the least squares
                solution could not be computed.
          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                factor T associated with B in the generalized QR
                factorization of the pair (A, B) is singular, so that
                rank( A B ) < N; the least squares solution could not
                be computed.

  ===================================================================

     .. Parameters ..
        Parameters:
        n -
        m -
        p -
        a -
        lda -
        b -
        ldb -
        d -
        x -
        y -
        work -
        lwork -
        info -

      • dgghrd

        public abstract void dgghrd(java.lang.String compq,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] q,
                            int ldq,
                            double[] z,
                            int ldz,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGHRD reduces a pair of real matrices (A,B) to generalized upper
  Hessenberg form using orthogonal transformations, where A is a
  general matrix and B is upper triangular.  The form of the
  generalized eigenvalue problem is
     A*x = lambda*B*x,
  and B is typically made upper triangular by computing its QR
  factorization and moving the orthogonal matrix Q to the left side
  of the equation.

  This subroutine simultaneously reduces A to a Hessenberg matrix H:
     Q**T*A*Z = H
  and transforms B to another upper triangular matrix T:
     Q**T*B*Z = T
  in order to reduce the problem to its standard form
     H*y = lambda*T*y
  where y = Z**T*x.

  The orthogonal matrices Q and Z are determined as products of Givens

  rotations.  They may either be formed explicitly, or they may be
  postmultiplied into input matrices Q1 and Z1, so that

       Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

       Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

  If Q1 is the orthogonal matrix from the QR factorization of B in the

  original equation A*x = lambda*B*x, then DGGHRD reduces the original

  problem to generalized Hessenberg form.

  Arguments
  =========

  COMPQ   (input) CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 orthogonal matrix Q is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry,
                 and the product Q1*Q is returned.

  COMPZ   (input) CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 orthogonal matrix Z is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry,
                 and the product Z1*Z is returned.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to SGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**T B Z.  The
          elements below the diagonal are set to zero.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
          typically from the QR factorization of B.
          On exit, if COMPQ='I', the orthogonal matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
          On exit, if COMPZ='I', the orthogonal matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and Van Loan (Johns Hopkins Press.)

  =====================================================================

     .. Parameters ..
        Parameters:
        compq -
        compz -
        n -
        ilo -
        ihi -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        info -

      • dgglse

        public abstract void dgglse(int m,
                            int n,
                            int p,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] c,
                            double[] d,
                            double[] x,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGLSE solves the linear equality-constrained least squares (LSE)
  problem:

          minimize || c - A*x ||_2   subject to   B*x = d

  where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
  M-vector, and d is a given P-vector. It is assumed that
  P <= N <= M+P, and

           rank(B) = P and  rank( (A) ) = N.
                                ( (B) )

  These conditions ensure that the LSE problem has a unique solution,
  which is obtained using a generalized RQ factorization of the
  matrices (B, A) given by

     B = (0 R)*Q,   A = Z*T*Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B. N >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B. 0 <= P <= N <= M+P.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(M,N)-by-N upper trapezoidal matrix T.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
          contains the P-by-P upper triangular matrix R.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  C       (input/output) DOUBLE PRECISION array, dimension (M)
          On entry, C contains the right hand side vector for the
          least squares part of the LSE problem.
          On exit, the residual sum of squares for the solution
          is given by the sum of squares of elements N-P+1 to M of
          vector C.

  D       (input/output) DOUBLE PRECISION array, dimension (P)
          On entry, D contains the right hand side vector for the
          constrained equation.
          On exit, D is destroyed.

  X       (output) DOUBLE PRECISION array, dimension (N)
          On exit, X is the solution of the LSE problem.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M+N+P).
          For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
          where NB is an upper bound for the optimal blocksizes for
          DGEQRF, SGERQF, DORMQR and SORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the upper triangular factor R associated with B in the

                generalized RQ factorization of the pair (B, A) is
                singular, so that rank(B) < P; the least squares
                solution could not be computed.
          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
                T associated with A in the generalized RQ factorization
                of the pair (B, A) is singular, so that
                rank( (A) ) < N; the least squares solution could not
                    ( (B) )
                be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        p -
        a -
        lda -
        b -
        ldb -
        c -
        d -
        x -
        work -
        lwork -
        info -

      • dggqrf

        public abstract void dggqrf(int n,
                            int m,
                            int p,
                            double[] a,
                            int lda,
                            double[] taua,
                            double[] b,
                            int ldb,
                            double[] taub,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGQRF computes a generalized QR factorization of an N-by-M matrix A

  and an N-by-P matrix B:

              A = Q*R,        B = Q*T*Z,

  where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
  matrix, and R and T assume one of the forms:

  if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                  (  0  ) N-M                         N   M-N
                     M

  where R11 is upper triangular, and

  if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                   P-N  N                           ( T21 ) P
                                                       P

  where T12 or T21 is upper triangular.

  In particular, if B is square and nonsingular, the GQR factorization

  of A and B implicitly gives the QR factorization of inv(B)*A:

               inv(B)*A = Z'*(inv(T)*R)

  where inv(B) denotes the inverse of the matrix B, and Z' denotes the

  transpose of the matrix Z.

  Arguments
  =========

  N       (input) INTEGER
          The number of rows of the matrices A and B. N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of columns of the matrix B.  P >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
          upper triangular if N >= M); the elements below the diagonal,
          with the array TAUA, represent the orthogonal matrix Q as a
          product of min(N,M) elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAUA    (output) DOUBLE PRECISION array, dimension (min(N,M))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)-th subdiagonal

          contain the N-by-P upper trapezoidal matrix T; the remaining

          elements, with the array TAUB, represent the orthogonal
          matrix Z as a product of elementary reflectors (see Further
          Details).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  TAUB    (output) DOUBLE PRECISION array, dimension (min(N,P))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the QR factorization
          of an N-by-M matrix, NB2 is the optimal blocksize for the
          RQ factorization of an N-by-P matrix, and NB3 is the optimal

          blocksize for a call of DORMQR.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(n,m).

  Each H(i) has the form

     H(i) = I - taua * v * v'

  where taua is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),

  and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine DORGQR.
  To use Q to update another matrix, use LAPACK subroutine DORMQR.

  The matrix Z is represented as a product of elementary reflectors

     Z = H(1) H(2) . . . H(k), where k = min(n,p).

  Each H(i) has the form

     H(i) = I - taub * v * v'

  where taub is a real scalar, and v is a real vector with
  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in

  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine DORGRQ.
  To use Z to update another matrix, use LAPACK subroutine DORMRQ.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        m -
        p -
        a -
        lda -
        taua -
        b -
        ldb -
        taub -
        work -
        lwork -
        info -

      • dggrqf

        public abstract void dggrqf(int m,
                            int p,
                            int n,
                            double[] a,
                            int lda,
                            double[] taua,
                            double[] b,
                            int ldb,
                            double[] taub,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGRQF computes a generalized RQ factorization of an M-by-N matrix A

  and a P-by-N matrix B:

              A = R*Q,        B = Z*T*Q,

  where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
  matrix, and R and T assume one of the forms:

  if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                   N-M  M                           ( R21 ) N
                                                       N

  where R12 or R21 is upper triangular, and

  if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                  (  0  ) P-N                         P   N-P
                     N

  where T11 is upper triangular.

  In particular, if B is square and nonsingular, the GRQ factorization

  of A and B implicitly gives the RQ factorization of A*inv(B):

               A*inv(B) = (R*inv(T))*Z'

  where inv(B) denotes the inverse of the matrix B, and Z' denotes the

  transpose of the matrix Z.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, if M <= N, the upper triangle of the subarray
          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
          if M > N, the elements on and above the (M-N)-th subdiagonal

          contain the M-by-N upper trapezoidal matrix R; the remaining

          elements, with the array TAUA, represent the orthogonal
          matrix Q as a product of elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  TAUA    (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the elements on and above the diagonal of the array

          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
          upper triangular if P >= N); the elements below the diagonal,
          with the array TAUB, represent the orthogonal matrix Z as a
          product of elementary reflectors (see Further Details).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TAUB    (output) DOUBLE PRECISION array, dimension (min(P,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the RQ factorization
          of an M-by-N matrix, NB2 is the optimal blocksize for the
          QR factorization of a P-by-N matrix, and NB3 is the optimal
          blocksize for a call of DORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INF0= -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - taua * v * v'

  where taua is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine DORGRQ.
  To use Q to update another matrix, use LAPACK subroutine DORMRQ.

  The matrix Z is represented as a product of elementary reflectors

     Z = H(1) H(2) . . . H(k), where k = min(p,n).

  Each H(i) has the form

     H(i) = I - taub * v * v'

  where taub is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),

  and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine DORGQR.
  To use Z to update another matrix, use LAPACK subroutine DORMQR.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        p -
        n -
        a -
        lda -
        taua -
        b -
        ldb -
        taub -
        work -
        lwork -
        info -

      • dggsvd

        public abstract void dggsvd(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int n,
                            int p,
                            org.netlib.util.intW k,
                            org.netlib.util.intW l,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] alpha,
                            double[] beta,
                            double[] u,
                            int ldu,
                            double[] v,
                            int ldv,
                            double[] q,
                            int ldq,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGSVD computes the generalized singular value decomposition (GSVD)
  of an M-by-N real matrix A and P-by-N real matrix B:

      U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

  where U, V and Q are orthogonal matrices, and Z' is the transpose
  of Z.  Let K+L = the effective numerical rank of the matrix (A',B')',
  then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
  D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
  following structures, respectively:

  If M-K-L >= 0,

                      K  L
         D1 =     K ( I  0 )
                  L ( 0  C )
              M-K-L ( 0  0 )

                    K  L
         D2 =   L ( 0  S )
              P-L ( 0  0 )

                  N-K-L  K    L
    ( 0 R ) = K (  0   R11  R12 )
              L (  0    0   R22 )

  where

    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
    S = diag( BETA(K+1),  ... , BETA(K+L) ),
    C**2 + S**2 = I.

    R is stored in A(1:K+L,N-K-L+1:N) on exit.

  If M-K-L < 0,

                    K M-K K+L-M
         D1 =   K ( I  0    0   )
              M-K ( 0  C    0   )

                      K M-K K+L-M
         D2 =   M-K ( 0  S    0  )
              K+L-M ( 0  0    I  )
                P-L ( 0  0    0  )

                     N-K-L  K   M-K  K+L-M
    ( 0 R ) =     K ( 0    R11  R12  R13  )
                M-K ( 0     0   R22  R23  )
              K+L-M ( 0     0    0   R33  )

  where

    C = diag( ALPHA(K+1), ... , ALPHA(M) ),
    S = diag( BETA(K+1),  ... , BETA(M) ),
    C**2 + S**2 = I.

    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
    ( 0  R22 R23 )
    in B(M-K+1:L,N+M-K-L+1:N) on exit.

  The routine computes C, S, R, and optionally the orthogonal
  transformation matrices U, V and Q.

  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
  A and B implicitly gives the SVD of A*inv(B):
                       A*inv(B) = U*(D1*inv(D2))*V'.
  If ( A',B')' has orthonormal columns, then the GSVD of A and B is
  also equal to the CS decomposition of A and B. Furthermore, the GSVD

  can be used to derive the solution of the eigenvalue problem:
                       A'*A x = lambda* B'*B x.
  In some literature, the GSVD of A and B is presented in the form
                   U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
  where U and V are orthogonal and X is nonsingular, D1 and D2 are
  ``diagonal''.  The former GSVD form can be converted to the latter
  form by taking the nonsingular matrix X as

                       X = Q*( I   0    )
                             ( 0 inv(R) ).

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  K       (output) INTEGER
  L       (output) INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in the Purpose section.
          K + L = effective numerical rank of (A',B')'.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular matrix R, or part of R.
          See Purpose for details.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix R if M-K-L < 0.
          See Purpose for details.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = C,
            BETA(K+1:K+L)  = S,
          or if M-K-L < 0,
            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
            BETA(K+1:M) =S, BETA(M+1:K+L) =1
          and
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0

  U       (output) DOUBLE PRECISION array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (output) DOUBLE PRECISION array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  WORK    (workspace) DOUBLE PRECISION array,
                      dimension (max(3*N,M,P)+N)

  IWORK   (workspace/output) INTEGER array, dimension (N)
          On exit, IWORK stores the sorting information. More
          precisely, the following loop will sort ALPHA
             for I = K+1, min(M,K+L)
                 swap ALPHA(I) and ALPHA(IWORK(I))
             endfor
          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, the Jacobi-type procedure failed to
                converge.  For further details, see subroutine DTGSJA.


  Internal Parameters
  ===================

  TOLA    DOUBLE PRECISION
  TOLB    DOUBLE PRECISION
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A',B')'. Generally, they are set to
                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

  Further Details
  ===============

  2-96 Based on modifications by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        n -
        p -
        k -
        l -
        a -
        lda -
        b -
        ldb -
        alpha -
        beta -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        work -
        iwork -
        info -

      • dggsvp

        public abstract void dggsvp(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int p,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double tola,
                            double tolb,
                            org.netlib.util.intW k,
                            org.netlib.util.intW l,
                            double[] u,
                            int ldu,
                            double[] v,
                            int ldv,
                            double[] q,
                            int ldq,
                            int[] iwork,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGGSVP computes orthogonal matrices U, V and Q such that

                   N-K-L  K    L
   U'*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                L ( 0     0   A23 )
            M-K-L ( 0     0    0  )

                   N-K-L  K    L
          =     K ( 0    A12  A13 )  if M-K-L < 0;
              M-K ( 0     0   A23 )

                 N-K-L  K    L
   V'*B*Q =   L ( 0     0   B13 )
            P-L ( 0     0    0  )

  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
  otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
  numerical rank of the (M+P)-by-N matrix (A',B')'.  Z' denotes the
  transpose of Z.

  This decomposition is the preprocessing step for computing the
  Generalized Singular Value Decomposition (GSVD), see subroutine
  DGGSVD.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular (or trapezoidal) matrix
          described in the Purpose section.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix described in
          the Purpose section.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TOLA    (input) DOUBLE PRECISION
  TOLB    (input) DOUBLE PRECISION
          TOLA and TOLB are the thresholds to determine the effective
          numerical rank of matrix B and a subblock of A. Generally,
          they are set to
             TOLA = MAX(M,N)*norm(A)*MAZHEPS,
             TOLB = MAX(P,N)*norm(B)*MAZHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

  K       (output) INTEGER
  L       (output) INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in Purpose.
          K + L = effective numerical rank of (A',B')'.

  U       (output) DOUBLE PRECISION array, dimension (LDU,M)
          If JOBU = 'U', U contains the orthogonal matrix U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (output) DOUBLE PRECISION array, dimension (LDV,M)
          If JOBV = 'V', V contains the orthogonal matrix V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  IWORK   (workspace) INTEGER array, dimension (N)

  TAU     (workspace) DOUBLE PRECISION array, dimension (N)

  WORK    (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))


  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.


  Further Details
  ===============

  The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
  with column pivoting to detect the effective numerical rank of the
  a matrix. It may be replaced by a better rank determination strategy.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        p -
        n -
        a -
        lda -
        b -
        ldb -
        tola -
        tolb -
        k -
        l -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        iwork -
        tau -
        work -
        info -

      • dgtcon

        public abstract void dgtcon(java.lang.String norm,
                            int n,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] du2,
                            int[] ipiv,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTCON estimates the reciprocal of the condition number of a real
  tridiagonal matrix A using the LU factorization as computed by
  DGTTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by DGTTRF.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

  DU2     (input) DOUBLE PRECISION array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  ANORM   (input) DOUBLE PRECISION
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        dl -
        d -
        du -
        du2 -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dgtrfs

        public abstract void dgtrfs(java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] dlf,
                            double[] df,
                            double[] duf,
                            double[] du2,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is tridiagonal, and provides
  error bounds and backward error estimates for the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

  DLF     (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by DGTTRF.

  DF      (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DUF     (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

  DU2     (input) DOUBLE PRECISION array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DGTTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        dlf -
        df -
        duf -
        du2 -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgtsv

        public abstract void dgtsv(int n,
                           int nrhs,
                           double[] dl,
                           double[] d,
                           double[] du,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTSV  solves the equation

     A*X = B,

  where A is an n by n tridiagonal matrix, by Gaussian elimination with
  partial pivoting.

  Note that the equation  A'*X = B  may be solved by interchanging the

  order of the arguments DU and DL.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of

          A.

          On exit, DL is overwritten by the (n-2) elements of the
          second super-diagonal of the upper triangular matrix U from
          the LU factorization of A, in DL(1), ..., DL(n-2).

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, D must contain the diagonal elements of A.

          On exit, D is overwritten by the n diagonal elements of U.

  DU      (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.

          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix of right hand side matrix B.
          On exit, if INFO = 0, the N by NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, U(i,i) is exactly zero, and the solution
               has not been computed.  The factorization has not been
               completed unless i = N.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        nrhs -
        dl -
        d -
        du -
        b -
        ldb -
        info -

      • dgtsvx

        public abstract void dgtsvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] dlf,
                            double[] df,
                            double[] duf,
                            double[] du2,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTSVX uses the LU factorization to compute the solution to a real
  system of linear equations A * X = B or A**T * X = B,
  where A is a tridiagonal matrix of order N and X and B are N-by-NRHS

  matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the LU decomposition is used to factor the matrix A
     as A = L * U, where L is a product of permutation and unit lower
     bidiagonal matrices and U is upper triangular with nonzeros in
     only the main diagonal and first two superdiagonals.

  2. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
                  will not be modified.
          = 'N':  The matrix will be copied to DLF, DF, and DUF
                  and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

  DLF     (input or output) DOUBLE PRECISION array, dimension (N-1)
          If FACT = 'F', then DLF is an input argument and on entry
          contains the (n-1) multipliers that define the matrix L from

          the LU factorization of A as computed by DGTTRF.

          If FACT = 'N', then DLF is an output argument and on exit
          contains the (n-1) multipliers that define the matrix L from

          the LU factorization of A.

  DF      (input or output) DOUBLE PRECISION array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.

          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.

  DUF     (input or output) DOUBLE PRECISION array, dimension (N-1)
          If FACT = 'F', then DUF is an input argument and on entry
          contains the (n-1) elements of the first superdiagonal of U.


          If FACT = 'N', then DUF is an output argument and on exit
          contains the (n-1) elements of the first superdiagonal of U.


  DU2     (input or output) DOUBLE PRECISION array, dimension (N-2)
          If FACT = 'F', then DU2 is an input argument and on entry
          contains the (n-2) elements of the second superdiagonal of
          U.

          If FACT = 'N', then DU2 is an output argument and on exit
          contains the (n-2) elements of the second superdiagonal of
          U.

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the LU factorization of A as

          computed by DGTTRF.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the LU factorization of A;
          row i of the matrix was interchanged with row IPIV(i).
          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
          a row interchange was not required.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has not been completed unless i = N, but the
                       factor U is exactly singular, so the solution
                       and error bounds could not be computed.
                       RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        dlf -
        df -
        duf -
        du2 -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dgttrf

        public abstract void dgttrf(int n,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] du2,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTTRF computes an LU factorization of a real tridiagonal matrix A
  using elimination with partial pivoting and row interchanges.

  The factorization has the form
     A = L * U
  where L is a product of permutation and unit lower bidiagonal
  matrices and U is upper triangular with nonzeros in only the main
  diagonal and first two superdiagonals.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  DL      (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of

          A.

          On exit, DL is overwritten by the (n-1) multipliers that
          define the matrix L from the LU factorization of A.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, D must contain the diagonal elements of A.

          On exit, D is overwritten by the n diagonal elements of the
          upper triangular matrix U from the LU factorization of A.

  DU      (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.

          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

  DU2     (output) DOUBLE PRECISION array, dimension (N-2)
          On exit, DU2 is overwritten by the (n-2) elements of the
          second super-diagonal of U.

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
          > 0:  if INFO = k, U(k,k) is exactly zero. The factorization

                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        dl -
        d -
        du -
        du2 -
        ipiv -
        info -

      • dgttrs

        public abstract void dgttrs(java.lang.String trans,
                            int n,
                            int nrhs,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] du2,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DGTTRS solves one of the systems of equations
     A*X = B  or  A'*X = B,
  with a tridiagonal matrix A using the LU factorization computed
  by DGTTRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

  DU2     (input) DOUBLE PRECISION array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        du2 -
        ipiv -
        b -
        ldb -
        info -

      • dgtts2

        public abstract void dgtts2(int itrans,
                            int n,
                            int nrhs,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] du2,
                            int[] ipiv,
                            double[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  DGTTS2 solves one of the systems of equations
     A*X = B  or  A'*X = B,
  with a tridiagonal matrix A using the LU factorization computed
  by DGTTRF.

  Arguments
  =========

  ITRANS  (input) INTEGER
          Specifies the form of the system of equations.
          = 0:  A * X = B  (No transpose)
          = 1:  A'* X = B  (Transpose)
          = 2:  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

  DU2     (input) DOUBLE PRECISION array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        itrans -
        n -
        nrhs -
        dl -
        d -
        du -
        du2 -
        ipiv -
        b -
        ldb -

      • dhgeqz

        public abstract void dhgeqz(java.lang.String job,
                            java.lang.String compq,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            double[] h,
                            int ldh,
                            double[] t,
                            int ldt,
                            double[] alphar,
                            double[] alphai,
                            double[] beta,
                            double[] q,
                            int ldq,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
  where H is an upper Hessenberg matrix and T is upper triangular,
  using the double-shift QZ method.
  Matrix pairs of this type are produced by the reduction to
  generalized upper Hessenberg form of a real matrix pair (A,B):

     A = Q1*H*Z1**T,  B = Q1*T*Z1**T,

  as computed by DGGHRD.

  If JOB='S', then the Hessenberg-triangular pair (H,T) is
  also reduced to generalized Schur form,
  
     H = Q*S*Z**T,  T = Q*P*Z**T,
  
  where Q and Z are orthogonal matrices, P is an upper triangular
  matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
  diagonal blocks.

  The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
  (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of

  eigenvalues.

  Additionally, the 2-by-2 upper triangular diagonal blocks of P
  corresponding to 2-by-2 blocks of S are reduced to positive diagonal

  form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
  P(j,j) > 0, and P(j+1,j+1) > 0.

  Optionally, the orthogonal matrix Q from the generalized Schur
  factorization may be postmultiplied into an input matrix Q1, and the

  orthogonal matrix Z may be postmultiplied into an input matrix Z1.
  If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
  the matrix pair (A,B) to generalized upper Hessenberg form, then the

  output matrices Q1*Q and Z1*Z are the orthogonal factors from the
  generalized Schur factorization of (A,B):

     A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
  
  To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently
  of (A,B)) are computed as a pair of values (alpha,beta), where alpha 
  complex and beta real.
  If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
  generalized nonsymmetric eigenvalue problem (GNEP)
     A*x = lambda*B*x
  and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
  alternate form of the GNEP
     mu*A*y = B*y.
  Real eigenvalues can be read directly from the generalized Schur
  form: 
    alpha = S(i,i), beta = P(i,i).

  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix

       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
       pp. 241--256.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form. 

  COMPQ   (input) CHARACTER*1
          = 'N': Left Schur vectors (Q) are not computed;
          = 'I': Q is initialized to the unit matrix and the matrix Q
                 of left Schur vectors of (H,T) is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry and
                 the product Q1*Q is returned.

  COMPZ   (input) CHARACTER*1
          = 'N': Right Schur vectors (Z) are not computed;
          = 'I': Z is initialized to the unit matrix and the matrix Z
                 of right Schur vectors of (H,T) is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry and
                 the product Z1*Z is returned.

  N       (input) INTEGER
          The order of the matrices H, T, Q, and Z.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI mark the rows and columns of H which are in
          Hessenberg form.  It is assumed that A is already upper
          triangular in rows and columns 1:ILO-1 and IHI+1:N.
          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

  H       (input/output) DOUBLE PRECISION array, dimension (LDH, N)
          On entry, the N-by-N upper Hessenberg matrix H.
          On exit, if JOB = 'S', H contains the upper quasi-triangular

          matrix S from the generalized Schur factorization;
          2-by-2 diagonal blocks (corresponding to complex conjugate
          pairs of eigenvalues) are returned in standard form, with
          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
          If JOB = 'E', the diagonal blocks of H match those of S, but

          the rest of H is unspecified.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max( 1, N ).

  T       (input/output) DOUBLE PRECISION array, dimension (LDT, N)
          On entry, the N-by-N upper triangular matrix T.
          On exit, if JOB = 'S', T contains the upper triangular
          matrix P from the generalized Schur factorization;
          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of
          are reduced to positive diagonal form, i.e., if H(j+1,j) is
          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
          T(j+1,j+1) > 0.
          If JOB = 'E', the diagonal blocks of T match those of P, but

          the rest of T is unspecified.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= max( 1, N ).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) DOUBLE PRECISION array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
          the reduction of (A,B) to generalized Hessenberg form.
          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur

          vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
          of left Schur vectors of (A,B).
          Not referenced if COMPZ = 'N'.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= 1.
          If COMPQ='V' or 'I', then LDQ >= N.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
          the reduction of (A,B) to generalized Hessenberg form.
          On exit, if COMPZ = 'I', the orthogonal matrix of
          right Schur vectors of (H,T), and if COMPZ = 'V', the
          orthogonal matrix of right Schur vectors of (A,B).
          Not referenced if COMPZ = 'N'.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If COMPZ='V' or 'I', then LDZ >= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO+1,...,N should be correct.
          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO-N+1,...,N should be correct.

  Further Details
  ===============

  Iteration counters:

  JITER  -- counts iterations.
  IITER  -- counts iterations run since ILAST was last
            changed.  This is therefore reset only when a 1-by-1 or
            2-by-2 block deflates off the bottom.

  =====================================================================

     .. Parameters ..
    $                     SAFETY = 1.0E+0 )
        Parameters:
        job -
        compq -
        compz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        t -
        ldt -
        alphar -
        alphai -
        beta -
        q -
        ldq -
        z -
        ldz -
        work -
        lwork -
        info -

      • dhsein

        public abstract void dhsein(java.lang.String side,
                            java.lang.String eigsrc,
                            java.lang.String initv,
                            boolean[] select,
                            int n,
                            double[] h,
                            int ldh,
                            double[] wr,
                            double[] wi,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            double[] work,
                            int[] ifaill,
                            int[] ifailr,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DHSEIN uses inverse iteration to find specified right and/or left
  eigenvectors of a real upper Hessenberg matrix H.

  The right eigenvector x and the left eigenvector y of the matrix H
  corresponding to an eigenvalue w are defined by:

               H * x = w * x,     y**h * H = w * y**h

  where y**h denotes the conjugate transpose of the vector y.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

  EIGSRC  (input) CHARACTER*1
          Specifies the source of eigenvalues supplied in (WR,WI):
          = 'Q': the eigenvalues were found using DHSEQR; thus, if
                 H has zero subdiagonal elements, and so is
                 block-triangular, then the j-th eigenvalue can be
                 assumed to be an eigenvalue of the block containing
                 the j-th row/column.  This property allows DHSEIN to
                 perform inverse iteration on just one diagonal block.

          = 'N': no assumptions are made on the correspondence
                 between eigenvalues and diagonal blocks.  In this
                 case, DHSEIN must always perform inverse iteration
                 using the whole matrix H.

  INITV   (input) CHARACTER*1
          = 'N': no initial vectors are supplied;
          = 'U': user-supplied initial vectors are stored in the arrays
                 VL and/or VR.

  SELECT  (input/output) LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the
          real eigenvector corresponding to a real eigenvalue WR(j),
          SELECT(j) must be set to .TRUE.. To select the complex
          eigenvector corresponding to a complex eigenvalue
          (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
          .FALSE..

  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  H       (input) DOUBLE PRECISION array, dimension (LDH,N)
          The upper Hessenberg matrix H.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

  WR      (input/output) DOUBLE PRECISION array, dimension (N)
  WI      (input) DOUBLE PRECISION array, dimension (N)
          On entry, the real and imaginary parts of the eigenvalues of

          H; a complex conjugate pair of eigenvalues must be stored in

          consecutive elements of WR and WI.
          On exit, WR may have been altered since close eigenvalues
          are perturbed slightly in searching for independent
          eigenvectors.

  VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
          contain starting vectors for the inverse iteration for the
          left eigenvectors; the starting vector for each eigenvector
          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'L' or 'B', the left eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VL, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is

          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'R', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.
          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

  VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
          contain starting vectors for the inverse iteration for the
          right eigenvectors; the starting vector for each eigenvector

          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'R' or 'B', the right eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VR, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is

          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'L', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.
          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR required to

          store the eigenvectors; each selected real eigenvector
          occupies one column and each selected complex eigenvector
          occupies two columns.

  WORK    (workspace) DOUBLE PRECISION array, dimension ((N+2)*N)

  IFAILL  (output) INTEGER array, dimension (MM)
          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
          eigenvector in the i-th column of VL (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VL hold a complex eigenvector, then IFAILL(i) and

          IFAILL(i+1) are set to the same value.
          If SIDE = 'R', IFAILL is not referenced.

  IFAILR  (output) INTEGER array, dimension (MM)
          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
          eigenvector in the i-th column of VR (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VR hold a complex eigenvector, then IFAILR(i) and

          IFAILR(i+1) are set to the same value.
          If SIDE = 'L', IFAILR is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, i is the number of eigenvectors which
                failed to converge; see IFAILL and IFAILR for further
                details.

  Further Details
  ===============

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x|+|y|.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        eigsrc -
        initv -
        select -
        n -
        h -
        ldh -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        ifaill -
        ifailr -
        info -

      • dhseqr

        public abstract void dhseqr(java.lang.String job,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            double[] h,
                            int ldh,
                            double[] wr,
                            double[] wi,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..
     Purpose
     =======

     DHSEQR computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     JOB   (input) CHARACTER*1
           = 'E':  compute eigenvalues only;
           = 'S':  compute eigenvalues and the Schur form T.

     COMPZ (input) CHARACTER*1
           = 'N':  no Schur vectors are computed;
           = 'I':  Z is initialized to the unit matrix and the matrix Z
                   of Schur vectors of H is returned;
           = 'V':  Z must contain an orthogonal matrix Q on entry, and

                   the product Q*Z is returned.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
           set by a previous call to DGEBAL, and then passed to DGEHRD

           when the matrix output by DGEBAL is reduced to Hessenberg
           form. Otherwise ILO and IHI should be set to 1 and N
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and JOB = 'S', then H contains the
           upper quasi-triangular matrix T from the Schur decomposition
           (the Schur form); 2-by-2 diagonal blocks (corresponding to
           complex conjugate pairs of eigenvalues) are returned in
           standard form, with H(i,i) = H(i+1,i+1) and
           H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
           contents of H are unspecified on exit.  (The output value of
           H when INFO.GT.0 is given under the description of INFO
           below.)

           Unlike earlier versions of DHSEQR, this subroutine may
           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
           or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) DOUBLE PRECISION array, dimension (N)
     WI    (output) DOUBLE PRECISION array, dimension (N)
           The real and imaginary parts, respectively, of the computed

           eigenvalues. If two eigenvalues are computed as a complex
           conjugate pair, they are stored in consecutive elements of
           WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
           WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in

           the same order as on the diagonal of the Schur form returned
           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     Z     (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
           If COMPZ = 'N', Z is not referenced.
           If COMPZ = 'I', on entry Z need not be set and on exit,
           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
           vectors of H.  If COMPZ = 'V', on entry Z must contain an
           N-by-N matrix Q, which is assumed to be equal to the unit
           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
           if INFO = 0, Z contains Q*Z.
           Normally Q is the orthogonal matrix generated by DORGHR
           after the call to DGEHRD which formed the Hessenberg matrix

           H. (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if COMPZ = 'I' or
           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
           On exit, if INFO = 0, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then DHSEQR does a workspace query.
           In this case, DHSEQR checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .LT. 0:  if INFO = -i, the i-th argument had an illegal
                    value
           .GT. 0:  if INFO = i, DHSEQR failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and JOB = 'E', then on exit, the
                remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and JOB   = 'S', then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and COMPZ = 'V', then on exit

                  (final value of Z)  =  (initial value of Z)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)

                If INFO .GT. 0 and COMPZ = 'I', then on exit
                      (final value of Z)  = U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)

                If INFO .GT. 0 and COMPZ = 'N', then Z is not
                accessed.

     ================================================================
             Default values supplied by
             ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
             It is suggested that these defaults be adjusted in order
             to attain best performance in each particular
             computational environment.

            ISPEC=1:  The DLAHQR vs DLAQR0 crossover point.
                      Default: 75. (Must be at least 11.)

            ISPEC=2:  Recommended deflation window size.
                      This depends on ILO, IHI and NS.  NS is the
                      number of simultaneous shifts returned
                      by ILAENV(ISPEC=4).  (See ISPEC=4 below.)
                      The default for (IHI-ILO+1).LE.500 is NS.
                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.

            ISPEC=3:  Nibble crossover point. (See ILAENV for
                      details.)  Default: 14% of deflation window
                      size.

            ISPEC=4:  Number of simultaneous shifts, NS, in
                      a multi-shift QR iteration.

                      If IHI-ILO+1 is ...

                      greater than      ...but less    ... the
                      or equal to ...      than        default is

                           1               30          NS -   2(+)
                          30               60          NS -   4(+)
                          60              150          NS =  10(+)
                         150              590          NS =  **
                         590             3000          NS =  64
                        3000             6000          NS = 128
                        6000             infinity      NS = 256

                  (+)  By default some or all matrices of this order 
                       are passed to the implicit double shift routine

                       DLAHQR and NS is ignored.  See ISPEC=1 above 
                       and comments in IPARM for details.

                       The asterisks (**) indicate an ad-hoc
                       function of N increasing from 10 to 64.

            ISPEC=5:  Select structured matrix multiply.
                      (See ILAENV for details.) Default: 3.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    DLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== NL allocates some local workspace to help small matrices
     .    through a rare DLAHQR failure.  NL .GT. NTINY = 11 is
     .    required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom-
     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
     .    allows up to six simultaneous shifts and a 16-by-16
     .    deflation window.  ====
        Parameters:
        job -
        compz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        z -
        ldz -
        work -
        lwork -
        info -

      • disnan

        public boolean disnan(double din)
        
     ..

  Purpose
  =======

  DISNAN returns .TRUE. if its argument is NaN, and .FALSE.
  otherwise.  To be replaced by the Fortran 2003 intrinsic in the
  future.

  Arguments
  =========

  DIN      (input) DOUBLE PRECISION
          Input to test for NaN.

  =====================================================================

  .. External Functions ..
        Parameters:
        din -

      • dlabad

        public abstract void dlabad(org.netlib.util.doubleW small,
                            org.netlib.util.doubleW large)
        
     ..

  Purpose
  =======

  DLABAD takes as input the values computed by DLAMCH for underflow and
  overflow, and returns the square root of each of these values if the

  log of LARGE is sufficiently large.  This subroutine is intended to
  identify machines with a large exponent range, such as the Crays, and
  redefine the underflow and overflow limits to be the square roots of

  the values computed by DLAMCH.  This subroutine is needed because
  DLAMCH does not compensate for poor arithmetic in the upper half of
  the exponent range, as is found on a Cray.

  Arguments
  =========

  SMALL   (input/output) DOUBLE PRECISION
          On entry, the underflow threshold as computed by DLAMCH.
          On exit, if LOG10(LARGE) is sufficiently large, the square
          root of SMALL, otherwise unchanged.

  LARGE   (input/output) DOUBLE PRECISION
          On entry, the overflow threshold as computed by DLAMCH.
          On exit, if LOG10(LARGE) is sufficiently large, the square
          root of LARGE, otherwise unchanged.

  =====================================================================

     .. Intrinsic Functions ..
        Parameters:
        small -
        large -

      • dlabrd

        public abstract void dlabrd(int m,
                            int n,
                            int nb,
                            double[] a,
                            int lda,
                            double[] d,
                            double[] e,
                            double[] tauq,
                            double[] taup,
                            double[] x,
                            int ldx,
                            double[] y,
                            int ldy)
        
     ..

  Purpose
  =======

  DLABRD reduces the first NB rows and columns of a real general
  m by n matrix A to upper or lower bidiagonal form by an orthogonal
  transformation Q' * A * P, and returns the matrices X and Y which
  are needed to apply the transformation to the unreduced part of A.

  If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower

  bidiagonal form.

  This is an auxiliary routine called by DGEBRD

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.

  N       (input) INTEGER
          The number of columns in the matrix A.

  NB      (input) INTEGER
          The number of leading rows and columns of A to be reduced.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit, the first NB rows and columns of the matrix are
          overwritten; the rest of the array is unchanged.
          If m >= n, elements on and below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the

            array TAUP, represent the orthogonal matrix P as a product

            of elementary reflectors.
          If m < n, elements below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors, and
            elements on and above the diagonal in the first NB rows,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) DOUBLE PRECISION array, dimension (NB)
          The diagonal elements of the first NB rows and columns of
          the reduced matrix.  D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (NB)
          The off-diagonal elements of the first NB rows and columns of
          the reduced matrix.

  TAUQ    (output) DOUBLE PRECISION array dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)
          The m-by-nb matrix X required to update the unreduced part
          of A.

  LDX     (input) INTEGER
          The leading dimension of the array X. LDX >= M.

  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
          The n-by-nb matrix Y required to update the unreduced part
          of A.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors.

  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  The elements of the vectors v and u together form the m-by-nb matrix

  V and the nb-by-n matrix U' which are needed, with X and Y, to apply

  the transformation to the unreduced part of the matrix, using a block
  update of the form:  A := A - V*Y' - X*U'.

  The contents of A on exit are illustrated by the following examples
  with nb = 2:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )

  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element

  of the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nb -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        x -
        ldx -
        y -
        ldy -

      • dlacn2

        public void dlacn2(int n,
                   double[] v,
                   double[] x,
                   int[] isgn,
                   org.netlib.util.doubleW est,
                   org.netlib.util.intW kase,
                   int[] isave)
        
     ..

  Purpose
  =======

  DLACN2 estimates the 1-norm of a square, real matrix A.
  Reverse communication is used for evaluating matrix-vector products.


  Arguments
  =========

  N      (input) INTEGER
         The order of the matrix.  N >= 1.

  V      (workspace) DOUBLE PRECISION array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

  X      (input/output) DOUBLE PRECISION array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A' * X,  if KASE=2,
         and DLACN2 must be re-called with all the other parameters
         unchanged.

  ISGN   (workspace) INTEGER array, dimension (N)

  EST    (input/output) DOUBLE PRECISION
         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
         unchanged from the previous call to DLACN2.
         On exit, EST is an estimate (a lower bound) for norm(A). 

  KASE   (input/output) INTEGER
         On the initial call to DLACN2, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A' * X.
         On the final return from DLACN2, KASE will again be 0.

  ISAVE  (input/output) INTEGER array, dimension (3)
         ISAVE is used to save variables between calls to DLACN2

  Further Details
  ======= =======

  Contributed by Nick Higham, University of Manchester.
  Originally named SONEST, dated March 16, 1988.

  Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of
  a real or complex matrix, with applications to condition estimation",
  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

  This is a thread safe version of DLACON, which uses the array ISAVE
  in place of a SAVE statement, as follows:

     DLACON     DLACN2
      JUMP     ISAVE(1)
      J        ISAVE(2)
      ITER     ISAVE(3)

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        v -
        x -
        isgn -
        est -
        kase -
        isave -

      • dlacon

        public abstract void dlacon(int n,
                            double[] v,
                            double[] x,
                            int[] isgn,
                            org.netlib.util.doubleW est,
                            org.netlib.util.intW kase)
        
     ..

  Purpose
  =======

  DLACON estimates the 1-norm of a square, real matrix A.
  Reverse communication is used for evaluating matrix-vector products.


  Arguments
  =========

  N      (input) INTEGER
         The order of the matrix.  N >= 1.

  V      (workspace) DOUBLE PRECISION array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

  X      (input/output) DOUBLE PRECISION array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A' * X,  if KASE=2,
         and DLACON must be re-called with all the other parameters
         unchanged.

  ISGN   (workspace) INTEGER array, dimension (N)

  EST    (input/output) DOUBLE PRECISION
         On entry with KASE = 1 or 2 and JUMP = 3, EST should be
         unchanged from the previous call to DLACON.
         On exit, EST is an estimate (a lower bound) for norm(A). 

  KASE   (input/output) INTEGER
         On the initial call to DLACON, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A' * X.
         On the final return from DLACON, KASE will again be 0.

  Further Details
  ======= =======

  Contributed by Nick Higham, University of Manchester.
  Originally named SONEST, dated March 16, 1988.

  Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of
  a real or complex matrix, with applications to condition estimation",
  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        v -
        x -
        isgn -
        est -
        kase -

      • dlacpy

        public abstract void dlacpy(java.lang.String uplo,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  DLACPY copies all or part of a two-dimensional matrix A to another
  matrix B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies the part of the matrix A to be copied to B.
          = 'U':      Upper triangular part
          = 'L':      Lower triangular part
          Otherwise:  All of the matrix A

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The m by n matrix A.  If UPLO = 'U', only the upper triangle

          or trapezoid is accessed; if UPLO = 'L', only the lower
          triangle or trapezoid is accessed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (output) DOUBLE PRECISION array, dimension (LDB,N)
          On exit, B = A in the locations specified by UPLO.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        m -
        n -
        a -
        lda -
        b -
        ldb -

      • dladiv

        public abstract void dladiv(double a,
                            double b,
                            double c,
                            double d,
                            org.netlib.util.doubleW p,
                            org.netlib.util.doubleW q)
        
     ..

  Purpose
  =======

  DLADIV performs complex division in  real arithmetic

                        a + i*b
             p + i*q = ---------
                        c + i*d

  The algorithm is due to Robert L. Smith and can be found
  in D. Knuth, The art of Computer Programming, Vol.2, p.195

  Arguments
  =========

  A       (input) DOUBLE PRECISION
  B       (input) DOUBLE PRECISION
  C       (input) DOUBLE PRECISION
  D       (input) DOUBLE PRECISION
          The scalars a, b, c, and d in the above expression.

  P       (output) DOUBLE PRECISION
  Q       (output) DOUBLE PRECISION
          The scalars p and q in the above expression.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        a -
        b -
        c -
        d -
        p -
        q -

      • dlae2

        public abstract void dlae2(double a,
                           double b,
                           double c,
                           org.netlib.util.doubleW rt1,
                           org.netlib.util.doubleW rt2)
        
     ..

  Purpose
  =======

  DLAE2  computes the eigenvalues of a 2-by-2 symmetric matrix
     [  A   B  ]
     [  B   C  ].
  On return, RT1 is the eigenvalue of larger absolute value, and RT2
  is the eigenvalue of smaller absolute value.

  Arguments
  =========

  A       (input) DOUBLE PRECISION
          The (1,1) element of the 2-by-2 matrix.

  B       (input) DOUBLE PRECISION
          The (1,2) and (2,1) elements of the 2-by-2 matrix.

  C       (input) DOUBLE PRECISION
          The (2,2) element of the 2-by-2 matrix.

  RT1     (output) DOUBLE PRECISION
          The eigenvalue of larger absolute value.

  RT2     (output) DOUBLE PRECISION
          The eigenvalue of smaller absolute value.

  Further Details
  ===============

  RT1 is accurate to a few ulps barring over/underflow.

  RT2 may be inaccurate if there is massive cancellation in the
  determinant A*C-B*B; higher precision or correctly rounded or
  correctly truncated arithmetic would be needed to compute RT2
  accurately in all cases.

  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  Underflow is harmless if the input data is 0 or exceeds
     underflow_threshold / macheps.

 =====================================================================


     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        rt1 -
        rt2 -

      • dlaebz

        public abstract void dlaebz(int ijob,
                            int nitmax,
                            int n,
                            int mmax,
                            int minp,
                            int nbmin,
                            double abstol,
                            double reltol,
                            double pivmin,
                            double[] d,
                            double[] e,
                            double[] e2,
                            int[] nval,
                            double[] ab,
                            double[] c,
                            org.netlib.util.intW mout,
                            int[] nab,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAEBZ contains the iteration loops which compute and use the
  function N(w), which is the count of eigenvalues of a symmetric
  tridiagonal matrix T less than or equal to its argument  w.  It
  performs a choice of two types of loops:

  IJOB=1, followed by
  IJOB=2: It takes as input a list of intervals and returns a list of
          sufficiently small intervals whose union contains the same
          eigenvalues as the union of the original intervals.
          The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
          The output interval (AB(j,1),AB(j,2)] will contain
          eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.

  IJOB=3: It performs a binary search in each input interval
          (AB(j,1),AB(j,2)] for a point  w(j)  such that
          N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
          the search.  If such a w(j) is found, then on output
          AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output

          (AB(j,1),AB(j,2)] will be a small interval containing the
          point where N(w) jumps through NVAL(j), unless that point
          lies outside the initial interval.

  Note that the intervals are in all cases half-open intervals,
  i.e., of the form  (a,b] , which includes  b  but not  a .

  To avoid underflow, the matrix should be scaled so that its largest
  element is no greater than  overflow**(1/2) * underflow**(1/4)
  in absolute value.  To assure the most accurate computation
  of small eigenvalues, the matrix should be scaled to be
  not much smaller than that, either.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966

  Note: the arguments are, in general, *not* checked for unreasonable
  values.

  Arguments
  =========

  IJOB    (input) INTEGER
          Specifies what is to be done:
          = 1:  Compute NAB for the initial intervals.
          = 2:  Perform bisection iteration to find eigenvalues of T.
          = 3:  Perform bisection iteration to invert N(w), i.e.,
                to find a point which has a specified number of
                eigenvalues of T to its left.
          Other values will cause DLAEBZ to return with INFO=-1.

  NITMAX  (input) INTEGER
          The maximum number of "levels" of bisection to be
          performed, i.e., an interval of width W will not be made
          smaller than 2^(-NITMAX) * W.  If not all intervals
          have converged after NITMAX iterations, then INFO is set
          to the number of non-converged intervals.

  N       (input) INTEGER
          The dimension n of the tridiagonal matrix T.  It must be at
          least 1.

  MMAX    (input) INTEGER
          The maximum number of intervals.  If more than MMAX intervals
          are generated, then DLAEBZ will quit with INFO=MMAX+1.

  MINP    (input) INTEGER
          The initial number of intervals.  It may not be greater than

          MMAX.

  NBMIN   (input) INTEGER
          The smallest number of intervals that should be processed
          using a vector loop.  If zero, then only the scalar loop
          will be used.

  ABSTOL  (input) DOUBLE PRECISION
          The minimum (absolute) width of an interval.  When an
          interval is narrower than ABSTOL, or than RELTOL times the
          larger (in magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  This must be at least
          zero.

  RELTOL  (input) DOUBLE PRECISION
          The minimum relative width of an interval.  When an interval

          is narrower than ABSTOL, or than RELTOL times the larger (in

          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum absolute value of a "pivot" in the Sturm
          sequence loop.  This *must* be at least  max |e(j)**2| *
          safe_min  and at least safe_min, where safe_min is at least
          the smallest number that can divide one without overflow.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (input) DOUBLE PRECISION array, dimension (N)
          The offdiagonal elements of the tridiagonal matrix T in
          positions 1 through N-1.  E(N) is arbitrary.

  E2      (input) DOUBLE PRECISION array, dimension (N)
          The squares of the offdiagonal elements of the tridiagonal
          matrix T.  E2(N) is ignored.

  NVAL    (input/output) INTEGER array, dimension (MINP)
          If IJOB=1 or 2, not referenced.
          If IJOB=3, the desired values of N(w).  The elements of NVAL

          will be reordered to correspond with the intervals in AB.
          Thus, NVAL(j) on output will not, in general be the same as
          NVAL(j) on input, but it will correspond with the interval
          (AB(j,1),AB(j,2)] on output.

  AB      (input/output) DOUBLE PRECISION array, dimension (MMAX,2)
          The endpoints of the intervals.  AB(j,1) is  a(j), the left
          endpoint of the j-th interval, and AB(j,2) is b(j), the
          right endpoint of the j-th interval.  The input intervals
          will, in general, be modified, split, and reordered by the
          calculation.

  C       (input/output) DOUBLE PRECISION array, dimension (MMAX)
          If IJOB=1, ignored.
          If IJOB=2, workspace.
          If IJOB=3, then on input C(j) should be initialized to the
          first search point in the binary search.

  MOUT    (output) INTEGER
          If IJOB=1, the number of eigenvalues in the intervals.
          If IJOB=2 or 3, the number of intervals output.
          If IJOB=3, MOUT will equal MINP.

  NAB     (input/output) INTEGER array, dimension (MMAX,2)
          If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
          If IJOB=2, then on input, NAB(i,j) should be set.  It must
             satisfy the condition:
             N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
             which means that in interval i only eigenvalues
             NAB(i,1)+1,...,NAB(i,2) will be considered.  Usually,
             NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with
             IJOB=1.
             On output, NAB(i,j) will contain
             max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of

             the input interval that the output interval
             (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
             the input values of NAB(k,1) and NAB(k,2).
          If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
             unless N(w) > NVAL(i) for all search points  w , in which

             case NAB(i,1) will not be modified, i.e., the output
             value will be the same as the input value (modulo
             reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)

             for all search points  w , in which case NAB(i,2) will
             not be modified.  Normally, NAB should be set to some
             distinctive value(s) before DLAEBZ is called.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MMAX)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (MMAX)
          Workspace.

  INFO    (output) INTEGER
          = 0:       All intervals converged.
          = 1--MMAX: The last INFO intervals did not converge.
          = MMAX+1:  More than MMAX intervals were generated.

  Further Details
  ===============

      This routine is intended to be called only by other LAPACK
  routines, thus the interface is less user-friendly.  It is intended
  for two purposes:

  (a) finding eigenvalues.  In this case, DLAEBZ should have one or
      more initial intervals set up in AB, and DLAEBZ should be called

      with IJOB=1.  This sets up NAB, and also counts the eigenvalues.

      Intervals with no eigenvalues would usually be thrown out at
      this point.  Also, if not all the eigenvalues in an interval i
      are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
      For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
      eigenvalue.  DLAEBZ is then called with IJOB=2 and MMAX
      no smaller than the value of MOUT returned by the call with
      IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1
      through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
      tolerance specified by ABSTOL and RELTOL.

  (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
      In this case, start with a Gershgorin interval  (a,b).  Set up
      AB to contain 2 search intervals, both initially (a,b).  One
      NVAL element should contain  f-1  and the other should contain  l
      , while C should contain a and b, resp.  NAB(i,1) should be -1
      and NAB(i,2) should be N+1, to flag an error if the desired
      interval does not lie in (a,b).  DLAEBZ is then called with
      IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals --

      j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
      if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
      >= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and
      N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and
      w(l-r)=...=w(l+k) are handled similarly.

  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        nitmax -
        n -
        mmax -
        minp -
        nbmin -
        abstol -
        reltol -
        pivmin -
        d -
        e -
        e2 -
        nval -
        ab -
        c -
        mout -
        nab -
        work -
        iwork -
        info -

      • dlaed0

        public abstract void dlaed0(int icompq,
                            int qsiz,
                            int n,
                            double[] d,
                            double[] e,
                            double[] q,
                            int ldq,
                            double[] qstore,
                            int ldqs,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED0 computes all eigenvalues and corresponding eigenvectors of a
  symmetric tridiagonal matrix using the divide and conquer method.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.
          = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                matrix.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry, the main diagonal of the tridiagonal matrix.
         On exit, its eigenvalues.

  E      (input) DOUBLE PRECISION array, dimension (N-1)
         The off-diagonal elements of the tridiagonal matrix.
         On exit, E has been destroyed.

  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, Q must contain an N-by-N orthogonal matrix.
         If ICOMPQ = 0    Q is not referenced.
         If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                          orthogonal matrix used to reduce the full
                          matrix to tridiagonal form corresponding to
                          the subset of the full matrix which is being

                          decomposed at this time.
         If ICOMPQ = 2    On entry, Q will be the identity matrix.
                          On exit, Q contains the eigenvectors of the
                          tridiagonal matrix.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  If eigenvectors are
         desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

  QSTORE (workspace) DOUBLE PRECISION array, dimension (LDQS, N)
         Referenced only when ICOMPQ = 1.  Used to store parts of
         the eigenvector matrix when the updating matrix multiplies
         take place.

  LDQS   (input) INTEGER
         The leading dimension of the array QSTORE.  If ICOMPQ = 1,
         then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

  WORK   (workspace) DOUBLE PRECISION array,
         If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                     1 + 3*N + 2*N*lg N + 2*N**2
                     ( lg( N ) = smallest integer k
                                 such that 2^k >= N )
         If ICOMPQ = 2, the dimension of WORK must be at least
                     4*N + N**2.

  IWORK  (workspace) INTEGER array,
         If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                        6 + 6*N + 5*N*lg N.
                        ( lg( N ) = smallest integer k
                                    such that 2^k >= N )
         If ICOMPQ = 2, the dimension of IWORK must be at least
                        3 + 5*N.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        qsiz -
        n -
        d -
        e -
        q -
        ldq -
        qstore -
        ldqs -
        work -
        iwork -
        info -

      • dlaed1

        public abstract void dlaed1(int n,
                            double[] d,
                            double[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.doubleW rho,
                            int cutpnt,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED1 computes the updated eigensystem of a diagonal
  matrix after modification by a rank-one symmetric matrix.  This
  routine is used only for the eigenproblem which requires all
  eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
  the case in which eigenvalues only or eigenvalues and eigenvectors
  of a full symmetric matrix (which was reduced to tridiagonal form)
  are desired.

    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

     where Z = Q'u, u is a vector of length N with ones in the
     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

     The eigenvectors of the original matrix are stored in Q, and the
     eigenvalues are in D.  The algorithm consists of three stages:

        The first stage consists of deflating the size of the problem
        when there are multiple eigenvalues or if there is a zero in
        the Z vector.  For each such occurence the dimension of the
        secular equation problem is reduced by one.  This stage is
        performed by the routine DLAED2.

        The second stage consists of calculating the updated
        eigenvalues. This is done by finding the roots of the secular
        equation via the routine DLAED4 (as called by DLAED3).
        This routine also calculates the eigenvectors of the current
        problem.

        The final stage consists of computing the updated eigenvectors

        directly using the updated eigenvalues.  The eigenvectors for
        the current problem are multiplied with the eigenvectors from
        the overall problem.

  Arguments
  =========

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.


  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input/output) INTEGER array, dimension (N)
         On entry, the permutation which separately sorts the two
         subproblems in D into ascending order.
         On exit, the permutation which will reintegrate the
         subproblems back into sorted order,
         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

  RHO    (input) DOUBLE PRECISION
         The subdiagonal entry used to create the rank-1 modification.


  CUTPNT (input) INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= CUTPNT <= N/2.

  WORK   (workspace) DOUBLE PRECISION array, dimension (4*N + N**2)

  IWORK  (workspace) INTEGER array, dimension (4*N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        work -
        iwork -
        info -

      • dlaed2

        public abstract void dlaed2(org.netlib.util.intW k,
                            int n,
                            int n1,
                            double[] d,
                            double[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.doubleW rho,
                            double[] z,
                            double[] dlamda,
                            double[] w,
                            double[] q2,
                            int[] indx,
                            int[] indxc,
                            int[] indxp,
                            int[] coltyp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED2 merges the two sets of eigenvalues together into a single
  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  eigenvalues are close together or if there is a tiny entry in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  Arguments
  =========

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation. 0 <= K <=N.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  N1     (input) INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= N1 <= N/2.

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.
         On exit, D contains the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in
         the two square blocks with corners at (1,1), (N1,N1)
         and (N1+1, N1+1), (N,N).
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input/output) INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have N1 added to their
         values. Destroyed on exit.

  RHO    (input/output) DOUBLE PRECISION
         On entry, the off-diagonal element associated with the rank-1

         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         DLAED3.

  Z      (input) DOUBLE PRECISION array, dimension (N)
         On entry, Z contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).
         On exit, the contents of Z have been destroyed by the updating
         process.

  DLAMDA (output) DOUBLE PRECISION array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         DLAED3 to form the secular equation.

  W      (output) DOUBLE PRECISION array, dimension (N)
         The first k values of the final deflation-altered z-vector
         which will be passed to DLAED3.

  Q2     (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
         A copy of the first K eigenvectors which will be used by
         DLAED3 in a matrix multiply (DGEMM) to solve for the new
         eigenvectors.

  INDX   (workspace) INTEGER array, dimension (N)
         The permutation used to sort the contents of DLAMDA into
         ascending order.

  INDXC  (output) INTEGER array, dimension (N)
         The permutation used to arrange the columns of the deflated
         Q matrix into three groups:  the first group contains non-zero
         elements only at and above N1, the second contains
         non-zero elements only below N1, and the third is dense.

  INDXP  (workspace) INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end

         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

  COLTYP (workspace/output) INTEGER array, dimension (N)
         During execution, a label which will indicate which of the
         following types a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : dense;
         3 : non-zero in the lower half only;
         4 : deflated.
         On exit, COLTYP(i) is the number of columns of type i,
         for i=1 to 4 only.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        k -
        n -
        n1 -
        d -
        q -
        ldq -
        indxq -
        rho -
        z -
        dlamda -
        w -
        q2 -
        indx -
        indxc -
        indxp -
        coltyp -
        info -

      • dlaed3

        public abstract void dlaed3(int k,
                            int n,
                            int n1,
                            double[] d,
                            double[] q,
                            int ldq,
                            double rho,
                            double[] dlamda,
                            double[] q2,
                            int[] indx,
                            int[] ctot,
                            double[] w,
                            double[] s,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED3 finds the roots of the secular equation, as defined by the
  values in D, W, and RHO, between 1 and K.  It makes the
  appropriate calls to DLAED4 and then updates the eigenvectors by
  multiplying the matrix of eigenvectors of the pair of eigensystems
  being combined by the matrix of eigenvectors of the K-by-K system
  which is solved here.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  Arguments
  =========

  K       (input) INTEGER
          The number of terms in the rational function to be solved by

          DLAED4.  K >= 0.

  N       (input) INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (deflation may result in N>K).

  N1      (input) INTEGER
          The location of the last eigenvalue in the leading submatrix.
          min(1,N) <= N1 <= N/2.

  D       (output) DOUBLE PRECISION array, dimension (N)
          D(I) contains the updated eigenvalues for
          1 <= I <= K.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          Initially the first K columns are used as workspace.
          On output the columns 1 to K contain
          the updated eigenvectors.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max(1,N).

  RHO     (input) DOUBLE PRECISION
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

  DLAMDA  (input/output) DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation. May be changed on output by
          having lowest order bit set to zero on Cray X-MP, Cray Y-MP,

          Cray-2, or Cray C-90, as described above.

  Q2      (input) DOUBLE PRECISION array, dimension (LDQ2, N)
          The first K columns of this matrix contain the non-deflated
          eigenvectors for the split problem.

  INDX    (input) INTEGER array, dimension (N)
          The permutation used to arrange the columns of the deflated
          Q matrix into three groups (see DLAED2).
          The rows of the eigenvectors found by DLAED4 must be likewise
          permuted before the matrix multiply can take place.

  CTOT    (input) INTEGER array, dimension (4)
          A count of the total number of the various types of columns
          in Q, as described in INDX.  The fourth column type is any
          column which has been deflated.

  W       (input/output) DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector. Destroyed on
          output.

  S       (workspace) DOUBLE PRECISION array, dimension (N1 + 1)*K
          Will contain the eigenvectors of the repaired matrix which
          will be multiplied by the previously accumulated eigenvectors
          to update the system.

  LDS     (input) INTEGER
          The leading dimension of S.  LDS >= max(1,K).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        k -
        n -
        n1 -
        d -
        q -
        ldq -
        rho -
        dlamda -
        q2 -
        indx -
        ctot -
        w -
        s -
        info -

      • dlaed4

        public abstract void dlaed4(int n,
                            int i,
                            double[] d,
                            double[] z,
                            double[] delta,
                            double rho,
                            org.netlib.util.doubleW dlam,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This subroutine computes the I-th updated eigenvalue of a symmetric
  rank-one modification to a diagonal matrix whose elements are
  given in the array d, and that

             D(i) < D(j)  for  i < j

  and that RHO > 0.  This is arranged by the calling routine, and is
  no loss in generality.  The rank-one modified system is thus

             diag( D )  +  RHO *  Z * Z_transpose.

  where we assume the Euclidean norm of Z is 1.

  The method consists of approximating the rational functions in the
  secular equation by simpler interpolating rational functions.

  Arguments
  =========

  N      (input) INTEGER
         The length of all arrays.

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

  D      (input) DOUBLE PRECISION array, dimension (N)
         The original eigenvalues.  It is assumed that they are in
         order, D(I) < D(J)  for I < J.

  Z      (input) DOUBLE PRECISION array, dimension (N)
         The components of the updating vector.

  DELTA  (output) DOUBLE PRECISION array, dimension (N)
         If N .GT. 2, DELTA contains (D(j) - lambda_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5

         for detail. The vector DELTA contains the information necessar
         to construct the eigenvectors by DLAED3 and DLAED9.

  RHO    (input) DOUBLE PRECISION
         The scalar in the symmetric updating formula.

  DLAM   (output) DOUBLE PRECISION
         The computed lambda_I, the I-th updated eigenvalue.

  INFO   (output) INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Internal Parameters
  ===================

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.

            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1

   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
   if we are working with THREE poles!

   MAXIT is the maximum number of iterations allowed for each
   eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        i -
        d -
        z -
        delta -
        rho -
        dlam -
        info -

      • dlaed5

        public abstract void dlaed5(int i,
                            double[] d,
                            double[] z,
                            double[] delta,
                            double rho,
                            org.netlib.util.doubleW dlam)
        
     ..

  Purpose
  =======

  This subroutine computes the I-th eigenvalue of a symmetric rank-one

  modification of a 2-by-2 diagonal matrix

             diag( D )  +  RHO *  Z * transpose(Z) .

  The diagonal elements in the array D are assumed to satisfy

             D(i) < D(j)  for  i < j .

  We also assume RHO > 0 and that the Euclidean norm of the vector
  Z is one.

  Arguments
  =========

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

  D      (input) DOUBLE PRECISION array, dimension (2)
         The original eigenvalues.  We assume D(1) < D(2).

  Z      (input) DOUBLE PRECISION array, dimension (2)
         The components of the updating vector.

  DELTA  (output) DOUBLE PRECISION array, dimension (2)
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

  RHO    (input) DOUBLE PRECISION
         The scalar in the symmetric updating formula.

  DLAM   (output) DOUBLE PRECISION
         The computed lambda_I, the I-th updated eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        i -
        d -
        z -
        delta -
        rho -
        dlam -

      • dlaed6

        public abstract void dlaed6(int kniter,
                            boolean orgati,
                            double rho,
                            double[] d,
                            double[] z,
                            double finit,
                            org.netlib.util.doubleW tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED6 computes the positive or negative root (closest to the origin)
  of
                   z(1)        z(2)        z(3)
  f(x) =   rho + --------- + ---------- + ---------
                  d(1)-x      d(2)-x      d(3)-x

  It is assumed that

        if ORGATI = .true. the root is between d(2) and d(3);
        otherwise it is between d(1) and d(2)

  This routine will be called by DLAED4 when necessary. In most cases,

  the root sought is the smallest in magnitude, though it might not be

  in some extremely rare situations.

  Arguments
  =========

  KNITER       (input) INTEGER
               Refer to DLAED4 for its significance.

  ORGATI       (input) LOGICAL
               If ORGATI is true, the needed root is between d(2) and
               d(3); otherwise it is between d(1) and d(2).  See
               DLAED4 for further details.

  RHO          (input) DOUBLE PRECISION
               Refer to the equation f(x) above.

  D            (input) DOUBLE PRECISION array, dimension (3)
               D satisfies d(1) < d(2) < d(3).

  Z            (input) DOUBLE PRECISION array, dimension (3)
               Each of the elements in z must be positive.

  FINIT        (input) DOUBLE PRECISION
               The value of f at 0. It is more accurate than the one
               evaluated inside this routine (if someone wants to do
               so).

  TAU          (output) DOUBLE PRECISION
               The root of the equation f(x).

  INFO         (output) INTEGER
               = 0: successful exit
               > 0: if INFO = 1, failure to converge

  Further Details
  ===============

  30/06/99: Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  10/02/03: This version has a few statements commented out for thread

  safety (machine parameters are computed on each entry). SJH.

  05/10/06: Modified from a new version of Ren-Cang Li, use
     Gragg-Thornton-Warner cubic convergent scheme for better stability

  =====================================================================

     .. Parameters ..
        Parameters:
        kniter -
        orgati -
        rho -
        d -
        z -
        finit -
        tau -
        info -

      • dlaed7

        public abstract void dlaed7(int icompq,
                            int n,
                            int qsiz,
                            int tlvls,
                            int curlvl,
                            int curpbm,
                            double[] d,
                            double[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.doubleW rho,
                            int cutpnt,
                            double[] qstore,
                            int[] qptr,
                            int[] prmptr,
                            int[] perm,
                            int[] givptr,
                            int[] givcol,
                            double[] givnum,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED7 computes the updated eigensystem of a diagonal
  matrix after modification by a rank-one symmetric matrix. This
  routine is used only for the eigenproblem which requires all
  eigenvalues and optionally eigenvectors of a dense symmetric matrix
  that has been reduced to tridiagonal form.  DLAED1 handles
  the case in which all eigenvalues and eigenvectors of a symmetric
  tridiagonal matrix are desired.

    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

     where Z = Q'u, u is a vector of length N with ones in the
     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

     The eigenvectors of the original matrix are stored in Q, and the
     eigenvalues are in D.  The algorithm consists of three stages:

        The first stage consists of deflating the size of the problem
        when there are multiple eigenvalues or if there is a zero in
        the Z vector.  For each such occurence the dimension of the
        secular equation problem is reduced by one.  This stage is
        performed by the routine DLAED8.

        The second stage consists of calculating the updated
        eigenvalues. This is done by finding the roots of the secular
        equation via the routine DLAED4 (as called by DLAED9).
        This routine also calculates the eigenvectors of the current
        problem.

        The final stage consists of computing the updated eigenvectors

        directly using the updated eigenvalues.  The eigenvectors for
        the current problem are multiplied with the eigenvectors from
        the overall problem.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  TLVLS  (input) INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

  CURLVL (input) INTEGER
         The current level in the overall merge routine,
         0 <= CURLVL <= TLVLS.

  CURPBM (input) INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.


  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (output) INTEGER array, dimension (N)
         The permutation which will reintegrate the subproblem just
         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
         will be in ascending order.

  RHO    (input) DOUBLE PRECISION
         The subdiagonal element used to create the rank-1
         modification.

  CUTPNT (input) INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

  QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.

  QPTR   (input/output) INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.

  PRMPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.

  PERM   (input) INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

  GIVPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)

         indicates the number of Givens rotations.

  GIVCOL (input) INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  WORK   (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N)

  IWORK  (workspace) INTEGER array, dimension (4*N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        n -
        qsiz -
        tlvls -
        curlvl -
        curpbm -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        qstore -
        qptr -
        prmptr -
        perm -
        givptr -
        givcol -
        givnum -
        work -
        iwork -
        info -

      • dlaed8

        public abstract void dlaed8(int icompq,
                            org.netlib.util.intW k,
                            int n,
                            int qsiz,
                            double[] d,
                            double[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.doubleW rho,
                            int cutpnt,
                            double[] z,
                            double[] dlamda,
                            double[] q2,
                            int ldq2,
                            double[] w,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            double[] givnum,
                            int[] indxp,
                            int[] indx,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED8 merges the two sets of eigenvalues together into a single
  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  eigenvalues are close together or if there is a tiny element in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be
         combined.  On exit, the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise,
         on entry, Q contains the eigenvectors of the partially solved

         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input) INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have CUTPNT added to
         their values in order to be accurate.

  RHO    (input/output) DOUBLE PRECISION
         On entry, the off-diagonal element associated with the rank-1

         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         DLAED3.

  CUTPNT (input) INTEGER
         The location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

  Z      (input) DOUBLE PRECISION array, dimension (N)
         On entry, Z contains the updating vector (the last row of
         the first sub-eigenvector matrix and the first row of the
         second sub-eigenvector matrix).
         On exit, the contents of Z are destroyed by the updating
         process.

  DLAMDA (output) DOUBLE PRECISION array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         DLAED3 to form the secular equation.

  Q2     (output) DOUBLE PRECISION array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         a copy of the first K eigenvectors which will be used by
         DLAED7 in a matrix multiply (DGEMM) to update the new
         eigenvectors.

  LDQ2   (input) INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max(1,N).

  W      (output) DOUBLE PRECISION array, dimension (N)
         The first k values of the final deflation-altered z-vector and
         will be passed to DLAED3.

  PERM   (output) INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied
         to each eigenblock.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem.

  GIVCOL (output) INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (output) DOUBLE PRECISION array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  INDXP  (workspace) INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end

         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

  INDX   (workspace) INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending

         order.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        k -
        n -
        qsiz -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        z -
        dlamda -
        q2 -
        ldq2 -
        w -
        perm -
        givptr -
        givcol -
        givnum -
        indxp -
        indx -
        info -

      • dlaed9

        public abstract void dlaed9(int k,
                            int kstart,
                            int kstop,
                            int n,
                            double[] d,
                            double[] q,
                            int ldq,
                            double rho,
                            double[] dlamda,
                            double[] w,
                            double[] s,
                            int lds,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAED9 finds the roots of the secular equation, as defined by the
  values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
  appropriate calls to DLAED4 and then stores the new matrix of
  eigenvectors for use in calculating the next level of Z vectors.

  Arguments
  =========

  K       (input) INTEGER
          The number of terms in the rational function to be solved by

          DLAED4.  K >= 0.

  KSTART  (input) INTEGER
  KSTOP   (input) INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed.  1 <= KSTART <= KSTOP <= K.

  N       (input) INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (delation may result in N > K).

  D       (output) DOUBLE PRECISION array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP.

  Q       (workspace) DOUBLE PRECISION array, dimension (LDQ,N)

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max( 1, N ).

  RHO     (input) DOUBLE PRECISION
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

  DLAMDA  (input) DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

  W       (input) DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector.

  S       (output) DOUBLE PRECISION array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system.

  LDS     (input) INTEGER
          The leading dimension of S.  LDS >= max( 1, K ).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        k -
        kstart -
        kstop -
        n -
        d -
        q -
        ldq -
        rho -
        dlamda -
        w -
        s -
        lds -
        info -

      • dlaeda

        public abstract void dlaeda(int n,
                            int tlvls,
                            int curlvl,
                            int curpbm,
                            int[] prmptr,
                            int[] perm,
                            int[] givptr,
                            int[] givcol,
                            double[] givnum,
                            double[] q,
                            int[] qptr,
                            double[] z,
                            double[] ztemp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAEDA computes the Z vector corresponding to the merge step in the
  CURLVLth step of the merge process with TLVLS steps for the CURPBMth

  problem.

  Arguments
  =========

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  TLVLS  (input) INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

  CURLVL (input) INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.

  CURPBM (input) INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

  PRMPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and incidentally the
         size of the full, non-deflated problem.

  PERM   (input) INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

  GIVPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)

         indicates the number of Givens rotations.

  GIVCOL (input) INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  Q      (input) DOUBLE PRECISION array, dimension (N**2)
         Contains the square eigenblocks from previous levels, the
         starting positions for blocks are given by QPTR.

  QPTR   (input) INTEGER array, dimension (N+2)
         Contains a list of pointers which indicate where in Q an
         eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
         the size of the block.

  Z      (output) DOUBLE PRECISION array, dimension (N)
         On output this vector contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).

  ZTEMP  (workspace) DOUBLE PRECISION array, dimension (N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        tlvls -
        curlvl -
        curpbm -
        prmptr -
        perm -
        givptr -
        givcol -
        givnum -
        q -
        qptr -
        z -
        ztemp -
        info -

      • dlaein

        public abstract void dlaein(boolean rightv,
                            boolean noinit,
                            int n,
                            double[] h,
                            int ldh,
                            double wr,
                            double wi,
                            double[] vr,
                            double[] vi,
                            double[] b,
                            int ldb,
                            double[] work,
                            double eps3,
                            double smlnum,
                            double bignum,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAEIN uses inverse iteration to find a right or left eigenvector
  corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
  matrix H.

  Arguments
  =========

  RIGHTV   (input) LOGICAL
          = .TRUE. : compute right eigenvector;
          = .FALSE.: compute left eigenvector.

  NOINIT   (input) LOGICAL
          = .TRUE. : no initial vector supplied in (VR,VI).
          = .FALSE.: initial vector supplied in (VR,VI).

  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  H       (input) DOUBLE PRECISION array, dimension (LDH,N)
          The upper Hessenberg matrix H.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

  WR      (input) DOUBLE PRECISION
  WI      (input) DOUBLE PRECISION
          The real and imaginary parts of the eigenvalue of H whose
          corresponding right or left eigenvector is to be computed.

  VR      (input/output) DOUBLE PRECISION array, dimension (N)
  VI      (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
          a real starting vector for inverse iteration using the real
          eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
          must contain the real and imaginary parts of a complex
          starting vector for inverse iteration using the complex
          eigenvalue (WR,WI); otherwise VR and VI need not be set.
          On exit, if WI = 0.0 (real eigenvalue), VR contains the
          computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
          VR and VI contain the real and imaginary parts of the
          computed complex eigenvector. The eigenvector is normalized
          so that the component of largest magnitude has magnitude 1;
          here the magnitude of a complex number (x,y) is taken to be
          |x| + |y|.
          VI is not referenced if WI = 0.0.

  B       (workspace) DOUBLE PRECISION array, dimension (LDB,N)

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= N+1.

  WORK   (workspace) DOUBLE PRECISION array, dimension (N)

  EPS3    (input) DOUBLE PRECISION
          A small machine-dependent value which is used to perturb
          close eigenvalues, and to replace zero pivots.

  SMLNUM  (input) DOUBLE PRECISION
          A machine-dependent value close to the underflow threshold.

  BIGNUM  (input) DOUBLE PRECISION
          A machine-dependent value close to the overflow threshold.

  INFO    (output) INTEGER
          = 0:  successful exit
          = 1:  inverse iteration did not converge; VR is set to the
                last iterate, and so is VI if WI.ne.0.0.

  =====================================================================

     .. Parameters ..
        Parameters:
        rightv -
        noinit -
        n -
        h -
        ldh -
        wr -
        wi -
        vr -
        vi -
        b -
        ldb -
        work -
        eps3 -
        smlnum -
        bignum -
        info -

      • dlaev2

        public abstract void dlaev2(double a,
                            double b,
                            double c,
                            org.netlib.util.doubleW rt1,
                            org.netlib.util.doubleW rt2,
                            org.netlib.util.doubleW cs1,
                            org.netlib.util.doubleW sn1)
        
     ..

  Purpose
  =======

  DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
     [  A   B  ]
     [  B   C  ].
  On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
  eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
  eigenvector for RT1, giving the decomposition

     [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
     [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].

  Arguments
  =========

  A       (input) DOUBLE PRECISION
          The (1,1) element of the 2-by-2 matrix.

  B       (input) DOUBLE PRECISION
          The (1,2) element and the conjugate of the (2,1) element of
          the 2-by-2 matrix.

  C       (input) DOUBLE PRECISION
          The (2,2) element of the 2-by-2 matrix.

  RT1     (output) DOUBLE PRECISION
          The eigenvalue of larger absolute value.

  RT2     (output) DOUBLE PRECISION
          The eigenvalue of smaller absolute value.

  CS1     (output) DOUBLE PRECISION
  SN1     (output) DOUBLE PRECISION
          The vector (CS1, SN1) is a unit right eigenvector for RT1.

  Further Details
  ===============

  RT1 is accurate to a few ulps barring over/underflow.

  RT2 may be inaccurate if there is massive cancellation in the
  determinant A*C-B*B; higher precision or correctly rounded or
  correctly truncated arithmetic would be needed to compute RT2
  accurately in all cases.

  CS1 and SN1 are accurate to a few ulps barring over/underflow.

  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  Underflow is harmless if the input data is 0 or exceeds
     underflow_threshold / macheps.

 =====================================================================


     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        rt1 -
        rt2 -
        cs1 -
        sn1 -

      • dlaexc

        public abstract void dlaexc(boolean wantq,
                            int n,
                            double[] t,
                            int ldt,
                            double[] q,
                            int ldq,
                            int j1,
                            int n1,
                            int n2,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in

  an upper quasi-triangular matrix T by an orthogonal similarity
  transformation.

  T must be in Schur canonical form, that is, block upper triangular
  with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
  has its diagonal elemnts equal and its off-diagonal elements of
  opposite sign.

  Arguments
  =========

  WANTQ   (input) LOGICAL
          = .TRUE. : accumulate the transformation in the matrix Q;
          = .FALSE.: do not accumulate the transformation.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, the updated matrix T, again in Schur canonical form.

  LDT     (input)  INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
          On exit, if WANTQ is .TRUE., the updated matrix Q.
          If WANTQ is .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

  J1      (input) INTEGER
          The index of the first row of the first block T11.

  N1      (input) INTEGER
          The order of the first block T11. N1 = 0, 1 or 2.

  N2      (input) INTEGER
          The order of the second block T22. N2 = 0, 1 or 2.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          = 1: the transformed matrix T would be too far from Schur
               form; the blocks are not swapped and T and Q are
               unchanged.

  =====================================================================

     .. Parameters ..
        Parameters:
        wantq -
        n -
        t -
        ldt -
        q -
        ldq -
        j1 -
        n1 -
        n2 -
        work -
        info -

      • dlag2

        public abstract void dlag2(double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double safmin,
                           org.netlib.util.doubleW scale1,
                           org.netlib.util.doubleW scale2,
                           org.netlib.util.doubleW wr1,
                           org.netlib.util.doubleW wr2,
                           org.netlib.util.doubleW wi)
        
     ..

  Purpose
  =======

  DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
  problem  A - w B, with scaling as necessary to avoid over-/underflow.

  The scaling factor "s" results in a modified eigenvalue equation

      s A - w B

  where  s  is a non-negative scaling factor chosen so that  w,  w B,
  and  s A  do not overflow and, if possible, do not underflow, either.

  Arguments
  =========

  A       (input) DOUBLE PRECISION array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm

          is less than 1/SAFMIN.  Entries less than
          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= 2.

  B       (input) DOUBLE PRECISION array, dimension (LDB, 2)
          On entry, the 2 x 2 upper triangular matrix B.  It is
          assumed that the one-norm of B is less than 1/SAFMIN.  The
          diagonals should be at least sqrt(SAFMIN) times the largest
          element of B (in absolute value); if a diagonal is smaller
          than that, then  +/- sqrt(SAFMIN) will be used instead of
          that diagonal.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= 2.

  SAFMIN  (input) DOUBLE PRECISION
          The smallest positive number s.t. 1/SAFMIN does not
          overflow.  (This should always be DLAMCH('S') -- it is an
          argument in order to avoid having to call DLAMCH frequently.)

  SCALE1  (output) DOUBLE PRECISION
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the first eigenvalue.  If
          the eigenvalues are complex, then the eigenvalues are
          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
          exponent range of the machine), SCALE1=SCALE2, and SCALE1
          will always be positive.  If the eigenvalues are real, then
          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
          overflow or underflow, and in fact, SCALE1 may be zero or
          less than the underflow threshhold if the exact eigenvalue
          is sufficiently large.

  SCALE2  (output) DOUBLE PRECISION
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the second eigenvalue.  If

          the eigenvalues are complex, then SCALE2=SCALE1.  If the
          eigenvalues are real, then the second (real) eigenvalue is
          WR2 / SCALE2 , but this may overflow or underflow, and in
          fact, SCALE2 may be zero or less than the underflow
          threshhold if the exact eigenvalue is sufficiently large.

  WR1     (output) DOUBLE PRECISION
          If the eigenvalue is real, then WR1 is SCALE1 times the
          eigenvalue closest to the (2,2) element of A B**(-1).  If the
          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real

          part of the eigenvalues.

  WR2     (output) DOUBLE PRECISION
          If the eigenvalue is real, then WR2 is SCALE2 times the
          other eigenvalue.  If the eigenvalue is complex, then
          WR1=WR2 is SCALE1 times the real part of the eigenvalues.

  WI      (output) DOUBLE PRECISION
          If the eigenvalue is real, then WI is zero.  If the
          eigenvalue is complex, then WI is SCALE1 times the imaginary

          part of the eigenvalues.  WI will always be non-negative.

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        lda -
        b -
        ldb -
        safmin -
        scale1 -
        scale2 -
        wr1 -
        wr2 -
        wi -

      • dlag2s

        public void dlag2s(int m,
                   int n,
                   double[] a,
                   int lda,
                   float[] sa,
                   int ldsa,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAG2S converts a DOUBLE PRECISION matrix, SA, to a SINGLE
  PRECISION matrix, A.

  RMAX is the overflow for the SINGLE PRECISION arithmetic
  DLAG2S checks that all the entries of A are between -RMAX and
  RMAX. If not the convertion is aborted and a flag is raised.

  This is a helper routine so there is no argument checking.

  Arguments
  =========

  M       (input) INTEGER
          The number of lines of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N coefficient matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  SA      (output) REAL array, dimension (LDSA,N)
          On exit, if INFO=0, the M-by-N coefficient matrix SA.

  LDSA    (input) INTEGER
          The leading dimension of the array SA.  LDSA >= max(1,M).

  INFO    (output) INTEGER
          = 0:  successful exit
          > 0:  if INFO = k, the (i,j) entry of the matrix A has
                overflowed when moving from DOUBLE PRECISION to SINGLE

                k is given by k = (i-1)*LDA+j

  =========

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        sa -
        ldsa -
        info -

      • dlags2

        public abstract void dlags2(boolean upper,
                            double a1,
                            double a2,
                            double a3,
                            double b1,
                            double b2,
                            double b3,
                            org.netlib.util.doubleW csu,
                            org.netlib.util.doubleW snu,
                            org.netlib.util.doubleW csv,
                            org.netlib.util.doubleW snv,
                            org.netlib.util.doubleW csq,
                            org.netlib.util.doubleW snq)
        
     ..

  Purpose
  =======

  DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
  that if ( UPPER ) then

            U'*A*Q = U'*( A1 A2 )*Q = ( x  0  )
                        ( 0  A3 )     ( x  x  )
  and
            V'*B*Q = V'*( B1 B2 )*Q = ( x  0  )
                        ( 0  B3 )     ( x  x  )

  or if ( .NOT.UPPER ) then

            U'*A*Q = U'*( A1 0  )*Q = ( x  x  )
                        ( A2 A3 )     ( 0  x  )
  and
            V'*B*Q = V'*( B1 0  )*Q = ( x  x  )
                        ( B2 B3 )     ( 0  x  )

  The rows of the transformed A and B are parallel, where

    U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
        ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )

  Z' denotes the transpose of Z.


  Arguments
  =========

  UPPER   (input) LOGICAL
          = .TRUE.: the input matrices A and B are upper triangular.
          = .FALSE.: the input matrices A and B are lower triangular.

  A1      (input) DOUBLE PRECISION
  A2      (input) DOUBLE PRECISION
  A3      (input) DOUBLE PRECISION
          On entry, A1, A2 and A3 are elements of the input 2-by-2
          upper (lower) triangular matrix A.

  B1      (input) DOUBLE PRECISION
  B2      (input) DOUBLE PRECISION
  B3      (input) DOUBLE PRECISION
          On entry, B1, B2 and B3 are elements of the input 2-by-2
          upper (lower) triangular matrix B.

  CSU     (output) DOUBLE PRECISION
  SNU     (output) DOUBLE PRECISION
          The desired orthogonal matrix U.

  CSV     (output) DOUBLE PRECISION
  SNV     (output) DOUBLE PRECISION
          The desired orthogonal matrix V.

  CSQ     (output) DOUBLE PRECISION
  SNQ     (output) DOUBLE PRECISION
          The desired orthogonal matrix Q.

  =====================================================================

     .. Parameters ..
        Parameters:
        upper -
        a1 -
        a2 -
        a3 -
        b1 -
        b2 -
        b3 -
        csu -
        snu -
        csv -
        snv -
        csq -
        snq -

      • dlagtf

        public abstract void dlagtf(int n,
                            double[] a,
                            double lambda,
                            double[] b,
                            double[] c,
                            double tol,
                            double[] d,
                            int[] in,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
  tridiagonal matrix and lambda is a scalar, as

     T - lambda*I = PLU,

  where P is a permutation matrix, L is a unit lower tridiagonal matrix
  with at most one non-zero sub-diagonal elements per column and U is
  an upper triangular matrix with at most two non-zero super-diagonal
  elements per column.

  The factorization is obtained by Gaussian elimination with partial
  pivoting and implicit row scaling.

  The parameter LAMBDA is included in the routine so that DLAGTF may
  be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
  inverse iteration.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix T.

  A       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, A must contain the diagonal elements of T.

          On exit, A is overwritten by the n diagonal elements of the
          upper triangular matrix U of the factorization of T.

  LAMBDA  (input) DOUBLE PRECISION
          On entry, the scalar lambda.

  B       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, B must contain the (n-1) super-diagonal elements of
          T.

          On exit, B is overwritten by the (n-1) super-diagonal
          elements of the matrix U of the factorization of T.

  C       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, C must contain the (n-1) sub-diagonal elements of
          T.

          On exit, C is overwritten by the (n-1) sub-diagonal elements

          of the matrix L of the factorization of T.

  TOL     (input) DOUBLE PRECISION
          On entry, a relative tolerance used to indicate whether or
          not the matrix (T - lambda*I) is nearly singular. TOL should

          normally be chose as approximately the largest relative error
          in the elements of T. For example, if the elements of T are
          correct to about 4 significant figures, then TOL should be
          set to about 5*10**(-4). If TOL is supplied as less than eps,
          where eps is the relative machine precision, then the value
          eps is used in place of TOL.

  D       (output) DOUBLE PRECISION array, dimension (N-2)
          On exit, D is overwritten by the (n-2) second super-diagonal

          elements of the matrix U of the factorization of T.

  IN      (output) INTEGER array, dimension (N)
          On exit, IN contains details of the permutation matrix P. If

          an interchange occurred at the kth step of the elimination,
          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
          returns the smallest positive integer j such that

             abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,

          where norm( A(j) ) denotes the sum of the absolute values of

          the jth row of the matrix A. If no such j exists then IN(n)
          is returned as zero. If IN(n) is returned as positive, then a
          diagonal element of U is small, indicating that
          (T - lambda*I) is singular or nearly singular,

  INFO    (output) INTEGER
          = 0   : successful exit
          .lt. 0: if INFO = -k, the kth argument had an illegal value

 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        a -
        lambda -
        b -
        c -
        tol -
        d -
        in -
        info -

      • dlagtm

        public abstract void dlagtm(java.lang.String trans,
                            int n,
                            int nrhs,
                            double alpha,
                            double[] dl,
                            double[] d,
                            double[] du,
                            double[] x,
                            int ldx,
                            double beta,
                            double[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  DLAGTM performs a matrix-vector product of the form

     B := alpha * A * X + beta * B

  where A is a tridiagonal matrix of order N, B and X are N by NRHS
  matrices, and alpha and beta are real scalars, each of which may be
  0., 1., or -1.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  No transpose, B := alpha * A * X + beta * B
          = 'T':  Transpose,    B := alpha * A'* X + beta * B
          = 'C':  Conjugate transpose = Transpose

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices X and B.

  ALPHA   (input) DOUBLE PRECISION
          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
          it is assumed to be 0.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) sub-diagonal elements of T.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of T.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) super-diagonal elements of T.

  X       (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
          The N by NRHS matrix X.
  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(N,1).

  BETA    (input) DOUBLE PRECISION
          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
          it is assumed to be 1.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix B.
          On exit, B is overwritten by the matrix expression
          B := alpha * A * X + beta * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(N,1).

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        alpha -
        dl -
        d -
        du -
        x -
        ldx -
        beta -
        b -
        ldb -

      • dlagts

        public abstract void dlagts(int job,
                            int n,
                            double[] a,
                            double[] b,
                            double[] c,
                            double[] d,
                            int[] in,
                            double[] y,
                            org.netlib.util.doubleW tol,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAGTS may be used to solve one of the systems of equations

     (T - lambda*I)*x = y   or   (T - lambda*I)'*x = y,

  where T is an n by n tridiagonal matrix, for x, following the
  factorization of (T - lambda*I) as

     (T - lambda*I) = P*L*U ,

  by routine DLAGTF. The choice of equation to be solved is
  controlled by the argument JOB, and in each case there is an option
  to perturb zero or very small diagonal elements of U, this option
  being intended for use in applications such as inverse iteration.

  Arguments
  =========

  JOB     (input) INTEGER
          Specifies the job to be performed by DLAGTS as follows:
          =  1: The equations  (T - lambda*I)x = y  are to be solved,
                but diagonal elements of U are not to be perturbed.
          = -1: The equations  (T - lambda*I)x = y  are to be solved
                and, if overflow would otherwise occur, the diagonal
                elements of U are to be perturbed. See argument TOL
                below.
          =  2: The equations  (T - lambda*I)'x = y  are to be solved,

                but diagonal elements of U are not to be perturbed.
          = -2: The equations  (T - lambda*I)'x = y  are to be solved
                and, if overflow would otherwise occur, the diagonal
                elements of U are to be perturbed. See argument TOL
                below.

  N       (input) INTEGER
          The order of the matrix T.

  A       (input) DOUBLE PRECISION array, dimension (N)
          On entry, A must contain the diagonal elements of U as
          returned from DLAGTF.

  B       (input) DOUBLE PRECISION array, dimension (N-1)
          On entry, B must contain the first super-diagonal elements of
          U as returned from DLAGTF.

  C       (input) DOUBLE PRECISION array, dimension (N-1)
          On entry, C must contain the sub-diagonal elements of L as
          returned from DLAGTF.

  D       (input) DOUBLE PRECISION array, dimension (N-2)
          On entry, D must contain the second super-diagonal elements
          of U as returned from DLAGTF.

  IN      (input) INTEGER array, dimension (N)
          On entry, IN must contain details of the matrix P as returned
          from DLAGTF.

  Y       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side vector y.
          On exit, Y is overwritten by the solution vector x.

  TOL     (input/output) DOUBLE PRECISION
          On entry, with  JOB .lt. 0, TOL should be the minimum
          perturbation to be made to very small diagonal elements of U.
          TOL should normally be chosen as about eps*norm(U), where eps
          is the relative machine precision, but if TOL is supplied as

          non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
          If  JOB .gt. 0  then TOL is not referenced.

          On exit, TOL is changed as described above, only if TOL is
          non-positive on entry. Otherwise TOL is unchanged.

  INFO    (output) INTEGER
          = 0   : successful exit
          .lt. 0: if INFO = -i, the i-th argument had an illegal value

          .gt. 0: overflow would occur when computing the INFO(th)
                  element of the solution vector x. This can only occur
                  when JOB is supplied as positive and either means
                  that a diagonal element of U is very small, or that
                  the elements of the right-hand side vector y are very
                  large.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        b -
        c -
        d -
        in -
        y -
        tol -
        info -

      • dlagv2

        public abstract void dlagv2(double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] alphar,
                            double[] alphai,
                            double[] beta,
                            org.netlib.util.doubleW csl,
                            org.netlib.util.doubleW snl,
                            org.netlib.util.doubleW csr,
                            org.netlib.util.doubleW snr)
        
     ..

  Purpose
  =======

  DLAGV2 computes the Generalized Schur factorization of a real 2-by-2

  matrix pencil (A,B) where B is upper triangular. This routine
  computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
  SNR such that

  1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
     types), then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],

  2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
     then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]

     where b11 >= b22 > 0.


  Arguments
  =========

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.
          On exit, A is overwritten by the ``A-part'' of the
          generalized Schur form.

  LDA     (input) INTEGER
          THe leading dimension of the array A.  LDA >= 2.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, 2)
          On entry, the upper triangular 2 x 2 matrix B.
          On exit, B is overwritten by the ``B-part'' of the
          generalized Schur form.

  LDB     (input) INTEGER
          THe leading dimension of the array B.  LDB >= 2.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (2)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (2)
  BETA    (output) DOUBLE PRECISION array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
          be zero.

  CSL     (output) DOUBLE PRECISION
          The cosine of the left rotation matrix.

  SNL     (output) DOUBLE PRECISION
          The sine of the left rotation matrix.

  CSR     (output) DOUBLE PRECISION
          The cosine of the right rotation matrix.

  SNR     (output) DOUBLE PRECISION
          The sine of the right rotation matrix.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        csl -
        snl -
        csr -
        snr -

      • dlahqr

        public abstract void dlahqr(boolean wantt,
                            boolean wantz,
                            int n,
                            int ilo,
                            int ihi,
                            double[] h,
                            int ldh,
                            double[] wr,
                            double[] wi,
                            int iloz,
                            int ihiz,
                            double[] z,
                            int ldz,
                            org.netlib.util.intW info)
        
     ..

     Purpose
     =======

     DLAHQR is an auxiliary routine called by DHSEQR to update the
     eigenvalues and Schur decomposition already computed by DHSEQR, by
     dealing with the Hessenberg submatrix in rows and columns ILO to
     IHI.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N       (input) INTEGER
          The order of the matrix H.  N >= 0.

     ILO     (input) INTEGER
     IHI     (input) INTEGER
          It is assumed that H is already upper quasi-triangular in
          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
          ILO = 1). DLAHQR works primarily with the Hessenberg
          submatrix in rows and columns ILO to IHI, but applies
          transformations to all of H if WANTT is .TRUE..
          1 <= ILO <= max(1,IHI); IHI <= N.

     H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.
          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
          quasi-triangular in rows and columns ILO:IHI, with any
          2-by-2 diagonal blocks in standard form. If INFO is zero
          and WANTT is .FALSE., the contents of H are unspecified on
          exit.  The output state of H if INFO is nonzero is given
          below under the description of INFO.

     LDH     (input) INTEGER
          The leading dimension of the array H. LDH >= max(1,N).

     WR      (output) DOUBLE PRECISION array, dimension (N)
     WI      (output) DOUBLE PRECISION array, dimension (N)
          The real and imaginary parts, respectively, of the computed
          eigenvalues ILO to IHI are stored in the corresponding
          elements of WR and WI. If two eigenvalues are computed as a
          complex conjugate pair, they are stored in consecutive
          elements of WR and WI, say the i-th and (i+1)th, with
          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
          eigenvalues are stored in the same order as on the diagonal
          of the Schur form returned in H, with WR(i) = H(i,i), and, if
          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE..
          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

     Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          If WANTZ is .TRUE., on entry Z must contain the current
          matrix Z of transformations accumulated by DHSEQR, and on
          exit Z has been updated; transformations are applied only to

          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
          If WANTZ is .FALSE., Z is not referenced.

     LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= max(1,N).

     INFO    (output) INTEGER
           =   0: successful exit
          .GT. 0: If INFO = i, DLAHQR failed to compute all the
                  eigenvalues ILO to IHI in a total of 30 iterations
                  per eigenvalue; elements i+1:ihi of WR and WI
                  contain those eigenvalues which have been
                  successfully computed.

                  If INFO .GT. 0 and WANTT is .FALSE., then on exit,
                  the remaining unconverged eigenvalues are the
                  eigenvalues of the upper Hessenberg matrix rows
                  and columns ILO thorugh INFO of the final, output
                  value of H.

                  If INFO .GT. 0 and WANTT is .TRUE., then on exit
          (*)       (initial value of H)*U  = U*(final value of H)
                  where U is an orthognal matrix.    The final
                  value of H is upper Hessenberg and triangular in
                  rows and columns INFO+1 through IHI.

                  If INFO .GT. 0 and WANTZ is .TRUE., then on exit
                      (final value of Z)  = (initial value of Z)*U
                  where U is the orthogonal matrix in (*)
                  (regardless of the value of WANTT.)

     Further Details
     ===============

     02-96 Based on modifications by
     David Day, Sandia National Laboratory, USA

     12-04 Further modifications by
     Ralph Byers, University of Kansas, USA

       This is a modified version of DLAHQR from LAPACK version 3.0.
       It is (1) more robust against overflow and underflow and
       (2) adopts the more conservative Ahues & Tisseur stopping
       criterion (LAWN 122, 1997).

     =========================================================

     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        info -

      • dlahr2

        public void dlahr2(int n,
                   int k,
                   int nb,
                   double[] a,
                   int lda,
                   double[] tau,
                   double[] t,
                   int ldt,
                   double[] y,
                   int ldy)
        
     ..

  Purpose
  =======

  DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
  matrix A so that elements below the k-th subdiagonal are zero. The
  reduction is performed by an orthogonal similarity transformation
  Q' * A * Q. The routine returns the matrices V and T which determine

  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.

  This is an auxiliary routine called by DGEHRD.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  K       (input) INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.

  NB      (input) INTEGER
          The number of columns to be reduced.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)

          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a

          product of elementary reflectors. The other columns of A are

          unchanged. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further

          Details.

  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
          The upper triangular matrix T.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= NB.

  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
          The n-by-nb matrix Y.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrix Q is represented as a product of nb elementary reflectors


     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix

  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V') * (A - Y*V').

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  This file is a slight modification of LAPACK-3.0's DLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK routine. This function is
  not backward compatible with LAPACK3.0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        k -
        nb -
        a -
        lda -
        tau -
        t -
        ldt -
        y -
        ldy -

      • dlahrd

        public abstract void dlahrd(int n,
                            int k,
                            int nb,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] t,
                            int ldt,
                            double[] y,
                            int ldy)
        
     ..

  Purpose
  =======

  DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
  matrix A so that elements below the k-th subdiagonal are zero. The
  reduction is performed by an orthogonal similarity transformation
  Q' * A * Q. The routine returns the matrices V and T which determine

  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.

  This is an OBSOLETE auxiliary routine. 
  This routine will be 'deprecated' in a  future release.
  Please use the new routine DLAHR2 instead.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  K       (input) INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.

  NB      (input) INTEGER
          The number of columns to be reduced.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)

          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a

          product of elementary reflectors. The other columns of A are

          unchanged. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further

          Details.

  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
          The upper triangular matrix T.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= NB.

  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
          The n-by-nb matrix Y.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrix Q is represented as a product of nb elementary reflectors


     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix

  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V') * (A - Y*V').

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        k -
        nb -
        a -
        lda -
        tau -
        t -
        ldt -
        y -
        ldy -

      • dlaic1

        public abstract void dlaic1(int job,
                            int j,
                            double[] x,
                            double sest,
                            double[] w,
                            double gamma,
                            org.netlib.util.doubleW sestpr,
                            org.netlib.util.doubleW s,
                            org.netlib.util.doubleW c)
        
     ..

  Purpose
  =======

  DLAIC1 applies one step of incremental condition estimation in
  its simplest version:

  Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
  lower triangular matrix L, such that
           twonorm(L*x) = sest
  Then DLAIC1 computes sestpr, s, c such that
  the vector
                  [ s*x ]
           xhat = [  c  ]
  is an approximate singular vector of
                  [ L     0  ]
           Lhat = [ w' gamma ]
  in the sense that
           twonorm(Lhat*xhat) = sestpr.

  Depending on JOB, an estimate for the largest or smallest singular
  value is computed.

  Note that [s c]' and sestpr**2 is an eigenpair of the system

      diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                            [ gamma ]

  where  alpha =  x'*w.

  Arguments
  =========

  JOB     (input) INTEGER
          = 1: an estimate for the largest singular value is computed.

          = 2: an estimate for the smallest singular value is computed.

  J       (input) INTEGER
          Length of X and W

  X       (input) DOUBLE PRECISION array, dimension (J)
          The j-vector x.

  SEST    (input) DOUBLE PRECISION
          Estimated singular value of j by j matrix L

  W       (input) DOUBLE PRECISION array, dimension (J)
          The j-vector w.

  GAMMA   (input) DOUBLE PRECISION
          The diagonal element gamma.

  SESTPR  (output) DOUBLE PRECISION
          Estimated singular value of (j+1) by (j+1) matrix Lhat.

  S       (output) DOUBLE PRECISION
          Sine needed in forming xhat.

  C       (output) DOUBLE PRECISION
          Cosine needed in forming xhat.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        j -
        x -
        sest -
        w -
        gamma -
        sestpr -
        s -
        c -

      • dlaisnan

        public boolean dlaisnan(double din1,
                        double din2)
        
     ..

  Purpose
  =======

  This routine is not for general use.  It exists solely to avoid
  over-optimization in DISNAN.

  DLAISNAN checks for NaNs by comparing its two arguments for
  inequality.  NaN is the only floating-point value where NaN != NaN
  returns .TRUE.  To check for NaNs, pass the same variable as both
  arguments.

  Strictly speaking, Fortran does not allow aliasing of function
  arguments. So a compiler must assume that the two arguments are
  not the same variable, and the test will not be optimized away.
  Interprocedural or whole-program optimization may delete this
  test.  The ISNAN functions will be replaced by the correct
  Fortran 03 intrinsic once the intrinsic is widely available.

  Arguments
  =========

  DIN1     (input) DOUBLE PRECISION
  DIN2     (input) DOUBLE PRECISION
          Two numbers to compare for inequality.

  =====================================================================

  .. Executable Statements ..
        Parameters:
        din1 -
        din2 -

      • dlaln2

        public abstract void dlaln2(boolean ltrans,
                            int na,
                            int nw,
                            double smin,
                            double ca,
                            double[] a,
                            int lda,
                            double d1,
                            double d2,
                            double[] b,
                            int ldb,
                            double wr,
                            double wi,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW scale,
                            org.netlib.util.doubleW xnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLALN2 solves a system of the form  (ca A - w D ) X = s B
  or (ca A' - w D) X = s B   with possible scaling ("s") and
  perturbation of A.  (A' means A-transpose.)

  A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
  real diagonal matrix, w is a real or complex value, and X and B are
  NA x 1 matrices -- real if w is real, complex if w is complex.  NA
  may be 1 or 2.

  If w is complex, X and B are represented as NA x 2 matrices,
  the first column of each being the real part and the second
  being the imaginary part.

  "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
  so chosen that X can be computed without overflow.  X is further
  scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
  than overflow.

  If both singular values of (ca A - w D) are less than SMIN,
  SMIN*identity will be used instead of (ca A - w D).  If only one
  singular value is less than SMIN, one element of (ca A - w D) will be
  perturbed enough to make the smallest singular value roughly SMIN.
  If both singular values are at least SMIN, (ca A - w D) will not be
  perturbed.  In any case, the perturbation will be at most some small

  multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
  are computed by infinity-norm approximations, and thus will only be
  correct to a factor of 2 or so.

  Note: all input quantities are assumed to be smaller than overflow
  by a reasonable factor.  (See BIGNUM.)

  Arguments
  ==========

  LTRANS  (input) LOGICAL
          =.TRUE.:  A-transpose will be used.
          =.FALSE.: A will be used (not transposed.)

  NA      (input) INTEGER
          The size of the matrix A.  It may (only) be 1 or 2.

  NW      (input) INTEGER
          1 if "w" is real, 2 if "w" is complex.  It may only be 1
          or 2.

  SMIN    (input) DOUBLE PRECISION
          The desired lower bound on the singular values of A.  This
          should be a safe distance away from underflow or overflow,
          say, between (underflow/machine precision) and  (machine
          precision * overflow ).  (See BIGNUM and ULP.)

  CA      (input) DOUBLE PRECISION
          The coefficient c, which A is multiplied by.

  A       (input) DOUBLE PRECISION array, dimension (LDA,NA)
          The NA x NA matrix A.

  LDA     (input) INTEGER
          The leading dimension of A.  It must be at least NA.

  D1      (input) DOUBLE PRECISION
          The 1,1 element in the diagonal matrix D.

  D2      (input) DOUBLE PRECISION
          The 2,2 element in the diagonal matrix D.  Not used if NW=1.


  B       (input) DOUBLE PRECISION array, dimension (LDB,NW)
          The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
          complex), column 1 contains the real part of B and column 2
          contains the imaginary part.

  LDB     (input) INTEGER
          The leading dimension of B.  It must be at least NA.

  WR      (input) DOUBLE PRECISION
          The real part of the scalar "w".

  WI      (input) DOUBLE PRECISION
          The imaginary part of the scalar "w".  Not used if NW=1.

  X       (output) DOUBLE PRECISION array, dimension (LDX,NW)
          The NA x NW matrix X (unknowns), as computed by DLALN2.
          If NW=2 ("w" is complex), on exit, column 1 will contain
          the real part of X and column 2 will contain the imaginary
          part.

  LDX     (input) INTEGER
          The leading dimension of X.  It must be at least NA.

  SCALE   (output) DOUBLE PRECISION
          The scale factor that B must be multiplied by to insure
          that overflow does not occur when computing X.  Thus,
          (ca A - w D) X  will be SCALE*B, not B (ignoring
          perturbations of A.)  It will be at most 1.

  XNORM   (output) DOUBLE PRECISION
          The infinity-norm of X, when X is regarded as an NA x NW
          real matrix.

  INFO    (output) INTEGER
          An error flag.  It will be set to zero if no error occurs,
          a negative number if an argument is in error, or a positive
          number if  ca A - w D  had to be perturbed.
          The possible values are:
          = 0: No error occurred, and (ca A - w D) did not have to be
                 perturbed.
          = 1: (ca A - w D) had to be perturbed to make its smallest
               (or only) singular value greater than SMIN.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltrans -
        na -
        nw -
        smin -
        ca -
        a -
        lda -
        d1 -
        d2 -
        b -
        ldb -
        wr -
        wi -
        x -
        ldx -
        scale -
        xnorm -
        info -

      • dlals0

        public abstract void dlals0(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            int nrhs,
                            double[] b,
                            int ldb,
                            double[] bx,
                            int ldbx,
                            int[] perm,
                            int givptr,
                            int[] givcol,
                            int ldgcol,
                            double[] givnum,
                            int ldgnum,
                            double[] poles,
                            double[] difl,
                            double[] difr,
                            double[] z,
                            int k,
                            double c,
                            double s,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLALS0 applies back the multiplying factors of either the left or the
  right singular vector matrix of a diagonal matrix appended by a row
  to the right hand side matrix B in solving the least squares problem

  using the divide-and-conquer SVD approach.

  For the left singular vector matrix, three types of orthogonal
  matrices are involved:

  (1L) Givens rotations: the number of such rotations is GIVPTR; the
       pairs of columns/rows they were applied to are stored in GIVCOL;
       and the C- and S-values of these rotations are stored in GIVNUM.

  (2L) Permutation. The (NL+1)-st row of B is to be moved to the first

       row, and for J=2:N, PERM(J)-th row of B is to be moved to the
       J-th row.

  (3L) The left singular vector matrix of the remaining matrix.

  For the right singular vector matrix, four types of orthogonal
  matrices are involved:

  (1R) The right singular vector matrix of the remaining matrix.

  (2R) If SQRE = 1, one extra Givens rotation to generate the right
       null space.

  (3R) The inverse transformation of (2L).

  (4R) The inverse transformation of (1L).

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Left singular vector matrix.
         = 1: Right singular vector matrix.

  NL     (input) INTEGER
         The row dimension of the upper block. NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block. NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  NRHS   (input) INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

  B      (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M. On output, B contains
         the solution X in rows 1 through N.

  LDB    (input) INTEGER
         The leading dimension of B. LDB must be at least
         max(1,MAX( M, N ) ).

  BX     (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS )

  LDBX   (input) INTEGER
         The leading dimension of BX.

  PERM   (input) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) applied
         to the two blocks.

  GIVPTR (input) INTEGER
         The number of Givens rotations which took place in this
         subproblem.

  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of rows/columns
         involved in a Givens rotation.

  LDGCOL (input) INTEGER
         The leading dimension of GIVCOL, must be at least N.

  GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value used in the
         corresponding Givens rotation.

  LDGNUM (input) INTEGER
         The leading dimension of arrays DIFR, POLES and
         GIVNUM, must be at least K.

  POLES  (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         On entry, POLES(1:K, 1) contains the new singular
         values obtained from solving the secular equation, and
         POLES(1:K, 2) is an array containing the poles in the secular

         equation.

  DIFL   (input) DOUBLE PRECISION array, dimension ( K ).
         On entry, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

  DIFR   (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
         On entry, DIFR(I, 1) contains the distances between I-th
         updated (undeflated) singular value and the I+1-th
         (undeflated) old singular value. And DIFR(I, 2) is the
         normalizing factor for the I-th right singular vector.

  Z      (input) DOUBLE PRECISION array, dimension ( K )
         Contain the components of the deflation-adjusted updating row

         vector.

  K      (input) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  C      (input) DOUBLE PRECISION
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (input) DOUBLE PRECISION
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  WORK   (workspace) DOUBLE PRECISION array, dimension ( K )

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        nrhs -
        b -
        ldb -
        bx -
        ldbx -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        poles -
        difl -
        difr -
        z -
        k -
        c -
        s -
        work -
        info -

      • dlalsa

        public abstract void dlalsa(int icompq,
                            int smlsiz,
                            int n,
                            int nrhs,
                            double[] b,
                            int ldb,
                            double[] bx,
                            int ldbx,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int[] k,
                            double[] difl,
                            double[] difr,
                            double[] z,
                            double[] poles,
                            int[] givptr,
                            int[] givcol,
                            int ldgcol,
                            int[] perm,
                            double[] givnum,
                            double[] c,
                            double[] s,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLALSA is an itermediate step in solving the least squares problem
  by computing the SVD of the coefficient matrix in compact form (The
  singular vectors are computed as products of simple orthorgonal
  matrices.).

  If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
  matrix of an upper bidiagonal matrix to the right hand side; and if
  ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
  right hand side. The singular vector matrices were generated in
  compact form by DLALSA.

  Arguments
  =========


  ICOMPQ (input) INTEGER
         Specifies whether the left or the right singular vector
         matrix is involved.
         = 0: Left singular vector matrix
         = 1: Right singular vector matrix

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The row and column dimensions of the upper bidiagonal matrix.


  NRHS   (input) INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

  B      (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M.
         On output, B contains the solution X in rows 1 through N.

  LDB    (input) INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,MAX( M, N ) ).

  BX     (output) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
         On exit, the result of applying the left or right singular
         vector matrix to B.

  LDBX   (input) INTEGER
         The leading dimension of BX.

  U      (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
         On entry, U contains the left singular vector matrices of all

         subproblems at the bottom level.

  LDU    (input) INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR,
         POLES, GIVNUM, and Z.

  VT     (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
         On entry, VT' contains the right singular vector matrices of
         all subproblems at the bottom level.

  K      (input) INTEGER array, dimension ( N ).

  DIFL   (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

  DIFR   (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
         distances between singular values on the I-th level and
         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
         record the normalizing factors of the right singular vectors
         matrices of subproblems on I-th level.

  Z      (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
         On entry, Z(1, I) contains the components of the deflation-
         adjusted updating row vector for subproblems on the I-th
         level.

  POLES  (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
         singular values involved in the secular equations on the I-th

         level.

  GIVPTR (input) INTEGER array, dimension ( N ).
         On entry, GIVPTR( I ) records the number of Givens
         rotations performed on the I-th problem on the computation
         tree.

  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the

         locations of Givens rotations performed on the I-th level on
         the computation tree.

  LDGCOL (input) INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

  PERM   (input) INTEGER array, dimension ( LDGCOL, NLVL ).
         On entry, PERM(*, I) records permutations done on the I-th
         level of the computation tree.

  GIVNUM (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
         values of Givens rotations performed on the I-th level on the

         computation tree.

  C      (input) DOUBLE PRECISION array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  S      (input) DOUBLE PRECISION array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         S( I ) contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  WORK   (workspace) DOUBLE PRECISION array.
         The dimension must be at least N.

  IWORK  (workspace) INTEGER array.
         The dimension must be at least 3 * N

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        smlsiz -
        n -
        nrhs -
        b -
        ldb -
        bx -
        ldbx -
        u -
        ldu -
        vt -
        k -
        difl -
        difr -
        z -
        poles -
        givptr -
        givcol -
        ldgcol -
        perm -
        givnum -
        c -
        s -
        work -
        iwork -
        info -

      • dlalsd

        public abstract void dlalsd(java.lang.String uplo,
                            int smlsiz,
                            int n,
                            int nrhs,
                            double[] d,
                            double[] e,
                            double[] b,
                            int ldb,
                            double rcond,
                            org.netlib.util.intW rank,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLALSD uses the singular value decomposition of A to solve the least

  squares problem of finding X to minimize the Euclidean norm of each
  column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
  are N-by-NRHS. The solution X overwrites B.

  The singular values of A smaller than RCOND times the largest
  singular value are treated as zero in solving the least squares
  problem; in this case a minimum norm solution is returned.
  The actual singular values are returned in D in ascending order.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  Arguments
  =========

  UPLO   (input) CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

  NRHS   (input) INTEGER
         The number of columns of B. NRHS must be at least 1.

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.


  E      (input/output) DOUBLE PRECISION array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.

         On exit, E has been destroyed.

  B      (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

  LDB    (input) INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

  RCOND  (input) DOUBLE PRECISION
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

  RANK   (output) INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

  WORK   (workspace) DOUBLE PRECISION array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

  IWORK  (workspace) INTEGER array, dimension at least
         (3*N*NLVL + 11*N)

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute an singular value while

               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        smlsiz -
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        rcond -
        rank -
        work -
        iwork -
        info -

      • dlamrg

        public abstract void dlamrg(int n1,
                            int n2,
                            double[] a,
                            int dtrd1,
                            int dtrd2,
                            int[] index)
        
     ..

  Purpose
  =======

  DLAMRG will create a permutation list which will merge the elements
  of A (which is composed of two independently sorted sets) into a
  single set which is sorted in ascending order.

  Arguments
  =========

  N1     (input) INTEGER
  N2     (input) INTEGER
         These arguements contain the respective lengths of the two
         sorted lists to be merged.

  A      (input) DOUBLE PRECISION array, dimension (N1+N2)
         The first N1 elements of A contain a list of numbers which
         are sorted in either ascending or descending order.  Likewise

         for the final N2 elements.

  DTRD1  (input) INTEGER
  DTRD2  (input) INTEGER
         These are the strides to be taken through the array A.
         Allowable strides are 1 and -1.  They indicate whether a
         subset of A is sorted in ascending (DTRDx = 1) or descending
         (DTRDx = -1) order.

  INDEX  (output) INTEGER array, dimension (N1+N2)
         On exit this array will contain a permutation such that
         if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
         sorted in ascending order.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n1 -
        n2 -
        a -
        dtrd1 -
        dtrd2 -
        index -

      • dlaneg

        public int dlaneg(int n,
                  double[] d,
                  double[] lld,
                  double sigma,
                  double pivmin,
                  int r)
        
     ..

  Purpose
  =======

  DLANEG computes the Sturm count, the number of negative pivots
  encountered while factoring tridiagonal T - sigma I = L D L^T.
  This implementation works directly on the factors without forming
  the tridiagonal matrix T.  The Sturm count is also the number of
  eigenvalues of T less than sigma.

  This routine is called from DLARRB.

  The current routine does not use the PIVMIN parameter but rather
  requires IEEE-754 propagation of Infinities and NaNs.  This
  routine also has no input range restrictions but does require
  default exception handling such that x/0 produces Inf when x is
  non-zero, and Inf/Inf produces NaN.  For more information, see:

    Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
    Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on
    Scientific Computing, v28, n5, 2006.  DOI 10.1137/050641624
    (Tech report version in LAWN 172 with the same title.)

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  LLD     (input) DOUBLE PRECISION array, dimension (N-1)
          The (N-1) elements L(i)*L(i)*D(i).

  SIGMA   (input) DOUBLE PRECISION
          Shift amount in T - sigma I = L D L^T.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence.  May be used
          when zero pivots are encountered on non-IEEE-754
          architectures.

  R       (input) INTEGER
          The twist index for the twisted factorization that is used
          for the negcount.

  Further Details
  ===============

  Based on contributions by
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Jason Riedy, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        lld -
        sigma -
        pivmin -
        r -

      • dlangb

        public double dlangb(java.lang.String norm,
                     int n,
                     int kl,
                     int ku,
                     double[] ab,
                     int ldab,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANGB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n band matrix  A,  with kl sub-diagonals and ku super-diagonals.

  Description
  ===========

  DLANGB returns the value

     DLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANGB as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANGB is
          set to zero.

  KL      (input) INTEGER
          The number of sub-diagonals of the matrix A.  KL >= 0.

  KU      (input) INTEGER
          The number of super-diagonals of the matrix A.  KU >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================



     .. Parameters ..
        Parameters:
        norm -
        n -
        kl -
        ku -
        ab -
        ldab -
        work -

      • dlange

        public double dlange(java.lang.String norm,
                     int m,
                     int n,
                     double[] a,
                     int lda,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANGE  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real matrix A.

  Description
  ===========

  DLANGE returns the value

     DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANGE as described
          above.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.  When M = 0,
          DLANGE is set to zero.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.  When N = 0,

          DLANGE is set to zero.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The m by n matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        m -
        n -
        a -
        lda -
        work -

      • dlangt

        public double dlangt(java.lang.String norm,
                     int n,
                     double[] dl,
                     double[] d,
                     double[] du)
        
     ..

  Purpose
  =======

  DLANGT  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real tridiagonal matrix A.

  Description
  ===========

  DLANGT returns the value

     DLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANGT as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANGT is
          set to zero.

  DL      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) sub-diagonal elements of A.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of A.

  DU      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) super-diagonal elements of A.

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        dl -
        d -
        du -

      • dlanhs

        public double dlanhs(java.lang.String norm,
                     int n,
                     double[] a,
                     int lda,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANHS  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  Hessenberg matrix A.

  Description
  ===========

  DLANHS returns the value

     DLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANHS as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANHS is
          set to zero.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The n by n upper Hessenberg matrix A; the part of A below the
          first sub-diagonal is not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        n -
        a -
        lda -
        work -

      • dlansb

        public double dlansb(java.lang.String norm,
                     java.lang.String uplo,
                     int n,
                     int k,
                     double[] ab,
                     int ldab,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANSB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n symmetric band matrix A,  with k super-diagonals.

  Description
  ===========

  DLANSB returns the value

     DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANSB as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          band matrix A is supplied.
          = 'U':  Upper triangular part is supplied
          = 'L':  Lower triangular part is supplied

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANSB is
          set to zero.

  K       (input) INTEGER
          The number of super-diagonals or sub-diagonals of the
          band matrix A.  K >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first K+1 rows of AB.  The j-th column of A is

          stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        k -
        ab -
        ldab -
        work -

      • dlansp

        public double dlansp(java.lang.String norm,
                     java.lang.String uplo,
                     int n,
                     double[] ap,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANSP  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric matrix A,  supplied in packed form.

  Description
  ===========

  DLANSP returns the value

     DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANSP as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is supplied.
          = 'U':  Upper triangular part of A is supplied
          = 'L':  Lower triangular part of A is supplied

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANSP is
          set to zero.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        ap -
        work -

      • dlanst

        public double dlanst(java.lang.String norm,
                     int n,
                     double[] d,
                     double[] e)
        
     ..

  Purpose
  =======

  DLANST  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric tridiagonal matrix A.

  Description
  ===========

  DLANST returns the value

     DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANST as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANST is
          set to zero.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of A.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) sub-diagonal or super-diagonal elements of A.

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        d -
        e -

      • dlansy

        public double dlansy(java.lang.String norm,
                     java.lang.String uplo,
                     int n,
                     double[] a,
                     int lda,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANSY  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric matrix A.

  Description
  ===========

  DLANSY returns the value

     DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANSY as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is to be referenced.
          = 'U':  Upper triangular part of A is referenced
          = 'L':  Lower triangular part of A is referenced

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANSY is
          set to zero.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading n by n
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading n by n lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        a -
        lda -
        work -

      • dlantb

        public double dlantb(java.lang.String norm,
                     java.lang.String uplo,
                     java.lang.String diag,
                     int n,
                     int k,
                     double[] ab,
                     int ldab,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANTB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n triangular band matrix A,  with ( k + 1 ) diagonals.

  Description
  ===========

  DLANTB returns the value

     DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANTB as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANTB is
          set to zero.

  K       (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals of the matrix A if UPLO = 'L'.
          K >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first k+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
          Note that when DIAG = 'U', the elements of the array AB
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        k -
        ab -
        ldab -
        work -

      • dlantp

        public double dlantp(java.lang.String norm,
                     java.lang.String uplo,
                     java.lang.String diag,
                     int n,
                     double[] ap,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANTP  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  triangular matrix A, supplied in packed form.

  Description
  ===========

  DLANTP returns the value

     DLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANTP as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, DLANTP is
          set to zero.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          Note that when DIAG = 'U', the elements of the array AP
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        ap -
        work -

      • dlantr

        public double dlantr(java.lang.String norm,
                     java.lang.String uplo,
                     java.lang.String diag,
                     int m,
                     int n,
                     double[] a,
                     int lda,
                     double[] work)
        
     ..

  Purpose
  =======

  DLANTR  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  trapezoidal or triangular matrix A.

  Description
  ===========

  DLANTR returns the value

     DLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in DLANTR as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower trapezoidal.
          = 'U':  Upper trapezoidal
          = 'L':  Lower trapezoidal
          Note that A is triangular instead of trapezoidal if M = N.

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A has unit diagonal.
          = 'N':  Non-unit diagonal
          = 'U':  Unit diagonal

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0, and if
          UPLO = 'U', M <= N.  When M = 0, DLANTR is set to zero.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0, and if
          UPLO = 'L', N <= M.  When N = 0, DLANTR is set to zero.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The trapezoidal matrix A (A is triangular if M = N).
          If UPLO = 'U', the leading m by n upper trapezoidal part of
          the array A contains the upper trapezoidal matrix, and the
          strictly lower triangular part of A is not referenced.
          If UPLO = 'L', the leading m by n lower trapezoidal part of
          the array A contains the lower trapezoidal matrix, and the
          strictly upper triangular part of A is not referenced.  Note

          that when DIAG = 'U', the diagonal elements of A are not
          referenced and are assumed to be one.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        m -
        n -
        a -
        lda -
        work -

      • dlanv2

        public abstract void dlanv2(org.netlib.util.doubleW a,
                            org.netlib.util.doubleW b,
                            org.netlib.util.doubleW c,
                            org.netlib.util.doubleW d,
                            org.netlib.util.doubleW rt1r,
                            org.netlib.util.doubleW rt1i,
                            org.netlib.util.doubleW rt2r,
                            org.netlib.util.doubleW rt2i,
                            org.netlib.util.doubleW cs,
                            org.netlib.util.doubleW sn)
        
     ..

  Purpose
  =======

  DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
  matrix in standard form:

       [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
       [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]

  where either
  1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
  2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
  conjugate eigenvalues.

  Arguments
  =========

  A       (input/output) DOUBLE PRECISION
  B       (input/output) DOUBLE PRECISION
  C       (input/output) DOUBLE PRECISION
  D       (input/output) DOUBLE PRECISION
          On entry, the elements of the input matrix.
          On exit, they are overwritten by the elements of the
          standardised Schur form.

  RT1R    (output) DOUBLE PRECISION
  RT1I    (output) DOUBLE PRECISION
  RT2R    (output) DOUBLE PRECISION
  RT2I    (output) DOUBLE PRECISION
          The real and imaginary parts of the eigenvalues. If the
          eigenvalues are a complex conjugate pair, RT1I > 0.

  CS      (output) DOUBLE PRECISION
  SN      (output) DOUBLE PRECISION
          Parameters of the rotation matrix.

  Further Details
  ===============

  Modified by V. Sima, Research Institute for Informatics, Bucharest,
  Romania, to reduce the risk of cancellation errors,
  when computing real eigenvalues, and to ensure, if possible, that
  abs(RT1R) >= abs(RT2R).

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        d -
        rt1r -
        rt1i -
        rt2r -
        rt2i -
        cs -
        sn -

      • dlapll

        public abstract void dlapll(int n,
                            double[] x,
                            int incx,
                            double[] y,
                            int incy,
                            org.netlib.util.doubleW ssmin)
        
     ..

  Purpose
  =======

  Given two column vectors X and Y, let

                       A = ( X Y ).

  The subroutine first computes the QR factorization of A = Q*R,
  and then computes the SVD of the 2-by-2 upper triangular matrix R.
  The smaller singular value of R is returned in SSMIN, which is used
  as the measurement of the linear dependency of the vectors X and Y.

  Arguments
  =========

  N       (input) INTEGER
          The length of the vectors X and Y.

  X       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          On entry, X contains the N-vector X.
          On exit, X is overwritten.

  INCX    (input) INTEGER
          The increment between successive elements of X. INCX > 0.

  Y       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          On entry, Y contains the N-vector Y.
          On exit, Y is overwritten.

  INCY    (input) INTEGER
          The increment between successive elements of Y. INCY > 0.

  SSMIN   (output) DOUBLE PRECISION
          The smallest singular value of the N-by-2 matrix A = ( X Y ).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        ssmin -

      • dlapmt

        public abstract void dlapmt(boolean forwrd,
                            int m,
                            int n,
                            double[] x,
                            int ldx,
                            int[] k)
        
     ..

  Purpose
  =======

  DLAPMT rearranges the columns of the M by N matrix X as specified
  by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
  If FORWRD = .TRUE.,  forward permutation:

       X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.

  If FORWRD = .FALSE., backward permutation:

       X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

  Arguments
  =========

  FORWRD  (input) LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

  M       (input) INTEGER
          The number of rows of the matrix X. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix X. N >= 0.

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X, LDX >= MAX(1,M).

  K       (input/output) INTEGER array, dimension (N)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        forwrd -
        m -
        n -
        x -
        ldx -
        k -

      • dlapy2

        public double dlapy2(double x,
                     double y)
        
     ..

  Purpose
  =======

  DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary

  overflow.

  Arguments
  =========

  X       (input) DOUBLE PRECISION
  Y       (input) DOUBLE PRECISION
          X and Y specify the values x and y.

  =====================================================================

     .. Parameters ..
        Parameters:
        x -
        y -

      • dlapy3

        public double dlapy3(double x,
                     double y,
                     double z)
        
     ..

  Purpose
  =======

  DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause
  unnecessary overflow.

  Arguments
  =========

  X       (input) DOUBLE PRECISION
  Y       (input) DOUBLE PRECISION
  Z       (input) DOUBLE PRECISION
          X, Y and Z specify the values x, y and z.

  =====================================================================

     .. Parameters ..
        Parameters:
        x -
        y -
        z -

      • dlaqgb

        public abstract void dlaqgb(int m,
                            int n,
                            int kl,
                            int ku,
                            double[] ab,
                            int ldab,
                            double[] r,
                            double[] c,
                            double rowcnd,
                            double colcnd,
                            double amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  DLAQGB equilibrates a general M by N band matrix A with KL
  subdiagonals and KU superdiagonals using the row and scaling factors

  in the vectors R and C.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, the equilibrated matrix, in the same storage format

          as A.  See EQUED for the form of the equilibrated matrix.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDA >= KL+KU+1.

  R       (input) DOUBLE PRECISION array, dimension (M)
          The row scale factors for A.

  C       (input) DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.

  ROWCND  (input) DOUBLE PRECISION
          Ratio of the smallest R(i) to the largest R(i).

  COLCND  (input) DOUBLE PRECISION
          Ratio of the smallest C(i) to the largest C(i).

  AMAX    (input) DOUBLE PRECISION
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if row or column scaling
  should be done based on the ratio of the row or column scaling
  factors.  If ROWCND < THRESH, row scaling is done, and if
  COLCND < THRESH, column scaling is done.

  LARGE and SMALL are threshold values used to decide if row scaling
  should be done based on the absolute size of the largest matrix
  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        equed -

      • dlaqge

        public abstract void dlaqge(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] r,
                            double[] c,
                            double rowcnd,
                            double colcnd,
                            double amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  DLAQGE equilibrates a general M by N matrix A using the row and
  column scaling factors in the vectors R and C.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M by N matrix A.
          On exit, the equilibrated matrix.  See EQUED for the form of

          the equilibrated matrix.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  R       (input) DOUBLE PRECISION array, dimension (M)
          The row scale factors for A.

  C       (input) DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.

  ROWCND  (input) DOUBLE PRECISION
          Ratio of the smallest R(i) to the largest R(i).

  COLCND  (input) DOUBLE PRECISION
          Ratio of the smallest C(i) to the largest C(i).

  AMAX    (input) DOUBLE PRECISION
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if row or column scaling
  should be done based on the ratio of the row or column scaling
  factors.  If ROWCND < THRESH, row scaling is done, and if
  COLCND < THRESH, column scaling is done.

  LARGE and SMALL are threshold values used to decide if row scaling
  should be done based on the absolute size of the largest matrix
  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        equed -

      • dlaqp2

        public abstract void dlaqp2(int m,
                            int n,
                            int offset,
                            double[] a,
                            int lda,
                            int[] jpvt,
                            double[] tau,
                            double[] vn1,
                            double[] vn2,
                            double[] work)
        
     ..

  Purpose
  =======

  DLAQP2 computes a QR factorization with column pivoting of
  the block A(OFFSET+1:M,1:N).
  The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0.

  OFFSET  (input) INTEGER
          The number of rows of the matrix A that must be pivoted
          but no factorized. OFFSET >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is 
          the triangular factor obtained; the elements in block
          A(OFFSET+1:M,1:N) below the diagonal, together with the
          array TAU, represent the orthogonal matrix Q as a product of

          elementary reflectors. Block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  VN1     (input/output) DOUBLE PRECISION array, dimension (N)
          The vector with the partial column norms.

  VN2     (input/output) DOUBLE PRECISION array, dimension (N)
          The vector with the exact column norms.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  Further Details
  ===============

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.
  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        offset -
        a -
        lda -
        jpvt -
        tau -
        vn1 -
        vn2 -
        work -

      • dlaqps

        public abstract void dlaqps(int m,
                            int n,
                            int offset,
                            int nb,
                            org.netlib.util.intW kb,
                            double[] a,
                            int lda,
                            int[] jpvt,
                            double[] tau,
                            double[] vn1,
                            double[] vn2,
                            double[] auxv,
                            double[] f,
                            int ldf)
        
     ..

  Purpose
  =======

  DLAQPS computes a step of QR factorization with column pivoting
  of a real M-by-N matrix A by using Blas-3.  It tries to factorize
  NB columns from A starting from the row OFFSET+1, and updates all
  of the matrix with Blas-3 xGEMM.

  In some cases, due to catastrophic cancellations, it cannot
  factorize NB columns.  Hence, the actual number of factorized
  columns is returned in KB.

  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0

  OFFSET  (input) INTEGER
          The number of rows of A that have been factorized in
          previous steps.

  NB      (input) INTEGER
          The number of columns to factorize.

  KB      (output) INTEGER
          The number of columns actually factorized.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, block A(OFFSET+1:M,1:KB) is the triangular
          factor obtained and block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.
          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
          been updated.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          JPVT(I) = K <==> Column K of the full matrix A has been
          permuted into position I in AP.

  TAU     (output) DOUBLE PRECISION array, dimension (KB)
          The scalar factors of the elementary reflectors.

  VN1     (input/output) DOUBLE PRECISION array, dimension (N)
          The vector with the partial column norms.

  VN2     (input/output) DOUBLE PRECISION array, dimension (N)
          The vector with the exact column norms.

  AUXV    (input/output) DOUBLE PRECISION array, dimension (NB)
          Auxiliar vector.

  F       (input/output) DOUBLE PRECISION array, dimension (LDF,NB)
          Matrix F' = L*Y'*A.

  LDF     (input) INTEGER
          The leading dimension of the array F. LDF >= max(1,N).

  Further Details
  ===============

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.
  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        offset -
        nb -
        kb -
        a -
        lda -
        jpvt -
        tau -
        vn1 -
        vn2 -
        auxv -
        f -
        ldf -

      • dlaqr0

        public void dlaqr0(boolean wantt,
                   boolean wantz,
                   int n,
                   int ilo,
                   int ihi,
                   double[] h,
                   int ldh,
                   double[] wr,
                   double[] wi,
                   int iloz,
                   int ihiz,
                   double[] z,
                   int ldz,
                   double[] work,
                   int lwork,
                   org.netlib.util.intW info)
        
     ..

     Purpose
     =======

     DLAQR0 computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to DGEBAL, and then passed to DGEHRD when the

           matrix output by DGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO.GT.0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) DOUBLE PRECISION array, dimension (IHI)
     WI    (output) DOUBLE PRECISION array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed

           eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     ILOZ     (input) INTEGER
     IHIZ     (input) INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

     Z     (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) DOUBLE PRECISION array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then DLAQR0 does a workspace query.
           In this case, DLAQR0 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .GT. 0:  if INFO = i, DLAQR0 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
                accessed.


     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================

     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    DLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== Exceptional deflation windows:  try to cure rare
     .    slow convergence by increasing the size of the
     .    deflation window after KEXNW iterations. =====

     ==== Exceptional shifts: try to cure rare slow convergence
     .    with ad-hoc exceptional shifts every KEXSH iterations.
     .    The constants WILK1 and WILK2 are used to form the
     .    exceptional shifts. ====
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        work -
        lwork -
        info -

      • dlaqr1

        public void dlaqr1(int n,
                   double[] h,
                   int ldh,
                   double sr1,
                   double si1,
                   double sr2,
                   double si2,
                   double[] v)
        
     ..

       Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
       scalar multiple of the first column of the product

       (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)

       scaling to avoid overflows and most underflows. It
       is assumed that either

               1) sr1 = sr2 and si1 = -si2
           or
               2) si1 = si2 = 0.

       This is useful for starting double implicit shift bulges
       in the QR algorithm.


       N      (input) integer
              Order of the matrix H. N must be either 2 or 3.

       H      (input) DOUBLE PRECISION array of dimension (LDH,N)
              The 2-by-2 or 3-by-3 matrix H in (*).

       LDH    (input) integer
              The leading dimension of H as declared in
              the calling procedure.  LDH.GE.N

       SR1    (input) DOUBLE PRECISION
       SI1    The shifts in (*).
       SR2
       SI2

       V      (output) DOUBLE PRECISION array of dimension N
              A scalar multiple of the first column of the
              matrix K in (*).

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================

     .. Parameters ..
        Parameters:
        n -
        h -
        ldh -
        sr1 -
        si1 -
        sr2 -
        si2 -
        v -

      • dlaqr2

        public void dlaqr2(boolean wantt,
                   boolean wantz,
                   int n,
                   int ktop,
                   int kbot,
                   int nw,
                   double[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   double[] z,
                   int ldz,
                   org.netlib.util.intW ns,
                   org.netlib.util.intW nd,
                   double[] sr,
                   double[] si,
                   double[] v,
                   int ldv,
                   int nh,
                   double[] t,
                   int ldt,
                   int nv,
                   double[] wv,
                   int ldwv,
                   double[] work,
                   int lwork)
        
     ..

     This subroutine is identical to DLAQR3 except that it avoids
     recursion by calling DLAHQR instead of DLAQR4.



     Aggressive early deflation:

     This subroutine accepts as input an upper Hessenberg matrix
     H and performs an orthogonal similarity transformation
     designed to detect and deflate fully converged eigenvalues from
     a trailing principal submatrix.  On output H has been over-
     written by a new Hessenberg matrix that is a perturbation of
     an orthogonal similarity transformation of H.  It is to be
     hoped that the final version of H has many zero subdiagonal
     entries.


     WANTT   (input) LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

     WANTZ   (input) LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

     N       (input) INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

     KTOP    (input) INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

     KBOT    (input) INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

     NW      (input) INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

     H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

     LDH     (input) integer
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

     Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

     LDZ     (input) integer
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

     NS      (output) integer
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

     ND      (output) integer
          The number of converged eigenvalues uncovered by this
          subroutine.

     SR      (output) DOUBLE PRECISION array, dimension KBOT
     SI      (output) DOUBLE PRECISION array, dimension KBOT
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

     V       (workspace) DOUBLE PRECISION array, dimension (LDV,NW)
          An NW-by-NW work array.

     LDV     (input) integer scalar
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

     NH      (input) integer scalar
          The number of columns of T.  NH.GE.NW.

     T       (workspace) DOUBLE PRECISION array, dimension (LDT,NW)

     LDT     (input) integer
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

     NV      (input) integer
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

     WV      (workspace) DOUBLE PRECISION array, dimension (LDWV,NW)

     LDWV    (input) integer
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

     WORK    (workspace) DOUBLE PRECISION array, dimension LWORK.
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

     LWORK   (input) integer
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; DLAQR2
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ktop -
        kbot -
        nw -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        ns -
        nd -
        sr -
        si -
        v -
        ldv -
        nh -
        t -
        ldt -
        nv -
        wv -
        ldwv -
        work -
        lwork -

      • dlaqr3

        public void dlaqr3(boolean wantt,
                   boolean wantz,
                   int n,
                   int ktop,
                   int kbot,
                   int nw,
                   double[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   double[] z,
                   int ldz,
                   org.netlib.util.intW ns,
                   org.netlib.util.intW nd,
                   double[] sr,
                   double[] si,
                   double[] v,
                   int ldv,
                   int nh,
                   double[] t,
                   int ldt,
                   int nv,
                   double[] wv,
                   int ldwv,
                   double[] work,
                   int lwork)
        
     ..


     Aggressive early deflation:

     This subroutine accepts as input an upper Hessenberg matrix
     H and performs an orthogonal similarity transformation
     designed to detect and deflate fully converged eigenvalues from
     a trailing principal submatrix.  On output H has been over-
     written by a new Hessenberg matrix that is a perturbation of
     an orthogonal similarity transformation of H.  It is to be
     hoped that the final version of H has many zero subdiagonal
     entries.


     WANTT   (input) LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

     WANTZ   (input) LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

     N       (input) INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

     KTOP    (input) INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

     KBOT    (input) INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

     NW      (input) INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

     H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

     LDH     (input) integer
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

     Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

     LDZ     (input) integer
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

     NS      (output) integer
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

     ND      (output) integer
          The number of converged eigenvalues uncovered by this
          subroutine.

     SR      (output) DOUBLE PRECISION array, dimension KBOT
     SI      (output) DOUBLE PRECISION array, dimension KBOT
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

     V       (workspace) DOUBLE PRECISION array, dimension (LDV,NW)
          An NW-by-NW work array.

     LDV     (input) integer scalar
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

     NH      (input) integer scalar
          The number of columns of T.  NH.GE.NW.

     T       (workspace) DOUBLE PRECISION array, dimension (LDT,NW)

     LDT     (input) integer
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

     NV      (input) integer
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

     WV      (workspace) DOUBLE PRECISION array, dimension (LDWV,NW)

     LDWV    (input) integer
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

     WORK    (workspace) DOUBLE PRECISION array, dimension LWORK.
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

     LWORK   (input) integer
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; DLAQR3
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ==================================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ktop -
        kbot -
        nw -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        ns -
        nd -
        sr -
        si -
        v -
        ldv -
        nh -
        t -
        ldt -
        nv -
        wv -
        ldwv -
        work -
        lwork -

      • dlaqr4

        public void dlaqr4(boolean wantt,
                   boolean wantz,
                   int n,
                   int ilo,
                   int ihi,
                   double[] h,
                   int ldh,
                   double[] wr,
                   double[] wi,
                   int iloz,
                   int ihiz,
                   double[] z,
                   int ldz,
                   double[] work,
                   int lwork,
                   org.netlib.util.intW info)
        
     ..

     This subroutine implements one level of recursion for DLAQR0.
     It is a complete implementation of the small bulge multi-shift
     QR algorithm.  It may be called by DLAQR0 and, for large enough
     deflation window size, it may be called by DLAQR3.  This
     subroutine is identical to DLAQR0 except that it calls DLAQR2
     instead of DLAQR3.

     Purpose
     =======

     DLAQR4 computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to DGEBAL, and then passed to DGEHRD when the

           matrix output by DGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO.GT.0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) DOUBLE PRECISION array, dimension (IHI)
     WI    (output) DOUBLE PRECISION array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed

           eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     ILOZ     (input) INTEGER
     IHIZ     (input) INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

     Z     (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) DOUBLE PRECISION array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then DLAQR4 does a workspace query.
           In this case, DLAQR4 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .GT. 0:  if INFO = i, DLAQR4 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
                accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    DLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== Exceptional deflation windows:  try to cure rare
     .    slow convergence by increasing the size of the
     .    deflation window after KEXNW iterations. =====

     ==== Exceptional shifts: try to cure rare slow convergence
     .    with ad-hoc exceptional shifts every KEXSH iterations.
     .    The constants WILK1 and WILK2 are used to form the
     .    exceptional shifts. ====
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        work -
        lwork -
        info -

      • dlaqr5

        public void dlaqr5(boolean wantt,
                   boolean wantz,
                   int kacc22,
                   int n,
                   int ktop,
                   int kbot,
                   int nshfts,
                   double[] sr,
                   double[] si,
                   double[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   double[] z,
                   int ldz,
                   double[] v,
                   int ldv,
                   double[] u,
                   int ldu,
                   int nv,
                   double[] wv,
                   int ldwv,
                   int nh,
                   double[] wh,
                   int ldwh)
        
     ..

     This auxiliary subroutine called by DLAQR0 performs a
     single small-bulge multi-shift QR sweep.

      WANTT  (input) logical scalar
             WANTT = .true. if the quasi-triangular Schur factor
             is being computed.  WANTT is set to .false. otherwise.

      WANTZ  (input) logical scalar
             WANTZ = .true. if the orthogonal Schur factor is being
             computed.  WANTZ is set to .false. otherwise.

      KACC22 (input) integer with value 0, 1, or 2.
             Specifies the computation mode of far-from-diagonal
             orthogonal updates.
        = 0: DLAQR5 does not accumulate reflections and does not
             use matrix-matrix multiply to update far-from-diagonal
             matrix entries.
        = 1: DLAQR5 accumulates reflections and uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries.
        = 2: DLAQR5 accumulates reflections, uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries,
             and takes advantage of 2-by-2 block structure during
             matrix multiplies.

      N      (input) integer scalar
             N is the order of the Hessenberg matrix H upon which this

             subroutine operates.

      KTOP   (input) integer scalar
      KBOT   (input) integer scalar
             These are the first and last rows and columns of an
             isolated diagonal block upon which the QR sweep is to be
             applied. It is assumed without a check that
                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0
             and
                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

      NSHFTS (input) integer scalar
             NSHFTS gives the number of simultaneous shifts.  NSHFTS
             must be positive and even.

      SR     (input) DOUBLE PRECISION array of size (NSHFTS)
      SI     (input) DOUBLE PRECISION array of size (NSHFTS)
             SR contains the real parts and SI contains the imaginary
             parts of the NSHFTS shifts of origin that define the
             multi-shift QR sweep.

      H      (input/output) DOUBLE PRECISION array of size (LDH,N)
             On input H contains a Hessenberg matrix.  On output a
             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied

             to the isolated diagonal block in rows and columns KTOP
             through KBOT.

      LDH    (input) integer scalar
             LDH is the leading dimension of H just as declared in the

             calling procedure.  LDH.GE.MAX(1,N).

      ILOZ   (input) INTEGER
      IHIZ   (input) INTEGER
             Specify the rows of Z to which transformations must be
             applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N

      Z      (input/output) DOUBLE PRECISION array of size (LDZ,IHI)
             If WANTZ = .TRUE., then the QR Sweep orthogonal
             similarity transformation is accumulated into
             Z(ILOZ:IHIZ,ILO:IHI) from the right.
             If WANTZ = .FALSE., then Z is unreferenced.

      LDZ    (input) integer scalar
             LDA is the leading dimension of Z just as declared in
             the calling procedure. LDZ.GE.N.

      V      (workspace) DOUBLE PRECISION array of size (LDV,NSHFTS/2)


      LDV    (input) integer scalar
             LDV is the leading dimension of V as declared in the
             calling procedure.  LDV.GE.3.

      U      (workspace) DOUBLE PRECISION array of size
             (LDU,3*NSHFTS-3)

      LDU    (input) integer scalar
             LDU is the leading dimension of U just as declared in the

             in the calling subroutine.  LDU.GE.3*NSHFTS-3.

      NH     (input) integer scalar
             NH is the number of columns in array WH available for
             workspace. NH.GE.1.

      WH     (workspace) DOUBLE PRECISION array of size (LDWH,NH)

      LDWH   (input) integer scalar
             Leading dimension of WH just as declared in the
             calling procedure.  LDWH.GE.3*NSHFTS-3.

      NV     (input) integer scalar
             NV is the number of rows in WV agailable for workspace.
             NV.GE.1.

      WV     (workspace) DOUBLE PRECISION array of size
             (LDWV,3*NSHFTS-3)

      LDWV   (input) integer scalar
             LDWV is the leading dimension of WV as declared in the
             in the calling subroutine.  LDWV.GE.NV.


     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ============================================================
     Reference:

     K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
     Algorithm Part I: Maintaining Well Focused Shifts, and
     Level 3 Performance, SIAM Journal of Matrix Analysis,
     volume 23, pages 929--947, 2002.

     ============================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        kacc22 -
        n -
        ktop -
        kbot -
        nshfts -
        sr -
        si -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        v -
        ldv -
        u -
        ldu -
        nv -
        wv -
        ldwv -
        nh -
        wh -
        ldwh -

      • dlaqsb

        public abstract void dlaqsb(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] s,
                            double scond,
                            double amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  DLAQSB equilibrates a symmetric band matrix A using the scaling
  factors in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U'*U or A = L*L' of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  S       (input) DOUBLE PRECISION array, dimension (N)
          The scale factors for A.

  SCOND   (input) DOUBLE PRECISION
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) DOUBLE PRECISION
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        s -
        scond -
        amax -
        equed -

      • dlaqsp

        public abstract void dlaqsp(java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] s,
                            double scond,
                            double amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  DLAQSP equilibrates a symmetric matrix A using the scaling factors
  in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in

          the same storage format as A.

  S       (input) DOUBLE PRECISION array, dimension (N)
          The scale factors for A.

  SCOND   (input) DOUBLE PRECISION
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) DOUBLE PRECISION
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        s -
        scond -
        amax -
        equed -

      • dlaqsy

        public abstract void dlaqsy(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] s,
                            double scond,
                            double amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  DLAQSY equilibrates a symmetric matrix A using the scaling factors
  in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if EQUED = 'Y', the equilibrated matrix:
          diag(S) * A * diag(S).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  S       (input) DOUBLE PRECISION array, dimension (N)
          The scale factors for A.

  SCOND   (input) DOUBLE PRECISION
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) DOUBLE PRECISION
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        s -
        scond -
        amax -
        equed -

      • dlaqtr

        public abstract void dlaqtr(boolean ltran,
                            boolean lreal,
                            int n,
                            double[] t,
                            int ldt,
                            double[] b,
                            double w,
                            org.netlib.util.doubleW scale,
                            double[] x,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAQTR solves the real quasi-triangular system

               op(T)*p = scale*c,               if LREAL = .TRUE.

  or the complex quasi-triangular systems

             op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.

  in real arithmetic, where T is upper quasi-triangular.
  If LREAL = .FALSE., then the first diagonal block of T must be
  1 by 1, B is the specially structured matrix

                 B = [ b(1) b(2) ... b(n) ]
                     [       w            ]
                     [           w        ]
                     [              .     ]
                     [                 w  ]

  op(A) = A or A', A' denotes the conjugate transpose of
  matrix A.

  On input, X = [ c ].  On output, X = [ p ].
                [ d ]                  [ q ]

  This subroutine is designed for the condition number estimation
  in routine DTRSNA.

  Arguments
  =========

  LTRAN   (input) LOGICAL
          On entry, LTRAN specifies the option of conjugate transpose:

             = .FALSE.,    op(T+i*B) = T+i*B,
             = .TRUE.,     op(T+i*B) = (T+i*B)'.

  LREAL   (input) LOGICAL
          On entry, LREAL specifies the input matrix structure:
             = .FALSE.,    the input is complex
             = .TRUE.,     the input is real

  N       (input) INTEGER
          On entry, N specifies the order of T+i*B. N >= 0.

  T       (input) DOUBLE PRECISION array, dimension (LDT,N)
          On entry, T contains a matrix in Schur canonical form.
          If LREAL = .FALSE., then the first diagonal block of T mu
          be 1 by 1.

  LDT     (input) INTEGER
          The leading dimension of the matrix T. LDT >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (N)
          On entry, B contains the elements to form the matrix
          B as described above.
          If LREAL = .TRUE., B is not referenced.

  W       (input) DOUBLE PRECISION
          On entry, W is the diagonal element of the matrix B.
          If LREAL = .TRUE., W is not referenced.

  SCALE   (output) DOUBLE PRECISION
          On exit, SCALE is the scale factor.

  X       (input/output) DOUBLE PRECISION array, dimension (2*N)
          On entry, X contains the right hand side of the system.
          On exit, X is overwritten by the solution.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          On exit, INFO is set to
             0: successful exit.
               1: the some diagonal 1 by 1 block has been perturbed by

                  a small number SMIN to keep nonsingularity.
               2: the some diagonal 2 by 2 block has been perturbed by

                  a small number in DLALN2 to keep nonsingularity.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltran -
        lreal -
        n -
        t -
        ldt -
        b -
        w -
        scale -
        x -
        work -
        info -

      • dlar1v

        public void dlar1v(int n,
                   int b1,
                   int bn,
                   double lambda,
                   double[] d,
                   double[] l,
                   double[] ld,
                   double[] lld,
                   double pivmin,
                   double gaptol,
                   double[] z,
                   boolean wantnc,
                   org.netlib.util.intW negcnt,
                   org.netlib.util.doubleW ztz,
                   org.netlib.util.doubleW mingma,
                   org.netlib.util.intW r,
                   int[] isuppz,
                   org.netlib.util.doubleW nrminv,
                   org.netlib.util.doubleW resid,
                   org.netlib.util.doubleW rqcorr,
                   double[] work)
        
     ..

  Purpose
  =======

  DLAR1V computes the (scaled) r-th column of the inverse of
  the sumbmatrix in rows B1 through BN of the tridiagonal matrix
  L D L^T - sigma I. When sigma is close to an eigenvalue, the
  computed vector is an accurate eigenvector. Usually, r corresponds
  to the index where the eigenvector is largest in magnitude.
  The following steps accomplish this computation :
  (a) Stationary qd transform,  L D L^T - sigma I = L(+) D(+) L(+)^T,
  (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T,
  (c) Computation of the diagonal elements of the inverse of
      L D L^T - sigma I by combining the above transforms, and choosing
      r as the index where the diagonal of the inverse is (one of the)

      largest in magnitude.
  (d) Computation of the (scaled) r-th column of the inverse using the

      twisted factorization obtained by combining the top part of the
      the stationary and the bottom part of the progressive transform.


  Arguments
  =========

  N        (input) INTEGER
           The order of the matrix L D L^T.

  B1       (input) INTEGER
           First index of the submatrix of L D L^T.

  BN       (input) INTEGER
           Last index of the submatrix of L D L^T.

  LAMBDA    (input) DOUBLE PRECISION
           The shift. In order to compute an accurate eigenvector,
           LAMBDA should be a good approximation to an eigenvalue
           of L D L^T.

  L        (input) DOUBLE PRECISION array, dimension (N-1)
           The (n-1) subdiagonal elements of the unit bidiagonal matrix
           L, in elements 1 to N-1.

  D        (input) DOUBLE PRECISION array, dimension (N)
           The n diagonal elements of the diagonal matrix D.

  LD       (input) DOUBLE PRECISION array, dimension (N-1)
           The n-1 elements L(i)*D(i).

  LLD      (input) DOUBLE PRECISION array, dimension (N-1)
           The n-1 elements L(i)*L(i)*D(i).

  PIVMIN   (input) DOUBLE PRECISION
           The minimum pivot in the Sturm sequence.

  GAPTOL   (input) DOUBLE PRECISION
           Tolerance that indicates when eigenvector entries are neglig
           w.r.t. their contribution to the residual.

  Z        (input/output) DOUBLE PRECISION array, dimension (N)
           On input, all entries of Z must be set to 0.
           On output, Z contains the (scaled) r-th column of the
           inverse. The scaling is such that Z(R) equals 1.

  WANTNC   (input) LOGICAL
           Specifies whether NEGCNT has to be computed.

  NEGCNT   (output) INTEGER
           If WANTNC is .TRUE. then NEGCNT = the number of pivots < piv
           in the  matrix factorization L D L^T, and NEGCNT = -1 otherw

  ZTZ      (output) DOUBLE PRECISION
           The square of the 2-norm of Z.

  MINGMA   (output) DOUBLE PRECISION
           The reciprocal of the largest (in magnitude) diagonal
           element of the inverse of L D L^T - sigma I.

  R        (input/output) INTEGER
           The twist index for the twisted factorization used to
           compute Z.
           On input, 0 <= R <= N. If R is input as 0, R is set to
           the index where (L D L^T - sigma I)^{-1} is largest
           in magnitude. If 1 <= R <= N, R is unchanged.
           On output, R contains the twist index used to compute Z.
           Ideally, R designates the position of the maximum entry in t
           eigenvector.

  ISUPPZ   (output) INTEGER array, dimension (2)
           The support of the vector in Z, i.e., the vector Z is
           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

  NRMINV   (output) DOUBLE PRECISION
           NRMINV = 1/SQRT( ZTZ )

  RESID    (output) DOUBLE PRECISION
           The residual of the FP vector.
           RESID = ABS( MINGMA )/SQRT( ZTZ )

  RQCORR   (output) DOUBLE PRECISION
           The Rayleigh Quotient correction to LAMBDA.
           RQCORR = MINGMA*TMP

  WORK     (workspace) DOUBLE PRECISION array, dimension (4*N)

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        b1 -
        bn -
        lambda -
        d -
        l -
        ld -
        lld -
        pivmin -
        gaptol -
        z -
        wantnc -
        negcnt -
        ztz -
        mingma -
        r -
        isuppz -
        nrminv -
        resid -
        rqcorr -
        work -

      • dlar2v

        public abstract void dlar2v(int n,
                            double[] x,
                            double[] y,
                            double[] z,
                            int incx,
                            double[] c,
                            double[] s,
                            int incc)
        
     ..

  Purpose
  =======

  DLAR2V applies a vector of real plane rotations from both sides to
  a sequence of 2-by-2 real symmetric matrices, defined by the elements
  of the vectors x, y and z. For i = 1,2,...,n

     ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )
     ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be applied.

  X       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector x.

  Y       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector y.

  Z       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector z.

  INCX    (input) INTEGER
          The increment between elements of X, Y and Z. INCX > 0.

  C       (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  S       (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C and S. INCC > 0.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        x -
        y -
        z -
        incx -
        c -
        s -
        incc -

      • dlarf

        public abstract void dlarf(java.lang.String side,
                           int m,
                           int n,
                           double[] v,
                           int incv,
                           double tau,
                           double[] c,
                           int Ldc,
                           double[] work)
        
     ..

  Purpose
  =======

  DLARF applies a real elementary reflector H to a real m by n matrix
  C, from either the left or the right. H is represented in the form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) DOUBLE PRECISION array, dimension
                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of H. V is not used if
          TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0.

  TAU     (input) DOUBLE PRECISION
          The value tau in the representation of H.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        incv -
        tau -
        c -
        Ldc -
        work -

      • dlarfb

        public abstract void dlarfb(java.lang.String side,
                            java.lang.String trans,
                            java.lang.String direct,
                            java.lang.String storev,
                            int m,
                            int n,
                            int k,
                            double[] v,
                            int ldv,
                            double[] t,
                            int ldt,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int ldwork)
        
     ..

  Purpose
  =======

  DLARFB applies a real block reflector H or its transpose H' to a
  real m by n matrix C, from either the left or the right.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply H or H' from the Left
          = 'R': apply H or H' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply H (No transpose)
          = 'T': apply H' (Transpose)

  DIRECT  (input) CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise
          = 'R': Rowwise

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  K       (input) INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

  V       (input) DOUBLE PRECISION array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'

                                (LDV,N) if STOREV = 'R' and SIDE = 'R'

          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
          if STOREV = 'R', LDV >= K.

  T       (input) DOUBLE PRECISION array, dimension (LDT,K)
          The triangular k by k matrix T in the representation of the
          block reflector.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by H*C or H'*C or C*H or C*H'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDA >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension (LDWORK,K)

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        direct -
        storev -
        m -
        n -
        k -
        v -
        ldv -
        t -
        ldt -
        c -
        Ldc -
        work -
        ldwork -

      • dlarfg

        public abstract void dlarfg(int n,
                            org.netlib.util.doubleW alpha,
                            double[] x,
                            int incx,
                            org.netlib.util.doubleW tau)
        
     ..

  Purpose
  =======

  DLARFG generates a real elementary reflector H of order n, such
  that

        H * ( alpha ) = ( beta ),   H' * H = I.
            (   x   )   (   0  )

  where alpha and beta are scalars, and x is an (n-1)-element real
  vector. H is represented in the form

        H = I - tau * ( 1 ) * ( 1 v' ) ,
                      ( v )

  where tau is a real scalar and v is a real (n-1)-element
  vector.

  If the elements of x are all zero, then tau = 0 and H is taken to be

  the unit matrix.

  Otherwise  1 <= tau <= 2.

  Arguments
  =========

  N       (input) INTEGER
          The order of the elementary reflector.

  ALPHA   (input/output) DOUBLE PRECISION
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

  X       (input/output) DOUBLE PRECISION array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  TAU     (output) DOUBLE PRECISION
          The value tau.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        alpha -
        x -
        incx -
        tau -

      • dlarft

        public abstract void dlarft(java.lang.String direct,
                            java.lang.String storev,
                            int n,
                            int k,
                            double[] v,
                            int ldv,
                            double[] tau,
                            double[] t,
                            int ldt)
        
     ..

  Purpose
  =======

  DLARFT forms the triangular factor T of a real block reflector H
  of order n, which is defined as a product of k elementary reflectors.

  If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;


  If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.


  If STOREV = 'C', the vector which defines the elementary reflector
  H(i) is stored in the i-th column of the array V, and

     H  =  I - V * T * V'

  If STOREV = 'R', the vector which defines the elementary reflector
  H(i) is stored in the i-th row of the array V, and

     H  =  I - V' * T * V

  Arguments
  =========

  DIRECT  (input) CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise
          = 'R': rowwise

  N       (input) INTEGER
          The order of the block reflector H. N >= 0.

  K       (input) INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

  V       (input/output) DOUBLE PRECISION array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.


  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

  T       (output) DOUBLE PRECISION array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  Further Details
  ===============

  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and

  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

               V = (  1       )                 V = (  1 v1 v1 v1 v1 )

                   ( v1  1    )                     (     1 v2 v2 v2 )

                   ( v1 v2  1 )                     (        1 v3 v3 )

                   ( v1 v2 v3 )
                   ( v1 v2 v3 )

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )

                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )

                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )

                   (     1 v3 )
                   (        1 )

  =====================================================================

     .. Parameters ..
        Parameters:
        direct -
        storev -
        n -
        k -
        v -
        ldv -
        tau -
        t -
        ldt -

      • dlarfx

        public abstract void dlarfx(java.lang.String side,
                            int m,
                            int n,
                            double[] v,
                            double tau,
                            double[] c,
                            int Ldc,
                            double[] work)
        
     ..

  Purpose
  =======

  DLARFX applies a real elementary reflector H to a real m by n
  matrix C, from either the left or the right. H is represented in the

  form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix

  This version uses inline code if H has order < 11.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
                                     or (N) if SIDE = 'R'
          The vector v in the representation of H.

  TAU     (input) DOUBLE PRECISION
          The value tau in the representation of H.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDA >= (1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                      (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'
          WORK is not referenced if H has order < 11.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        tau -
        c -
        Ldc -
        work -

      • dlargv

        public abstract void dlargv(int n,
                            double[] x,
                            int incx,
                            double[] y,
                            int incy,
                            double[] c,
                            int incc)
        
     ..

  Purpose
  =======

  DLARGV generates a vector of real plane rotations, determined by
  elements of the real vectors x and y. For i = 1,2,...,n

     (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
     ( -s(i)  c(i) ) ( y(i) ) = (   0  )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be generated.

  X       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          On entry, the vector x.
          On exit, x(i) is overwritten by a(i), for i = 1,...,n.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  Y       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          On entry, the vector y.
          On exit, the sines of the plane rotations.

  INCY    (input) INTEGER
          The increment between elements of Y. INCY > 0.

  C       (output) DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C. INCC > 0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        c -
        incc -

      • dlarnv

        public abstract void dlarnv(int idist,
                            int[] iseed,
                            int n,
                            double[] x)
        
     ..

  Purpose
  =======

  DLARNV returns a vector of n random real numbers from a uniform or
  normal distribution.

  Arguments
  =========

  IDIST   (input) INTEGER
          Specifies the distribution of the random numbers:
          = 1:  uniform (0,1)
          = 2:  uniform (-1,1)
          = 3:  normal (0,1)

  ISEED   (input/output) INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array

          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

  N       (input) INTEGER
          The number of random numbers to be generated.

  X       (output) DOUBLE PRECISION array, dimension (N)
          The generated random numbers.

  Further Details
  ===============

  This routine calls the auxiliary routine DLARUV to generate random
  real numbers from a uniform (0,1) distribution, in batches of up to
  128 using vectorisable code. The Box-Muller method is used to
  transform numbers from a uniform to a normal distribution.

  =====================================================================

     .. Parameters ..
        Parameters:
        idist -
        iseed -
        n -
        x -

      • dlarra

        public void dlarra(int n,
                   double[] d,
                   double[] e,
                   double[] e2,
                   double spltol,
                   double tnrm,
                   org.netlib.util.intW nsplit,
                   int[] isplit,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Compute the splitting points with threshold SPLTOL.
  DLARRA sets any "small" off-diagonal elements to zero.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix. N > 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal
          matrix T.

  E       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) need not be set.
          On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT,
          are set to zero, the other entries of E are untouched.

  E2      (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the first (N-1) entries contain the SQUARES of the

          subdiagonal elements of the tridiagonal matrix T;
          E2(N) need not be set.
          On exit, the entries E2( ISPLIT( I ) ),
          1 <= I <= NSPLIT, have been set to zero

  SPLTOL (input) DOUBLE PRECISION
          The threshold for splitting. Two criteria can be used:
          SPLTOL<0 : criterion based on absolute off-diagonal value
          SPLTOL>0 : criterion that preserves relative accuracy

  TNRM (input) DOUBLE PRECISION
          The norm of the matrix.

  NSPLIT  (output) INTEGER
          The number of blocks T splits into. 1 <= NSPLIT <= N.

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to ISPLIT(1),
          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.


  INFO    (output) INTEGER
          = 0:  successful exit

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        e2 -
        spltol -
        tnrm -
        nsplit -
        isplit -
        info -

      • dlarrb

        public void dlarrb(int n,
                   double[] d,
                   double[] lld,
                   int ifirst,
                   int ilast,
                   double rtol1,
                   double rtol2,
                   int offset,
                   double[] w,
                   double[] wgap,
                   double[] werr,
                   double[] work,
                   int[] iwork,
                   double pivmin,
                   double spdiam,
                   int twist,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the relatively robust representation(RRR) L D L^T, DLARRB
  does "limited" bisection to refine the eigenvalues of L D L^T,
  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initi
  guesses for these eigenvalues are input in W, the corresponding estim
  of the error in these guesses and their gaps are input in WERR
  and WGAP, respectively. During bisection, intervals
  [left, right] are maintained by storing their mid-points and
  semi-widths in the arrays W and WERR respectively.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  LLD     (input) DOUBLE PRECISION array, dimension (N-1)
          The (N-1) elements L(i)*L(i)*D(i).

  IFIRST  (input) INTEGER
          The index of the first eigenvalue to be computed.

  ILAST   (input) INTEGER
          The index of the last eigenvalue to be computed.

  RTOL1   (input) DOUBLE PRECISION
  RTOL2   (input) DOUBLE PRECISION
          Tolerance for the convergence of the bisection intervals.
          An interval [LEFT,RIGHT] has converged if
          RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
          where GAP is the (estimated) distance to the nearest
          eigenvalue.

  OFFSET  (input) INTEGER
          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFS
          through ILAST-OFFSET elements of these arrays are to be used.

  W       (input/output) DOUBLE PRECISION array, dimension (N)
          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
          estimates of the eigenvalues of L D L^T indexed IFIRST throug
          ILAST.
          On output, these estimates are refined.

  WGAP    (input/output) DOUBLE PRECISION array, dimension (N-1)
          On input, the (estimated) gaps between consecutive
          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap betwe
          eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
          then WGAP(IFIRST-OFFSET) must be set to ZERO.
          On output, these gaps are refined.

  WERR    (input/output) DOUBLE PRECISION array, dimension (N)
          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) 
          the errors in the estimates of the corresponding elements in 
          On output, these errors are refined.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (2*N)
          Workspace.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence.

  SPDIAM  (input) DOUBLE PRECISION
          The spectral diameter of the matrix.

  TWIST   (input) INTEGER
          The twist index for the twisted factorization that is used
          for the negcount.
          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L
          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U
          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(

  INFO    (output) INTEGER
          Error flag.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        lld -
        ifirst -
        ilast -
        rtol1 -
        rtol2 -
        offset -
        w -
        wgap -
        werr -
        work -
        iwork -
        pivmin -
        spdiam -
        twist -
        info -

      • dlarrc

        public void dlarrc(java.lang.String jobt,
                   int n,
                   double vl,
                   double vu,
                   double[] d,
                   double[] e,
                   double pivmin,
                   org.netlib.util.intW eigcnt,
                   org.netlib.util.intW lcnt,
                   org.netlib.util.intW rcnt,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Find the number of eigenvalues of the symmetric tridiagonal matrix T

  that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T
  if JOBT = 'L'.

  Arguments
  =========

  JOBT    (input) CHARACTER*1
          = 'T':  Compute Sturm count for matrix T.
          = 'L':  Compute Sturm count for matrix L D L^T.

  N       (input) INTEGER
          The order of the matrix. N > 0.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          The lower and upper bounds for the eigenvalues.

  D       (input) DOUBLE PRECISION array, dimension (N)
          JOBT = 'T': The N diagonal elements of the tridiagonal matrix
          JOBT = 'L': The N diagonal elements of the diagonal matrix D.

  E       (input) DOUBLE PRECISION array, dimension (N)
          JOBT = 'T': The N-1 offdiagonal elements of the matrix T.
          JOBT = 'L': The N-1 offdiagonal elements of the matrix L.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  EIGCNT  (output) INTEGER
          The number of eigenvalues of the symmetric tridiagonal matrix
          that are in the interval (VL,VU]

  LCNT    (output) INTEGER
  RCNT    (output) INTEGER
          The left and right negcounts of the interval.

  INFO    (output) INTEGER

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobt -
        n -
        vl -
        vu -
        d -
        e -
        pivmin -
        eigcnt -
        lcnt -
        rcnt -
        info -

      • dlarrd

        public void dlarrd(java.lang.String range,
                   java.lang.String order,
                   int n,
                   double vl,
                   double vu,
                   int il,
                   int iu,
                   double[] gers,
                   double reltol,
                   double[] d,
                   double[] e,
                   double[] e2,
                   double pivmin,
                   int nsplit,
                   int[] isplit,
                   org.netlib.util.intW m,
                   double[] w,
                   double[] werr,
                   org.netlib.util.doubleW wl,
                   org.netlib.util.doubleW wu,
                   int[] iblock,
                   int[] indexw,
                   double[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLARRD computes the eigenvalues of a symmetric tridiagonal
  matrix T to suitable accuracy. This is an auxiliary code to be
  called from DSTEMR.
  The user may ask for all eigenvalues, all eigenvalues
  in the half-open interval (VL, VU], or the IL-th through IU-th
  eigenvalues.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  RANGE   (input) CHARACTER
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  ORDER   (input) CHARACTER
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and

                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal

          to VL, or greater than VU, will not be returned.  VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  GERS    (input) DOUBLE PRECISION array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)).

  RELTOL  (input) DOUBLE PRECISION
          The minimum relative width of an interval.  When an interval

          is narrower than RELTOL times the larger (in
          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.


  E2      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) squared off-diagonal elements of the tridiagonal ma

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence for T.

  NSPLIT  (input) INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to ISPLIT(1),

          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

  M       (output) INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

  W       (output) DOUBLE PRECISION array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalue approximations. DLARRD computes an interval
          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
          approximation is given as the interval midpoint
          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
          WERR(j) = abs( a_j - b_j)/2

  WERR    (output) DOUBLE PRECISION array, dimension (N)
          The error bound on the corresponding eigenvalue approximation
          in W.

  WL      (output) DOUBLE PRECISION
  WU      (output) DOUBLE PRECISION
          The interval (WL, WU] contains all the wanted eigenvalues.
          If RANGE='V', then WL=VL and WU=VU.
          If RANGE='A', then WL and WU are the global Gerschgorin bound
                        on the spectrum.
          If RANGE='I', then WL and WU are computed by DLAEBZ from the

                        index range specified.

  IBLOCK  (output) INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (DLARRD may use the remaining N-M elements as
          workspace.)

  INDEXW  (output) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
          i-th eigenvalue W(i) is the j-th eigenvalue in block k.

  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the

                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick

                                out eigenvalues IL:IU.  In some cases,

                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  Internal Parameters
  ===================

  FUDGE   DOUBLE PRECISION, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

  Based on contributions by
     W. Kahan, University of California, Berkeley, USA
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        order -
        n -
        vl -
        vu -
        il -
        iu -
        gers -
        reltol -
        d -
        e -
        e2 -
        pivmin -
        nsplit -
        isplit -
        m -
        w -
        werr -
        wl -
        wu -
        iblock -
        indexw -
        work -
        iwork -
        info -

      • dlarre

        public void dlarre(java.lang.String range,
                   int n,
                   org.netlib.util.doubleW vl,
                   org.netlib.util.doubleW vu,
                   int il,
                   int iu,
                   double[] d,
                   double[] e,
                   double[] e2,
                   double rtol1,
                   double rtol2,
                   double spltol,
                   org.netlib.util.intW nsplit,
                   int[] isplit,
                   org.netlib.util.intW m,
                   double[] w,
                   double[] werr,
                   double[] wgap,
                   int[] iblock,
                   int[] indexw,
                   double[] gers,
                   org.netlib.util.doubleW pivmin,
                   double[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  To find the desired eigenvalues of a given real symmetric
  tridiagonal matrix T, DLARRE sets any "small" off-diagonal
  elements to zero, and for each unreduced block T_i, it finds
  (a) a suitable shift at one end of the block's spectrum,
  (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
  (c) eigenvalues of each L_i D_i L_i^T.
  The representations and eigenvalues found are then used by
  DSTEMR to compute the eigenvectors of T.
  The accuracy varies depending on whether bisection is used to
  find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
  conpute all and then discard any unwanted one.
  As an added benefit, DLARRE also outputs the n
  Gerschgorin intervals for the matrices L_i D_i L_i^T.

  Arguments
  =========

  RANGE   (input) CHARACTER
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  N       (input) INTEGER
          The order of the matrix. N > 0.

  VL      (input/output) DOUBLE PRECISION
  VU      (input/output) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds for the eigenvalues.
          Eigenvalues less than or equal to VL, or greater than VU,
          will not be returned.  VL < VU.
          If RANGE='I' or ='A', DLARRE computes bounds on the desired
          part of the spectrum.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal
          matrix T.
          On exit, the N diagonal elements of the diagonal
          matrices D_i.

  E       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) need not be set.
          On exit, E contains the subdiagonal elements of the unit
          bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
          1 <= I <= NSPLIT, contain the base points sigma_i on output.


  E2      (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the first (N-1) entries contain the SQUARES of the

          subdiagonal elements of the tridiagonal matrix T;
          E2(N) need not be set.
          On exit, the entries E2( ISPLIT( I ) ),
          1 <= I <= NSPLIT, have been set to zero

  RTOL1   (input) DOUBLE PRECISION
  RTOL2   (input) DOUBLE PRECISION
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

  SPLTOL (input) DOUBLE PRECISION
          The threshold for splitting.

  NSPLIT  (output) INTEGER
          The number of blocks T splits into. 1 <= NSPLIT <= N.

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to ISPLIT(1),
          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

  M       (output) INTEGER
          The total number of eigenvalues (of all L_i D_i L_i^T)
          found.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the eigenvalues. The
          eigenvalues of each of the blocks, L_i D_i L_i^T, are
          sorted in ascending order ( DLARRE may use the
          remaining N-M elements as workspace).

  WERR    (output) DOUBLE PRECISION array, dimension (N)
          The error bound on the corresponding eigenvalue in W.

  WGAP    (output) DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W.
          The gap is only with respect to the eigenvalues of the same b
          as each block has its own representation tree.
          Exception: at the right end of a block we store the left gap


  IBLOCK  (output) INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

  INDEXW  (output) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in block 2

  GERS    (output) DOUBLE PRECISION array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)).

  PIVMIN  (output) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  WORK    (workspace) DOUBLE PRECISION array, dimension (6*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (5*N)
          Workspace.

  INFO    (output) INTEGER
          = 0:  successful exit
          > 0:  A problem occured in DLARRE.
          < 0:  One of the called subroutines signaled an internal prob
                Needs inspection of the corresponding parameter IINFO
                for further information.

          =-1:  Problem in DLARRD.
          = 2:  No base representation could be found in MAXTRY iterati
                Increasing MAXTRY and recompilation might be a remedy.

          =-3:  Problem in DLARRB when computing the refined root
                representation for DLASQ2.
          =-4:  Problem in DLARRB when preforming bisection on the
                desired part of the spectrum.
          =-5:  Problem in DLASQ2.
          =-6:  Problem in DLASQ2.

  Further Details
  The base representations are required to suffer very little
  element growth and consequently define all their eigenvalues to
  high relative accuracy.
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        n -
        vl -
        vu -
        il -
        iu -
        d -
        e -
        e2 -
        rtol1 -
        rtol2 -
        spltol -
        nsplit -
        isplit -
        m -
        w -
        werr -
        wgap -
        iblock -
        indexw -
        gers -
        pivmin -
        work -
        iwork -
        info -

      • dlarrf

        public void dlarrf(int n,
                   double[] d,
                   double[] l,
                   double[] ld,
                   int clstrt,
                   int clend,
                   double[] w,
                   double[] wgap,
                   double[] werr,
                   double spdiam,
                   double clgapl,
                   double clgapr,
                   double pivmin,
                   org.netlib.util.doubleW sigma,
                   double[] dplus,
                   double[] lplus,
                   double[] work,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the initial representation L D L^T and its cluster of close
  eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...

  W( CLEND ), DLARRF finds a new relatively robust representation
  L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
  eigenvalues of L(+) D(+) L(+)^T is relatively isolated.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix (subblock, if the matrix splitted).

  D       (input) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  L       (input) DOUBLE PRECISION array, dimension (N-1)
          The (N-1) subdiagonal elements of the unit bidiagonal
          matrix L.

  LD      (input) DOUBLE PRECISION array, dimension (N-1)
          The (N-1) elements L(i)*D(i).

  CLSTRT  (input) INTEGER
          The index of the first eigenvalue in the cluster.

  CLEND   (input) INTEGER
          The index of the last eigenvalue in the cluster.

  W       (input) DOUBLE PRECISION array, dimension >=  (CLEND-CLSTRT+1
          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.

          W( CLSTRT ) through W( CLEND ) form the cluster of relatively
          close eigenalues.

  WGAP    (input/output) DOUBLE PRECISION array, dimension >=  (CLEND-C
          The separation from the right neighbor eigenvalue in W.

  WERR    (input) DOUBLE PRECISION array, dimension >=  (CLEND-CLSTRT+1
          WERR contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue APPROXIMATION in W

  SPDIAM (input) estimate of the spectral diameter obtained from the
          Gerschgorin intervals

  CLGAPL, CLGAPR (input) absolute gap on each end of the cluster.
          Set by the calling routine to protect against shifts too clos
          to eigenvalues outside the cluster.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

  SIGMA   (output) DOUBLE PRECISION
          The shift used to form L(+) D(+) L(+)^T.

  DPLUS   (output) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of the diagonal matrix D(+).

  LPLUS   (output) DOUBLE PRECISION array, dimension (N-1)
          The first (N-1) elements of LPLUS contain the subdiagonal
          elements of the unit bidiagonal matrix L(+).

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
          Workspace.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        l -
        ld -
        clstrt -
        clend -
        w -
        wgap -
        werr -
        spdiam -
        clgapl -
        clgapr -
        pivmin -
        sigma -
        dplus -
        lplus -
        work -
        info -

      • dlarrj

        public void dlarrj(int n,
                   double[] d,
                   double[] e2,
                   int ifirst,
                   int ilast,
                   double rtol,
                   int offset,
                   double[] w,
                   double[] werr,
                   double[] work,
                   int[] iwork,
                   double pivmin,
                   double spdiam,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the initial eigenvalue approximations of T, DLARRJ
  does  bisection to refine the eigenvalues of T,
  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initi
  guesses for these eigenvalues are input in W, the corresponding estim
  of the error in these guesses in WERR. During bisection, intervals
  [left, right] are maintained by storing their mid-points and
  semi-widths in the arrays W and WERR respectively.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of T.

  E2      (input) DOUBLE PRECISION array, dimension (N-1)
          The Squares of the (N-1) subdiagonal elements of T.

  IFIRST  (input) INTEGER
          The index of the first eigenvalue to be computed.

  ILAST   (input) INTEGER
          The index of the last eigenvalue to be computed.

  RTOL   (input) DOUBLE PRECISION
          Tolerance for the convergence of the bisection intervals.
          An interval [LEFT,RIGHT] has converged if
          RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).

  OFFSET  (input) INTEGER
          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
          through ILAST-OFFSET elements of these arrays are to be used.

  W       (input/output) DOUBLE PRECISION array, dimension (N)
          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
          estimates of the eigenvalues of L D L^T indexed IFIRST throug
          ILAST.
          On output, these estimates are refined.

  WERR    (input/output) DOUBLE PRECISION array, dimension (N)
          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) 
          the errors in the estimates of the corresponding elements in 
          On output, these errors are refined.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (2*N)
          Workspace.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  SPDIAM  (input) DOUBLE PRECISION
          The spectral diameter of T.

  INFO    (output) INTEGER
          Error flag.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e2 -
        ifirst -
        ilast -
        rtol -
        offset -
        w -
        werr -
        work -
        iwork -
        pivmin -
        spdiam -
        info -

      • dlarrk

        public void dlarrk(int n,
                   int iw,
                   double gl,
                   double gu,
                   double[] d,
                   double[] e2,
                   double pivmin,
                   double reltol,
                   org.netlib.util.doubleW w,
                   org.netlib.util.doubleW werr,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLARRK computes one eigenvalue of a symmetric tridiagonal
  matrix T to suitable accuracy. This is an auxiliary code to be
  called from DSTEMR.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  IW      (input) INTEGER
          The index of the eigenvalues to be returned.

  GL      (input) DOUBLE PRECISION
  GU      (input) DOUBLE PRECISION
          An upper and a lower bound on the eigenvalue.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E2      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) squared off-diagonal elements of the tridiagonal ma

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence for T.

  RELTOL  (input) DOUBLE PRECISION
          The minimum relative width of an interval.  When an interval

          is narrower than RELTOL times the larger (in
          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  W       (output) DOUBLE PRECISION

  WERR    (output) DOUBLE PRECISION
          The error bound on the corresponding eigenvalue approximation
          in W.

  INFO    (output) INTEGER
          = 0:       Eigenvalue converged
          = -1:      Eigenvalue did NOT converge

  Internal Parameters
  ===================

  FUDGE   DOUBLE PRECISION, default = 2
          A "fudge factor" to widen the Gershgorin intervals.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        iw -
        gl -
        gu -
        d -
        e2 -
        pivmin -
        reltol -
        w -
        werr -
        info -

      • dlarrr

        public void dlarrr(int n,
                   double[] d,
                   double[] e,
                   org.netlib.util.intW info)
        
     ..


  Purpose
  =======

  Perform tests to decide whether the symmetric tridiagonal matrix T
  warrants expensive computations which guarantee high relative accurac
  in the eigenvalues.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix. N > 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The N diagonal elements of the tridiagonal matrix T.

  E       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) is set to ZERO.

  INFO    (output) INTEGER
          INFO = 0(default) : the matrix warrants computations preservi
                              relative accuracy.
          INFO = 1          : the matrix warrants computations guarante
                              only absolute accuracy.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • dlarrv

        public void dlarrv(int n,
                   double vl,
                   double vu,
                   double[] d,
                   double[] l,
                   double pivmin,
                   int[] isplit,
                   int m,
                   int dol,
                   int dou,
                   double minrgp,
                   org.netlib.util.doubleW rtol1,
                   org.netlib.util.doubleW rtol2,
                   double[] w,
                   double[] werr,
                   double[] wgap,
                   int[] iblock,
                   int[] indexw,
                   double[] gers,
                   double[] z,
                   int ldz,
                   int[] isuppz,
                   double[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLARRV computes the eigenvectors of the tridiagonal matrix
  T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L
  The input eigenvalues should have been computed by DLARRE.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          Lower and upper bounds of the interval that contains the desi
          eigenvalues. VL < VU. Needed to compute gaps on the left or r
          end of the extremal eigenvalues in the desired RANGE.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D.
          On exit, D may be overwritten.

  L       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not splitted.) At the end of each block
          is stored the corresponding shift as given by DLARRE.
          On exit, L is overwritten.

  PIVMIN  (in) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.

  M       (input) INTEGER
          The total number of input eigenvalues.  0 <= M <= N.

  DOL     (input) INTEGER
  DOU     (input) INTEGER
          If the user wants to compute only selected eigenvectors from 
          the eigenvalues supplied, he can specify an index range DOL:D
          Or else the setting DOL=1, DOU=M should be applied.
          Note that DOL and DOU refer to the order in which the eigenva
          are stored in W.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain
          computed eigenvectors. All other columns of Z are set to zero

  MINRGP  (input) DOUBLE PRECISION

  RTOL1   (input) DOUBLE PRECISION
  RTOL2   (input) DOUBLE PRECISION
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

  W       (input/output) DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from DLARRE is expected here ). Furthermore, they are with

          respect to the shift of the corresponding root representation
          for their block. On exit, W holds the eigenvalues of the
          UNshifted matrix.

  WERR    (input/output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W

  WGAP    (input/output) DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W.

  IBLOCK  (input) INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

  INDEXW  (input) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second bl

  GERS    (input) DOUBLE PRECISION array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
          be computed from the original UNshifted matrix.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The I-th eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I ).

  WORK    (workspace) DOUBLE PRECISION array, dimension (12*N)

  IWORK   (workspace) INTEGER array, dimension (7*N)

  INFO    (output) INTEGER
          = 0:  successful exit

          > 0:  A problem occured in DLARRV.
          < 0:  One of the called subroutines signaled an internal prob
                Needs inspection of the corresponding parameter IINFO
                for further information.

          =-1:  Problem in DLARRB when refining a child's eigenvalues.

          =-2:  Problem in DLARRF when computing the RRR of a child.
                When a child is inside a tight cluster, it can be diffi
                to find an RRR. A partial remedy from the user's point 
                view is to make the parameter MINRGP smaller and recomp
                However, as the orthogonality of the computed vectors i
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINR
          =-3:  Problem in DLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        vl -
        vu -
        d -
        l -
        pivmin -
        isplit -
        m -
        dol -
        dou -
        minrgp -
        rtol1 -
        rtol2 -
        w -
        werr -
        wgap -
        iblock -
        indexw -
        gers -
        z -
        ldz -
        isuppz -
        work -
        iwork -
        info -

      • dlartg

        public abstract void dlartg(double f,
                            double g,
                            org.netlib.util.doubleW cs,
                            org.netlib.util.doubleW sn,
                            org.netlib.util.doubleW r)
        
     ..

  Purpose
  =======

  DLARTG generate a plane rotation so that

     [  CS  SN  ]  .  [ F ]  =  [ R ]   where CS**2 + SN**2 = 1.
     [ -SN  CS  ]     [ G ]     [ 0 ]

  This is a slower, more accurate version of the BLAS1 routine DROTG,
  with the following other differences:
     F and G are unchanged on return.
     If G=0, then CS=1 and SN=0.
     If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
        floating point operations (saves work in DBDSQR when
        there are zeros on the diagonal).

  If F exceeds G in magnitude, CS will be positive.

  Arguments
  =========

  F       (input) DOUBLE PRECISION
          The first component of vector to be rotated.

  G       (input) DOUBLE PRECISION
          The second component of vector to be rotated.

  CS      (output) DOUBLE PRECISION
          The cosine of the rotation.

  SN      (output) DOUBLE PRECISION
          The sine of the rotation.

  R       (output) DOUBLE PRECISION
          The nonzero component of the rotated vector.

  This version has a few statements commented out for thread safety
  (machine parameters are computed on each entry). 10 feb 03, SJH.

  =====================================================================

     .. Parameters ..
        Parameters:
        f -
        g -
        cs -
        sn -
        r -

      • dlartv

        public abstract void dlartv(int n,
                            double[] x,
                            int incx,
                            double[] y,
                            int incy,
                            double[] c,
                            double[] s,
                            int incc)
        
     ..

  Purpose
  =======

  DLARTV applies a vector of real plane rotations to elements of the
  real vectors x and y. For i = 1,2,...,n

     ( x(i) ) := (  c(i)  s(i) ) ( x(i) )
     ( y(i) )    ( -s(i)  c(i) ) ( y(i) )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be applied.

  X       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector x.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  Y       (input/output) DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          The vector y.

  INCY    (input) INTEGER
          The increment between elements of Y. INCY > 0.

  C       (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  S       (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C and S. INCC > 0.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        c -
        s -
        incc -

      • dlaruv

        public abstract void dlaruv(int[] iseed,
                            int n,
                            double[] x)
        
     ..

  Purpose
  =======

  DLARUV returns a vector of n random real numbers from a uniform (0,1)
  distribution (n <= 128).

  This is an auxiliary routine called by DLARNV and ZLARNV.

  Arguments
  =========

  ISEED   (input/output) INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array

          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

  N       (input) INTEGER
          The number of random numbers to be generated. N <= 128.

  X       (output) DOUBLE PRECISION array, dimension (N)
          The generated random numbers.

  Further Details
  ===============

  This routine uses a multiplicative congruential method with modulus
  2**48 and multiplier 33952834046453 (see G.S.Fishman,
  'Multiplicative congruential random number generators with modulus
  2**b: an exhaustive analysis for b = 32 and a partial analysis for
  b = 48', Math. Comp. 189, pp 331-344, 1990).

  48-bit integers are stored in 4 integer array elements with 12 bits
  per element. Hence the routine is portable across machines with
  integers of 32 bits or more.

  =====================================================================

     .. Parameters ..
        Parameters:
        iseed -
        n -
        x -

      • dlarz

        public abstract void dlarz(java.lang.String side,
                           int m,
                           int n,
                           int l,
                           double[] v,
                           int incv,
                           double tau,
                           double[] c,
                           int Ldc,
                           double[] work)
        
     ..

  Purpose
  =======

  DLARZ applies a real elementary reflector H to a real M-by-N
  matrix C, from either the left or the right. H is represented in the

  form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix.


  H is a product of k elementary reflectors as returned by DTZRZF.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  L       (input) INTEGER
          The number of entries of the vector V containing
          the meaningful part of the Householder vectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  V       (input) DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
          The vector v in the representation of H as returned by
          DTZRZF. V is not used if TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0.

  TAU     (input) DOUBLE PRECISION
          The value tau in the representation of H.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        l -
        v -
        incv -
        tau -
        c -
        Ldc -
        work -

      • dlarzb

        public abstract void dlarzb(java.lang.String side,
                            java.lang.String trans,
                            java.lang.String direct,
                            java.lang.String storev,
                            int m,
                            int n,
                            int k,
                            int l,
                            double[] v,
                            int ldv,
                            double[] t,
                            int ldt,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int ldwork)
        
     ..

  Purpose
  =======

  DLARZB applies a real block reflector H or its transpose H**T to
  a real distributed M-by-N  C from the left or the right.

  Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply H or H' from the Left
          = 'R': apply H or H' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H' (Transpose)

  DIRECT  (input) CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)

          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise                        (not supported yet)

          = 'R': Rowwise

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  K       (input) INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

  L       (input) INTEGER
          The number of columns of the matrix V containing the
          meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  V       (input) DOUBLE PRECISION array, dimension (LDV,NV).
          If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.

  T       (input) DOUBLE PRECISION array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by H*C or H'*C or C*H or C*H'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension (LDWORK,K)

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        direct -
        storev -
        m -
        n -
        k -
        l -
        v -
        ldv -
        t -
        ldt -
        c -
        Ldc -
        work -
        ldwork -

      • dlarzt

        public abstract void dlarzt(java.lang.String direct,
                            java.lang.String storev,
                            int n,
                            int k,
                            double[] v,
                            int ldv,
                            double[] tau,
                            double[] t,
                            int ldt)
        
     ..

  Purpose
  =======

  DLARZT forms the triangular factor T of a real block reflector
  H of order > n, which is defined as a product of k elementary
  reflectors.

  If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;


  If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.


  If STOREV = 'C', the vector which defines the elementary reflector
  H(i) is stored in the i-th column of the array V, and

     H  =  I - V * T * V'

  If STOREV = 'R', the vector which defines the elementary reflector
  H(i) is stored in the i-th row of the array V, and

     H  =  I - V' * T * V

  Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

  Arguments
  =========

  DIRECT  (input) CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)

          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise                        (not supported yet)

          = 'R': rowwise

  N       (input) INTEGER
          The order of the block reflector H. N >= 0.

  K       (input) INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

  V       (input/output) DOUBLE PRECISION array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.


  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

  T       (output) DOUBLE PRECISION array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and

  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

                                              ______V_____
         ( v1 v2 v3 )                        /            \
         ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
     V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
         ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
         ( v1 v2 v3 )
            .  .  .
            .  .  .
            1  .  .
               1  .
                  1

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

                                                        ______V_____
            1                                          /            \
            .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
            .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
            .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
            .  .  .
         ( v1 v2 v3 )
         ( v1 v2 v3 )
     V = ( v1 v2 v3 )
         ( v1 v2 v3 )
         ( v1 v2 v3 )

  =====================================================================

     .. Parameters ..
        Parameters:
        direct -
        storev -
        n -
        k -
        v -
        ldv -
        tau -
        t -
        ldt -

      • dlas2

        public abstract void dlas2(double f,
                           double g,
                           double h,
                           org.netlib.util.doubleW ssmin,
                           org.netlib.util.doubleW ssmax)
        
     ..

  Purpose
  =======

  DLAS2  computes the singular values of the 2-by-2 matrix
     [  F   G  ]
     [  0   H  ].
  On return, SSMIN is the smaller singular value and SSMAX is the
  larger singular value.

  Arguments
  =========

  F       (input) DOUBLE PRECISION
          The (1,1) element of the 2-by-2 matrix.

  G       (input) DOUBLE PRECISION
          The (1,2) element of the 2-by-2 matrix.

  H       (input) DOUBLE PRECISION
          The (2,2) element of the 2-by-2 matrix.

  SSMIN   (output) DOUBLE PRECISION
          The smaller singular value.

  SSMAX   (output) DOUBLE PRECISION
          The larger singular value.

  Further Details
  ===============

  Barring over/underflow, all output quantities are correct to within
  a few units in the last place (ulps), even in the absence of a guard

  digit in addition/subtraction.

  In IEEE arithmetic, the code works correctly if one matrix element is
  infinite.

  Overflow will not occur unless the largest singular value itself
  overflows, or is within a few ulps of overflow. (On machines with
  partial overflow, like the Cray, overflow may occur if the largest
  singular value is within a factor of 2 of overflow.)

  Underflow is harmless if underflow is gradual. Otherwise, results
  may correspond to a matrix modified by perturbations of size near
  the underflow threshold.

  ====================================================================


     .. Parameters ..
        Parameters:
        f -
        g -
        h -
        ssmin -
        ssmax -

      • dlascl

        public abstract void dlascl(java.lang.String type,
                            int kl,
                            int ku,
                            double cfrom,
                            double cto,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASCL multiplies the M by N real matrix A by the real scalar
  CTO/CFROM.  This is done without over/underflow as long as the final

  result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that

  A may be full, upper triangular, lower triangular, upper Hessenberg,

  or banded.

  Arguments
  =========

  TYPE    (input) CHARACTER*1
          TYPE indices the storage type of the input matrix.
          = 'G':  A is a full matrix.
          = 'L':  A is a lower triangular matrix.
          = 'U':  A is an upper triangular matrix.
          = 'H':  A is an upper Hessenberg matrix.
          = 'B':  A is a symmetric band matrix with lower bandwidth KL

                  and upper bandwidth KU and with the only the lower
                  half stored.
          = 'Q':  A is a symmetric band matrix with lower bandwidth KL

                  and upper bandwidth KU and with the only the upper
                  half stored.
          = 'Z':  A is a band matrix with lower bandwidth KL and upper

                  bandwidth KU.

  KL      (input) INTEGER
          The lower bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

  KU      (input) INTEGER
          The upper bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

  CFROM   (input) DOUBLE PRECISION
  CTO     (input) DOUBLE PRECISION
          The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
          without over/underflow if the final result CTO*A(I,J)/CFROM
          can be represented without over/underflow.  CFROM must be
          nonzero.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          The matrix to be multiplied by CTO/CFROM.  See TYPE for the
          storage type.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  INFO    (output) INTEGER
          0  - successful exit
          <0 - if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        type -
        kl -
        ku -
        cfrom -
        cto -
        m -
        n -
        a -
        lda -
        info -

      • dlasd0

        public abstract void dlasd0(int n,
                            int sqre,
                            double[] d,
                            double[] e,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            int smlsiz,
                            int[] iwork,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Using a divide and conquer approach, DLASD0 computes the singular
  value decomposition (SVD) of a real upper bidiagonal N-by-M
  matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
  The algorithm computes orthogonal matrices U and VT such that
  B = U * S * VT. The singular values S are overwritten on D.

  A related subroutine, DLASDA, computes only the singular values,
  and optionally, the singular vectors in compact form.

  Arguments
  =========

  N      (input) INTEGER
         On entry, the row dimension of the upper bidiagonal matrix.
         This is also the dimension of the main diagonal array D.

  SQRE   (input) INTEGER
         Specifies the column dimension of the bidiagonal matrix.
         = 0: The bidiagonal matrix has column dimension M = N;
         = 1: The bidiagonal matrix has column dimension M = N+1;

  D      (input/output) DOUBLE PRECISION array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix.
         On exit D, if INFO = 0, contains its singular values.

  E      (input) DOUBLE PRECISION array, dimension (M-1)
         Contains the subdiagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

  U      (output) DOUBLE PRECISION array, dimension at least (LDQ, N)
         On exit, U contains the left singular vectors.

  LDU    (input) INTEGER
         On entry, leading dimension of U.

  VT     (output) DOUBLE PRECISION array, dimension at least (LDVT, M)

         On exit, VT' contains the right singular vectors.

  LDVT   (input) INTEGER
         On entry, leading dimension of VT.

  SMLSIZ (input) INTEGER
         On entry, maximum size of the subproblems at the
         bottom of the computation tree.

  IWORK  (workspace) INTEGER work array.
         Dimension must be at least (8 * N)

  WORK   (workspace) DOUBLE PRECISION work array.
         Dimension must be at least (3 * M**2 + 2 * M)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        sqre -
        d -
        e -
        u -
        ldu -
        vt -
        ldvt -
        smlsiz -
        iwork -
        work -
        info -

      • dlasd1

        public abstract void dlasd1(int nl,
                            int nr,
                            int sqre,
                            double[] d,
                            org.netlib.util.doubleW alpha,
                            org.netlib.util.doubleW beta,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            int[] idxq,
                            int[] iwork,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
  where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.

  A related subroutine DLASD7 handles the case in which the singular
  values (and the singular vectors in factored form) are desired.

  DLASD1 computes the SVD as follows:

                ( D1(in)  0    0     0 )
    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                (   0     0   D2(in) 0 )

      = U(out) * ( D(out) 0) * VT(out)

  where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
  elsewhere; and the entry b is empty if SQRE = 0.

  The left singular vectors of the original matrix are stored in U, and
  the transpose of the right singular vectors are stored in VT, and the
  singular values are in D.  The algorithm consists of three stages:

     The first stage consists of deflating the size of the problem
     when there are multiple singular values or when there are zeros in
     the Z vector.  For each such occurence the dimension of the
     secular equation problem is reduced by one.  This stage is
     performed by the routine DLASD2.

     The second stage consists of calculating the updated
     singular values. This is done by finding the square roots of the
     roots of the secular equation via the routine DLASD4 (as called
     by DLASD3). This routine also calculates the singular vectors of
     the current problem.

     The final stage consists of computing the updated singular vectors
     directly using the updated singular values.  The singular vectors

     for the current problem are multiplied with the singular vectors
     from the overall problem.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  D      (input/output) DOUBLE PRECISION array,
                        dimension (N = NL+NR+1).
         On entry D(1:NL,1:NL) contains the singular values of the
         upper block; and D(NL+2:N) contains the singular values of
         the lower block. On exit D(1:N) contains the singular values
         of the modified matrix.

  ALPHA  (input/output) DOUBLE PRECISION
         Contains the diagonal element associated with the added row.

  BETA   (input/output) DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.

  U      (input/output) DOUBLE PRECISION array, dimension(LDU,N)
         On entry U(1:NL, 1:NL) contains the left singular vectors of
         the upper block; U(NL+2:N, NL+2:N) contains the left singular

         vectors of the lower block. On exit U contains the left
         singular vectors of the bidiagonal matrix.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= max( 1, N ).

  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
         where M = N + SQRE.
         On entry VT(1:NL+1, 1:NL+1)' contains the right singular
         vectors of the upper block; VT(NL+2:M, NL+2:M)' contains
         the right singular vectors of the lower block. On exit
         VT' contains the right singular vectors of the
         bidiagonal matrix.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= max( 1, M ).

  IDXQ  (output) INTEGER array, dimension(N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order, i.e.
         D( IDXQ( I = 1, N ) ) will be in ascending order.

  IWORK  (workspace) INTEGER array, dimension( 4 * N )

  WORK   (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )


  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        d -
        alpha -
        beta -
        u -
        ldu -
        vt -
        ldvt -
        idxq -
        iwork -
        work -
        info -

      • dlasd2

        public abstract void dlasd2(int nl,
                            int nr,
                            int sqre,
                            org.netlib.util.intW k,
                            double[] d,
                            double[] z,
                            double alpha,
                            double beta,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int ldvt,
                            double[] dsigma,
                            double[] u2,
                            int ldu2,
                            double[] vt2,
                            int ldvt2,
                            int[] idxp,
                            int[] idx,
                            int[] idxc,
                            int[] idxq,
                            int[] coltyp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD2 merges the two sets of singular values together into a single

  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  singular values are close together or if there is a tiny entry in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  DLASD2 is called from DLASD1.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  D      (input/output) DOUBLE PRECISION array, dimension(N)
         On entry D contains the singular values of the two submatrices
         to be combined.  On exit D contains the trailing (N-K) updated
         singular values (those which were deflated) sorted into
         increasing order.

  Z      (output) DOUBLE PRECISION array, dimension(N)
         On exit Z contains the updating row vector in the secular
         equation.

  ALPHA  (input) DOUBLE PRECISION
         Contains the diagonal element associated with the added row.

  BETA   (input) DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.

  U      (input/output) DOUBLE PRECISION array, dimension(LDU,N)
         On entry U contains the left singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL, NL), and (NL+2, NL+2), (N,N).
         On exit U contains the trailing (N-K) updated left singular
         vectors (those which were deflated) in its last N-K columns.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= N.

  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
         On entry VT' contains the right singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
         On exit VT' contains the trailing (N-K) updated right singular
         vectors (those which were deflated) in its last N-K columns.
         In case SQRE =1, the last row of VT spans the right null
         space.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= M.

  DSIGMA (output) DOUBLE PRECISION array, dimension (N)
         Contains a copy of the diagonal elements (K-1 singular values

         and one zero) in the secular equation.

  U2     (output) DOUBLE PRECISION array, dimension(LDU2,N)
         Contains a copy of the first K-1 left singular vectors which
         will be used by DLASD3 in a matrix multiply (DGEMM) to solve
         for the new left singular vectors. U2 is arranged into four
         blocks. The first block contains a column with 1 at NL+1 and
         zero everywhere else; the second block contains non-zero
         entries only at and above NL; the third contains non-zero
         entries only below NL+1; and the fourth is dense.

  LDU2   (input) INTEGER
         The leading dimension of the array U2.  LDU2 >= N.

  VT2    (output) DOUBLE PRECISION array, dimension(LDVT2,N)
         VT2' contains a copy of the first K right singular vectors
         which will be used by DLASD3 in a matrix multiply (DGEMM) to
         solve for the new right singular vectors. VT2 is arranged into
         three blocks. The first block contains a row that corresponds

         to the special 0 diagonal element in SIGMA; the second block
         contains non-zeros only at and before NL +1; the third block
         contains non-zeros only at and after  NL +2.

  LDVT2  (input) INTEGER
         The leading dimension of the array VT2.  LDVT2 >= M.

  IDXP   (workspace) INTEGER array dimension(N)
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output IDXP(2:K)
         points to the nondeflated D-values and IDXP(K+1:N)
         points to the deflated singular values.

  IDX    (workspace) INTEGER array dimension(N)
         This will contain the permutation used to sort the contents of
         D into ascending order.

  IDXC   (output) INTEGER array dimension(N)
         This will contain the permutation used to arrange the columns

         of the deflated U matrix into three groups:  the first group
         contains non-zero entries only at and above NL, the second
         contains non-zero entries only below NL+2, and the third is
         dense.

  IDXQ   (input/output) INTEGER array dimension(N)
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that entries in

         the first hlaf of this permutation must first be moved one
         position backward; and entries in the second half
         must first have NL+1 added to their values.

  COLTYP (workspace/output) INTEGER array dimension(N)
         As workspace, this will contain a label which will indicate
         which of the following types a column in the U2 matrix or a
         row in the VT2 matrix is:
         1 : non-zero in the upper half only
         2 : non-zero in the lower half only
         3 : dense
         4 : deflated

         On exit, it is an array of dimension 4, with COLTYP(I) being
         the dimension of the I-th type columns.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        k -
        d -
        z -
        alpha -
        beta -
        u -
        ldu -
        vt -
        ldvt -
        dsigma -
        u2 -
        ldu2 -
        vt2 -
        ldvt2 -
        idxp -
        idx -
        idxc -
        idxq -
        coltyp -
        info -

      • dlasd3

        public abstract void dlasd3(int nl,
                            int nr,
                            int sqre,
                            int k,
                            double[] d,
                            double[] q,
                            int ldq,
                            double[] dsigma,
                            double[] u,
                            int ldu,
                            double[] u2,
                            int ldu2,
                            double[] vt,
                            int ldvt,
                            double[] vt2,
                            int ldvt2,
                            int[] idxc,
                            int[] ctot,
                            double[] z,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD3 finds all the square roots of the roots of the secular
  equation, as defined by the values in D and Z.  It makes the
  appropriate calls to DLASD4 and then updates the singular
  vectors by matrix multiplication.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  DLASD3 is called from DLASD1.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (input) INTEGER
         The size of the secular equation, 1 =< K = < N.

  D      (output) DOUBLE PRECISION array, dimension(K)
         On exit the square roots of the roots of the secular equation,
         in ascending order.

  Q      (workspace) DOUBLE PRECISION array,
                     dimension at least (LDQ,K).

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= K.

  DSIGMA (input) DOUBLE PRECISION array, dimension(K)
         The first K elements of this array contain the old roots
         of the deflated updating problem.  These are the poles
         of the secular equation.

  U      (output) DOUBLE PRECISION array, dimension (LDU, N)
         The last N - K columns of this matrix contain the deflated
         left singular vectors.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= N.

  U2     (input/output) DOUBLE PRECISION array, dimension (LDU2, N)
         The first K columns of this matrix contain the non-deflated
         left singular vectors for the split problem.

  LDU2   (input) INTEGER
         The leading dimension of the array U2.  LDU2 >= N.

  VT     (output) DOUBLE PRECISION array, dimension (LDVT, M)
         The last M - K columns of VT' contain the deflated
         right singular vectors.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= N.

  VT2    (input/output) DOUBLE PRECISION array, dimension (LDVT2, N)
         The first K columns of VT2' contain the non-deflated
         right singular vectors for the split problem.

  LDVT2  (input) INTEGER
         The leading dimension of the array VT2.  LDVT2 >= N.

  IDXC   (input) INTEGER array, dimension ( N )
         The permutation used to arrange the columns of U (and rows of

         VT) into three groups:  the first group contains non-zero
         entries only at and above (or before) NL +1; the second
         contains non-zero entries only at and below (or after) NL+2;
         and the third is dense. The first column of U and the row of
         VT are treated separately, however.

         The rows of the singular vectors found by DLASD4
         must be likewise permuted before the matrix multiplies can
         take place.

  CTOT   (input) INTEGER array, dimension ( 4 )
         A count of the total number of the various types of columns
         in U (or rows in VT), as described in IDXC. The fourth column

         type is any column which has been deflated.

  Z      (input) DOUBLE PRECISION array, dimension (K)
         The first K elements of this array contain the components
         of the deflation-adjusted updating row vector.

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        k -
        d -
        q -
        ldq -
        dsigma -
        u -
        ldu -
        u2 -
        ldu2 -
        vt -
        ldvt -
        vt2 -
        ldvt2 -
        idxc -
        ctot -
        z -
        info -

      • dlasd4

        public abstract void dlasd4(int n,
                            int i,
                            double[] d,
                            double[] z,
                            double[] delta,
                            double rho,
                            org.netlib.util.doubleW sigma,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This subroutine computes the square root of the I-th updated
  eigenvalue of a positive symmetric rank-one modification to
  a positive diagonal matrix whose entries are given as the squares
  of the corresponding entries in the array d, and that

         0 <= D(i) < D(j)  for  i < j

  and that RHO > 0. This is arranged by the calling routine, and is
  no loss in generality.  The rank-one modified system is thus

         diag( D ) * diag( D ) +  RHO *  Z * Z_transpose.

  where we assume the Euclidean norm of Z is 1.

  The method consists of approximating the rational functions in the
  secular equation by simpler interpolating rational functions.

  Arguments
  =========

  N      (input) INTEGER
         The length of all arrays.

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

  D      (input) DOUBLE PRECISION array, dimension ( N )
         The original eigenvalues.  It is assumed that they are in
         order, 0 <= D(I) < D(J)  for I < J.

  Z      (input) DOUBLE PRECISION array, dimension ( N )
         The components of the updating vector.

  DELTA  (output) DOUBLE PRECISION array, dimension ( N )
         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
         contains the information necessary to construct the
         (singular) eigenvectors.

  RHO    (input) DOUBLE PRECISION
         The scalar in the symmetric updating formula.

  SIGMA  (output) DOUBLE PRECISION
         The computed sigma_I, the I-th updated eigenvalue.

  WORK   (workspace) DOUBLE PRECISION array, dimension ( N )
         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
         component.  If N = 1, then WORK( 1 ) = 1.

  INFO   (output) INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Internal Parameters
  ===================

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.

            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1

  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
  if we are working with THREE poles!

  MAXIT is the maximum number of iterations allowed for each
  eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        i -
        d -
        z -
        delta -
        rho -
        sigma -
        work -
        info -

      • dlasd5

        public abstract void dlasd5(int i,
                            double[] d,
                            double[] z,
                            double[] delta,
                            double rho,
                            org.netlib.util.doubleW dsigma,
                            double[] work)
        
     ..

  Purpose
  =======

  This subroutine computes the square root of the I-th eigenvalue
  of a positive symmetric rank-one modification of a 2-by-2 diagonal
  matrix

             diag( D ) * diag( D ) +  RHO *  Z * transpose(Z) .

  The diagonal entries in the array D are assumed to satisfy

             0 <= D(i) < D(j)  for  i < j .

  We also assume RHO > 0 and that the Euclidean norm of the vector
  Z is one.

  Arguments
  =========

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

  D      (input) DOUBLE PRECISION array, dimension ( 2 )
         The original eigenvalues.  We assume 0 <= D(1) < D(2).

  Z      (input) DOUBLE PRECISION array, dimension ( 2 )
         The components of the updating vector.

  DELTA  (output) DOUBLE PRECISION array, dimension ( 2 )
         Contains (D(j) - sigma_I) in its  j-th component.
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

  RHO    (input) DOUBLE PRECISION
         The scalar in the symmetric updating formula.

  DSIGMA (output) DOUBLE PRECISION
         The computed sigma_I, the I-th updated eigenvalue.

  WORK   (workspace) DOUBLE PRECISION array, dimension ( 2 )
         WORK contains (D(j) + sigma_I) in its  j-th component.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        i -
        d -
        z -
        delta -
        rho -
        dsigma -
        work -

      • dlasd6

        public abstract void dlasd6(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            double[] d,
                            double[] vf,
                            double[] vl,
                            org.netlib.util.doubleW alpha,
                            org.netlib.util.doubleW beta,
                            int[] idxq,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            int ldgcol,
                            double[] givnum,
                            int ldgnum,
                            double[] poles,
                            double[] difl,
                            double[] difr,
                            double[] z,
                            org.netlib.util.intW k,
                            org.netlib.util.doubleW c,
                            org.netlib.util.doubleW s,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD6 computes the SVD of an updated upper bidiagonal matrix B
  obtained by merging two smaller ones by appending a row. This
  routine is used only for the problem which requires all singular
  values and optionally singular vector matrices in factored form.
  B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
  A related subroutine, DLASD1, handles the case in which all singular

  values and singular vectors of the bidiagonal matrix are desired.

  DLASD6 computes the SVD as follows:

                ( D1(in)  0    0     0 )
    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                (   0     0   D2(in) 0 )

      = U(out) * ( D(out) 0) * VT(out)

  where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
  elsewhere; and the entry b is empty if SQRE = 0.

  The singular values of B can be computed using D1, D2, the first
  components of all the right singular vectors of the lower block, and

  the last components of all the right singular vectors of the upper
  block. These components are stored and updated in VF and VL,
  respectively, in DLASD6. Hence U and VT are not explicitly
  referenced.

  The singular values are stored in D. The algorithm consists of two
  stages:

        The first stage consists of deflating the size of the problem
        when there are multiple singular values or if there is a zero
        in the Z vector. For each such occurence the dimension of the
        secular equation problem is reduced by one. This stage is
        performed by the routine DLASD7.

        The second stage consists of calculating the updated
        singular values. This is done by finding the roots of the
        secular equation via the routine DLASD4 (as called by DLASD8).

        This routine also updates VF and VL and computes the distances

        between the updated singular values and the old singular
        values.

  DLASD6 is called from DLASDA.

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Compute singular values only.
         = 1: Compute singular vectors in factored form as well.

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  D      (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ).

         On entry D(1:NL,1:NL) contains the singular values of the
         upper block, and D(NL+2:N) contains the singular values
         of the lower block. On exit D(1:N) contains the singular
         values of the modified matrix.

  VF     (input/output) DOUBLE PRECISION array, dimension ( M )
         On entry, VF(1:NL+1) contains the first components of all
         right singular vectors of the upper block; and VF(NL+2:M)
         contains the first components of all right singular vectors
         of the lower block. On exit, VF contains the first components

         of all right singular vectors of the bidiagonal matrix.

  VL     (input/output) DOUBLE PRECISION array, dimension ( M )
         On entry, VL(1:NL+1) contains the  last components of all
         right singular vectors of the upper block; and VL(NL+2:M)
         contains the last components of all right singular vectors of

         the lower block. On exit, VL contains the last components of
         all right singular vectors of the bidiagonal matrix.

  ALPHA  (input/output) DOUBLE PRECISION
         Contains the diagonal element associated with the added row.

  BETA   (input/output) DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.

  IDXQ   (output) INTEGER array, dimension ( N )
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order, i.e.
         D( IDXQ( I = 1, N ) ) will be in ascending order.

  PERM   (output) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) to be applied
         to each block. Not referenced if ICOMPQ = 0.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem. Not referenced if ICOMPQ = 0.

  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation. Not referenced if ICOMPQ = 0.

  LDGCOL (input) INTEGER
         leading dimension of GIVCOL, must be at least N.

  GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value to be used in the
         corresponding Givens rotation. Not referenced if ICOMPQ = 0.

  LDGNUM (input) INTEGER
         The leading dimension of GIVNUM and POLES, must be at least N.

  POLES  (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         On exit, POLES(1,*) is an array containing the new singular
         values obtained from solving the secular equation, and
         POLES(2,*) is an array containing the poles in the secular
         equation. Not referenced if ICOMPQ = 0.

  DIFL   (output) DOUBLE PRECISION array, dimension ( N )
         On exit, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

  DIFR   (output) DOUBLE PRECISION array,
                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         On exit, DIFR(I, 1) is the distance between I-th updated
         (undeflated) singular value and the I+1-th (undeflated) old
         singular value.

         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
         normalizing factors for the right singular vector matrix.

         See DLASD8 for details on DIFL and DIFR.

  Z      (output) DOUBLE PRECISION array, dimension ( M )
         The first elements of this array contain the components
         of the deflation-adjusted updating row vector.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  C      (output) DOUBLE PRECISION
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (output) DOUBLE PRECISION
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  WORK   (workspace) DOUBLE PRECISION array, dimension ( 4 * M )

  IWORK  (workspace) INTEGER array, dimension ( 3 * N )

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        d -
        vf -
        vl -
        alpha -
        beta -
        idxq -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        poles -
        difl -
        difr -
        z -
        k -
        c -
        s -
        work -
        iwork -
        info -

      • dlasd7

        public abstract void dlasd7(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            org.netlib.util.intW k,
                            double[] d,
                            double[] z,
                            double[] zw,
                            double[] vf,
                            double[] vfw,
                            double[] vl,
                            double[] vlw,
                            double alpha,
                            double beta,
                            double[] dsigma,
                            int[] idx,
                            int[] idxp,
                            int[] idxq,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            int ldgcol,
                            double[] givnum,
                            int ldgnum,
                            org.netlib.util.doubleW c,
                            org.netlib.util.doubleW s,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD7 merges the two sets of singular values together into a single

  sorted set. Then it tries to deflate the size of the problem. There
  are two ways in which deflation can occur:  when two or more singular
  values are close together or if there is a tiny entry in the Z
  vector. For each such occurrence the order of the related
  secular equation problem is reduced by one.

  DLASD7 is called from DLASD6.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          Specifies whether singular vectors are to be computed
          in compact form, as follows:
          = 0: Compute singular values only.
          = 1: Compute singular vectors of upper
               bidiagonal matrix in compact form.

  NL     (input) INTEGER
         The row dimension of the upper block. NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block. NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has
         N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix, this is
         the order of the related secular equation. 1 <= K <=N.

  D      (input/output) DOUBLE PRECISION array, dimension ( N )
         On entry D contains the singular values of the two submatrices
         to be combined. On exit D contains the trailing (N-K) updated

         singular values (those which were deflated) sorted into
         increasing order.

  Z      (output) DOUBLE PRECISION array, dimension ( M )
         On exit Z contains the updating row vector in the secular
         equation.

  ZW     (workspace) DOUBLE PRECISION array, dimension ( M )
         Workspace for Z.

  VF     (input/output) DOUBLE PRECISION array, dimension ( M )
         On entry, VF(1:NL+1) contains the first components of all
         right singular vectors of the upper block; and VF(NL+2:M)
         contains the first components of all right singular vectors
         of the lower block. On exit, VF contains the first components

         of all right singular vectors of the bidiagonal matrix.

  VFW    (workspace) DOUBLE PRECISION array, dimension ( M )
         Workspace for VF.

  VL     (input/output) DOUBLE PRECISION array, dimension ( M )
         On entry, VL(1:NL+1) contains the  last components of all
         right singular vectors of the upper block; and VL(NL+2:M)
         contains the last components of all right singular vectors
         of the lower block. On exit, VL contains the last components
         of all right singular vectors of the bidiagonal matrix.

  VLW    (workspace) DOUBLE PRECISION array, dimension ( M )
         Workspace for VL.

  ALPHA  (input) DOUBLE PRECISION
         Contains the diagonal element associated with the added row.

  BETA   (input) DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.

  DSIGMA (output) DOUBLE PRECISION array, dimension ( N )
         Contains a copy of the diagonal elements (K-1 singular values

         and one zero) in the secular equation.

  IDX    (workspace) INTEGER array, dimension ( N )
         This will contain the permutation used to sort the contents of
         D into ascending order.

  IDXP   (workspace) INTEGER array, dimension ( N )
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output IDXP(2:K)
         points to the nondeflated D-values and IDXP(K+1:N)
         points to the deflated singular values.

  IDXQ   (input) INTEGER array, dimension ( N )
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that entries in

         the first half of this permutation must first be moved one
         position backward; and entries in the second half
         must first have NL+1 added to their values.

  PERM   (output) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) to be applied
         to each singular block. Not referenced if ICOMPQ = 0.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem. Not referenced if ICOMPQ = 0.

  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation. Not referenced if ICOMPQ = 0.

  LDGCOL (input) INTEGER
         The leading dimension of GIVCOL, must be at least N.

  GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value to be used in the
         corresponding Givens rotation. Not referenced if ICOMPQ = 0.

  LDGNUM (input) INTEGER
         The leading dimension of GIVNUM, must be at least N.

  C      (output) DOUBLE PRECISION
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (output) DOUBLE PRECISION
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        k -
        d -
        z -
        zw -
        vf -
        vfw -
        vl -
        vlw -
        alpha -
        beta -
        dsigma -
        idx -
        idxp -
        idxq -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        c -
        s -
        info -

      • dlasd8

        public abstract void dlasd8(int icompq,
                            int k,
                            double[] d,
                            double[] z,
                            double[] vf,
                            double[] vl,
                            double[] difl,
                            double[] difr,
                            int lddifr,
                            double[] dsigma,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASD8 finds the square roots of the roots of the secular equation,
  as defined by the values in DSIGMA and Z. It makes the appropriate
  calls to DLASD4, and stores, for each  element in D, the distance
  to its two nearest poles (elements in DSIGMA). It also updates
  the arrays VF and VL, the first and last components of all the
  right singular vectors of the original bidiagonal matrix.

  DLASD8 is called from DLASD6.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          Specifies whether singular vectors are to be computed in
          factored form in the calling routine:
          = 0: Compute singular values only.
          = 1: Compute singular vectors in factored form as well.

  K       (input) INTEGER
          The number of terms in the rational function to be solved
          by DLASD4.  K >= 1.

  D       (output) DOUBLE PRECISION array, dimension ( K )
          On output, D contains the updated singular values.

  Z       (input) DOUBLE PRECISION array, dimension ( K )
          The first K elements of this array contain the components
          of the deflation-adjusted updating row vector.

  VF      (input/output) DOUBLE PRECISION array, dimension ( K )
          On entry, VF contains  information passed through DBEDE8.
          On exit, VF contains the first K components of the first
          components of all right singular vectors of the bidiagonal
          matrix.

  VL      (input/output) DOUBLE PRECISION array, dimension ( K )
          On entry, VL contains  information passed through DBEDE8.
          On exit, VL contains the first K components of the last
          components of all right singular vectors of the bidiagonal
          matrix.

  DIFL    (output) DOUBLE PRECISION array, dimension ( K )
          On exit, DIFL(I) = D(I) - DSIGMA(I).

  DIFR    (output) DOUBLE PRECISION array,
                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
                   dimension ( K ) if ICOMPQ = 0.
          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
          defined and will not be referenced.

          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
          normalizing factors for the right singular vector matrix.

  LDDIFR  (input) INTEGER
          The leading dimension of DIFR, must be at least K.

  DSIGMA  (input) DOUBLE PRECISION array, dimension ( K )
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

  WORK    (workspace) DOUBLE PRECISION array, dimension at least 3 * K


  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        k -
        d -
        z -
        vf -
        vl -
        difl -
        difr -
        lddifr -
        dsigma -
        work -
        info -

      • dlasda

        public abstract void dlasda(int icompq,
                            int smlsiz,
                            int n,
                            int sqre,
                            double[] d,
                            double[] e,
                            double[] u,
                            int ldu,
                            double[] vt,
                            int[] k,
                            double[] difl,
                            double[] difr,
                            double[] z,
                            double[] poles,
                            int[] givptr,
                            int[] givcol,
                            int ldgcol,
                            int[] perm,
                            double[] givnum,
                            double[] c,
                            double[] s,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Using a divide and conquer approach, DLASDA computes the singular
  value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
  B with diagonal D and offdiagonal E, where M = N + SQRE. The
  algorithm computes the singular values in the SVD B = U * S * VT.
  The orthogonal matrices U and VT are optionally computed in
  compact form.

  A related subroutine, DLASD0, computes the singular values and
  the singular vectors in explicit form.

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed
         in compact form, as follows
         = 0: Compute singular values only.
         = 1: Compute singular vectors of upper bidiagonal
              matrix in compact form.

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The row dimension of the upper bidiagonal matrix. This is
         also the dimension of the main diagonal array D.

  SQRE   (input) INTEGER
         Specifies the column dimension of the bidiagonal matrix.
         = 0: The bidiagonal matrix has column dimension M = N;
         = 1: The bidiagonal matrix has column dimension M = N + 1.

  D      (input/output) DOUBLE PRECISION array, dimension ( N )
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit D, if INFO = 0, contains its singular values.


  E      (input) DOUBLE PRECISION array, dimension ( M-1 )
         Contains the subdiagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

  U      (output) DOUBLE PRECISION array,
         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
         singular vector matrices of all subproblems at the bottom
         level.

  LDU    (input) INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
         GIVNUM, and Z.

  VT     (output) DOUBLE PRECISION array,
         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced

         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right

         singular vector matrices of all subproblems at the bottom
         level.

  K      (output) INTEGER array,
         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
         secular equation on the computation tree.

  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
         where NLVL = floor(log_2 (N/SMLSIZ))).

  DIFR   (output) DOUBLE PRECISION array,
                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)

         record distances between singular values on the I-th
         level and singular values on the (I -1)-th level, and
         DIFR(1:N, 2 * I ) contains the normalizing factors for
         the right singular vector matrix. See DLASD8 for details.

  Z      (output) DOUBLE PRECISION array,
                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         The first K elements of Z(1, I) contain the components of
         the deflation-adjusted updating row vector for subproblems
         on the I-th level.

  POLES  (output) DOUBLE PRECISION array,
         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced

         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
         POLES(1, 2*I) contain  the new and old singular values
         involved in the secular equations on the I-th level.

  GIVPTR (output) INTEGER array,
         dimension ( N ) if ICOMPQ = 1, and not referenced if
         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
         the number of Givens rotations performed on the I-th
         problem on the computation tree.

  GIVCOL (output) INTEGER array,
         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
         of Givens rotations performed on the I-th level on the
         computation tree.

  LDGCOL (input) INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

  PERM   (output) INTEGER array,
         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
         permutations done on the I-th level of the computation tree.

  GIVNUM (output) DOUBLE PRECISION array,
         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
         values of Givens rotations performed on the I-th level on
         the computation tree.

  C      (output) DOUBLE PRECISION array,
         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.

         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,

         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  S      (output) DOUBLE PRECISION array, dimension ( N ) if
         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
         and the I-th subproblem is not square, on exit, S( I )
         contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  WORK   (workspace) DOUBLE PRECISION array, dimension
         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

  IWORK  (workspace) INTEGER array.
         Dimension must be at least (7 * N).

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        smlsiz -
        n -
        sqre -
        d -
        e -
        u -
        ldu -
        vt -
        k -
        difl -
        difr -
        z -
        poles -
        givptr -
        givcol -
        ldgcol -
        perm -
        givnum -
        c -
        s -
        work -
        iwork -
        info -

      • dlasdq

        public abstract void dlasdq(java.lang.String uplo,
                            int sqre,
                            int n,
                            int ncvt,
                            int nru,
                            int ncc,
                            double[] d,
                            double[] e,
                            double[] vt,
                            int ldvt,
                            double[] u,
                            int ldu,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASDQ computes the singular value decomposition (SVD) of a real
  (upper or lower) bidiagonal matrix with diagonal D and offdiagonal
  E, accumulating the transformations if desired. Letting B denote
  the input bidiagonal matrix, the algorithm computes orthogonal
  matrices Q and P such that B = Q * S * P' (P' denotes the transpose
  of P). The singular values S are overwritten on D.

  The input matrix U  is changed to U  * Q  if desired.
  The input matrix VT is changed to P' * VT if desired.
  The input matrix C  is changed to Q' * C  if desired.

  See "Computing  Small Singular Values of Bidiagonal Matrices With
  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
  LAPACK Working Note #3, for a detailed description of the algorithm.


  Arguments
  =========

  UPLO  (input) CHARACTER*1
        On entry, UPLO specifies whether the input bidiagonal matrix
        is upper or lower bidiagonal, and wether it is square are
        not.
           UPLO = 'U' or 'u'   B is upper bidiagonal.
           UPLO = 'L' or 'l'   B is lower bidiagonal.

  SQRE  (input) INTEGER
        = 0: then the input matrix is N-by-N.
        = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and
             (N+1)-by-N if UPLU = 'L'.

        The bidiagonal matrix has
        N = NL + NR + 1 rows and
        M = N + SQRE >= N columns.

  N     (input) INTEGER
        On entry, N specifies the number of rows and columns
        in the matrix. N must be at least 0.

  NCVT  (input) INTEGER
        On entry, NCVT specifies the number of columns of
        the matrix VT. NCVT must be at least 0.

  NRU   (input) INTEGER
        On entry, NRU specifies the number of rows of
        the matrix U. NRU must be at least 0.

  NCC   (input) INTEGER
        On entry, NCC specifies the number of columns of
        the matrix C. NCC must be at least 0.

  D     (input/output) DOUBLE PRECISION array, dimension (N)
        On entry, D contains the diagonal entries of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in ascending order.

  E     (input/output) DOUBLE PRECISION array.
        dimension is (N-1) if SQRE = 0 and N if SQRE = 1.
        On entry, the entries of E contain the offdiagonal entries
        of the bidiagonal matrix whose SVD is desired. On normal
        exit, E will contain 0. If the algorithm does not converge,
        D and E will contain the diagonal and superdiagonal entries
        of a bidiagonal matrix orthogonally equivalent to the one
        given as input.

  VT    (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT)
        On entry, contains a matrix which on exit has been
        premultiplied by P', dimension N-by-NCVT if SQRE = 0
        and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).

  LDVT  (input) INTEGER
        On entry, LDVT specifies the leading dimension of VT as
        declared in the calling (sub) program. LDVT must be at
        least 1. If NCVT is nonzero LDVT must also be at least N.

  U     (input/output) DOUBLE PRECISION array, dimension (LDU, N)
        On entry, contains a  matrix which on exit has been
        postmultiplied by Q, dimension NRU-by-N if SQRE = 0
        and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).

  LDU   (input) INTEGER
        On entry, LDU  specifies the leading dimension of U as
        declared in the calling (sub) program. LDU must be at
        least max( 1, NRU ) .

  C     (input/output) DOUBLE PRECISION array, dimension (LDC, NCC)
        On entry, contains an N-by-NCC matrix which on exit
        has been premultiplied by Q'  dimension N-by-NCC if SQRE = 0
        and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).

  LDC   (input) INTEGER
        On entry, LDC  specifies the leading dimension of C as
        declared in the calling (sub) program. LDC must be at
        least 1. If NCC is nonzero, LDC must also be at least N.

  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N)
        Workspace. Only referenced if one of NCVT, NRU, or NCC is
        nonzero, and if N is at least 2.

  INFO  (output) INTEGER
        On exit, a value of 0 indicates a successful exit.
        If INFO < 0, argument number -INFO is illegal.
        If INFO > 0, the algorithm did not converge, and INFO
        specifies how many superdiagonals did not converge.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        sqre -
        n -
        ncvt -
        nru -
        ncc -
        d -
        e -
        vt -
        ldvt -
        u -
        ldu -
        c -
        Ldc -
        work -
        info -

      • dlasdt

        public abstract void dlasdt(int n,
                            org.netlib.util.intW lvl,
                            org.netlib.util.intW nd,
                            int[] inode,
                            int[] ndiml,
                            int[] ndimr,
                            int msub)
        
     ..

  Purpose
  =======

  DLASDT creates a tree of subproblems for bidiagonal divide and
  conquer.

  Arguments
  =========

   N      (input) INTEGER
          On entry, the number of diagonal elements of the
          bidiagonal matrix.

   LVL    (output) INTEGER
          On exit, the number of levels on the computation tree.

   ND     (output) INTEGER
          On exit, the number of nodes on the tree.

   INODE  (output) INTEGER array, dimension ( N )
          On exit, centers of subproblems.

   NDIML  (output) INTEGER array, dimension ( N )
          On exit, row dimensions of left children.

   NDIMR  (output) INTEGER array, dimension ( N )
          On exit, row dimensions of right children.

   MSUB   (input) INTEGER.
          On entry, the maximum row dimension each subproblem at the
          bottom of the tree can be of.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        lvl -
        nd -
        inode -
        ndiml -
        ndimr -
        msub -

      • dlaset

        public abstract void dlaset(java.lang.String uplo,
                            int m,
                            int n,
                            double alpha,
                            double beta,
                            double[] a,
                            int lda)
        
     ..

  Purpose
  =======

  DLASET initializes an m-by-n matrix A to BETA on the diagonal and
  ALPHA on the offdiagonals.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies the part of the matrix A to be set.
          = 'U':      Upper triangular part is set; the strictly lower

                      triangular part of A is not changed.
          = 'L':      Lower triangular part is set; the strictly upper

                      triangular part of A is not changed.
          Otherwise:  All of the matrix A is set.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  ALPHA   (input) DOUBLE PRECISION
          The constant to which the offdiagonal elements are to be set.

  BETA    (input) DOUBLE PRECISION
          The constant to which the diagonal elements are to be set.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On exit, the leading m-by-n submatrix of A is set as follows:

          if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
          if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
          otherwise,     A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,

          and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

 =====================================================================


     .. Local Scalars ..
        Parameters:
        uplo -
        m -
        n -
        alpha -
        beta -
        a -
        lda -

      • dlasq1

        public abstract void dlasq1(int n,
                            double[] d,
                            double[] e,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASQ1 computes the singular values of a real N-by-N bidiagonal
  matrix with diagonal D and off-diagonal E. The singular values
  are computed to high relative accuracy, in the absence of
  denormalization, underflow and overflow. The algorithm was first
  presented in

  "Accurate singular values and differential qd algorithms" by K. V.
  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
  1994,

  and the present implementation is described in "An implementation of

  the dqds Algorithm (Positive Case)", LAPACK Working Note.

  Arguments
  =========

  N     (input) INTEGER
        The number of rows and columns in the matrix. N >= 0.

  D     (input/output) DOUBLE PRECISION array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in decreasing order.

  E     (input/output) DOUBLE PRECISION array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired.
        On exit, E is overwritten.

  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N)

  INFO  (output) INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 30*N
                  iterations (in inner while loop)
             = 3, termination criterion of outer while loop not met 
                  (program created more than N unreduced blocks)

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        work -
        info -

      • dlasq2

        public abstract void dlasq2(int n,
                            double[] z,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASQ2 computes all the eigenvalues of the symmetric positive 
  definite tridiagonal matrix associated with the qd array Z to high
  relative accuracy are computed to high relative accuracy, in the
  absence of denormalization, underflow and overflow.

  To see the relation of Z to the tridiagonal matrix, let L be a
  unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
  let U be an upper bidiagonal matrix with 1's above and diagonal
  Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
  symmetric tridiagonal to which it is similar.

  Note : DLASQ2 defines a logical variable, IEEE, which is true
  on machines which follow ieee-754 floating-point standard in their
  handling of infinities and NaNs, and false otherwise. This variable
  is passed to DLAZQ3.

  Arguments
  =========

  N     (input) INTEGER
        The number of rows and columns in the matrix. N >= 0.

  Z     (workspace) DOUBLE PRECISION array, dimension ( 4*N )
        On entry Z holds the qd array. On exit, entries 1 to N hold
        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
        shifts that failed.

  INFO  (output) INTEGER
        = 0: successful exit
        < 0: if the i-th argument is a scalar and had an illegal
             value, then INFO = -i, if the i-th argument is an
             array and the j-entry had an illegal value, then
             INFO = -(i*100+j)
        > 0: the algorithm failed
              = 1, a split was marked by a positive value in E
              = 2, current block of Z not diagonalized after 30*N
                   iterations (in inner while loop)
              = 3, termination criterion of outer while loop not met 
                   (program created more than N unreduced blocks)

  Further Details
  ===============
  Local Variables: I0:N0 defines a current unreduced segment of Z.
  The shifts are accumulated in SIGMA. Iteration count is in ITER.
  Ping-pong is controlled by PP (alternates between 0 and 1).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        z -
        info -

      • dlasq3

        public abstract void dlasq3(int i0,
                            org.netlib.util.intW n0,
                            double[] z,
                            int pp,
                            org.netlib.util.doubleW dmin,
                            org.netlib.util.doubleW sigma,
                            org.netlib.util.doubleW desig,
                            org.netlib.util.doubleW qmax,
                            org.netlib.util.intW nfail,
                            org.netlib.util.intW iter,
                            org.netlib.util.intW ndiv,
                            boolean ieee)
        
     ..

  Purpose
  =======

  DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
  In case of failure it changes shifts, and tries again until output
  is positive.

  Arguments
  =========

  I0     (input) INTEGER
         First index.

  N0     (input) INTEGER
         Last index.

  Z      (input) DOUBLE PRECISION array, dimension ( 4*N )
         Z holds the qd array.

  PP     (input) INTEGER
         PP=0 for ping, PP=1 for pong.

  DMIN   (output) DOUBLE PRECISION
         Minimum value of d.

  SIGMA  (output) DOUBLE PRECISION
         Sum of shifts used in current segment.

  DESIG  (input/output) DOUBLE PRECISION
         Lower order part of SIGMA

  QMAX   (input) DOUBLE PRECISION
         Maximum value of q.

  NFAIL  (output) INTEGER
         Number of times shift was too big.

  ITER   (output) INTEGER
         Number of iterations.

  NDIV   (output) INTEGER
         Number of divisions.

  TTYPE  (output) INTEGER
         Shift type.

  IEEE   (input) LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        sigma -
        desig -
        qmax -
        nfail -
        iter -
        ndiv -
        ieee -

      • dlasq4

        public abstract void dlasq4(int i0,
                            int n0,
                            double[] z,
                            int pp,
                            int n0in,
                            double dmin,
                            double dmin1,
                            double dmin2,
                            double dn,
                            double dn1,
                            double dn2,
                            org.netlib.util.doubleW tau,
                            org.netlib.util.intW ttype)
        
     ..

  Purpose
  =======

  DLASQ4 computes an approximation TAU to the smallest eigenvalue 
  using values of d from the previous transform.

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  N0IN  (input) INTEGER
        The value of N0 at start of EIGTEST.

  DMIN  (input) DOUBLE PRECISION
        Minimum value of d.

  DMIN1 (input) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

  DMIN2 (input) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (input) DOUBLE PRECISION
        d(N)

  DN1   (input) DOUBLE PRECISION
        d(N-1)

  DN2   (input) DOUBLE PRECISION
        d(N-2)

  TAU   (output) DOUBLE PRECISION
        This is the shift.

  TTYPE (output) INTEGER
        Shift type.

  Further Details
  ===============
  CNST1 = 9/16

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        n0in -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -
        ttype -

      • dlasq5

        public abstract void dlasq5(int i0,
                            int n0,
                            double[] z,
                            int pp,
                            double tau,
                            org.netlib.util.doubleW dmin,
                            org.netlib.util.doubleW dmin1,
                            org.netlib.util.doubleW dmin2,
                            org.netlib.util.doubleW dn,
                            org.netlib.util.doubleW dnm1,
                            org.netlib.util.doubleW dnm2,
                            boolean ieee)
        
     ..

  Purpose
  =======

  DLASQ5 computes one dqds transform in ping-pong form, one
  version for IEEE machines another for non IEEE machines.

  Arguments
  =========

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  TAU   (input) DOUBLE PRECISION
        This is the shift.

  DMIN  (output) DOUBLE PRECISION
        Minimum value of d.

  DMIN1 (output) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

  DMIN2 (output) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (output) DOUBLE PRECISION
        d(N0), the last value of d.

  DNM1  (output) DOUBLE PRECISION
        d(N0-1).

  DNM2  (output) DOUBLE PRECISION
        d(N0-2).

  IEEE  (input) LOGICAL
        Flag for IEEE or non IEEE arithmetic.

  =====================================================================

     .. Parameter ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        tau -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dnm1 -
        dnm2 -
        ieee -

      • dlasq6

        public abstract void dlasq6(int i0,
                            int n0,
                            double[] z,
                            int pp,
                            org.netlib.util.doubleW dmin,
                            org.netlib.util.doubleW dmin1,
                            org.netlib.util.doubleW dmin2,
                            org.netlib.util.doubleW dn,
                            org.netlib.util.doubleW dnm1,
                            org.netlib.util.doubleW dnm2)
        
     ..

  Purpose
  =======

  DLASQ6 computes one dqd (shift equal to zero) transform in
  ping-pong form, with protection against underflow and overflow.

  Arguments
  =========

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  DMIN  (output) DOUBLE PRECISION
        Minimum value of d.

  DMIN1 (output) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

  DMIN2 (output) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (output) DOUBLE PRECISION
        d(N0), the last value of d.

  DNM1  (output) DOUBLE PRECISION
        d(N0-1).

  DNM2  (output) DOUBLE PRECISION
        d(N0-2).

  =====================================================================

     .. Parameter ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dnm1 -
        dnm2 -

      • dlasr

        public abstract void dlasr(java.lang.String side,
                           java.lang.String pivot,
                           java.lang.String direct,
                           int m,
                           int n,
                           double[] c,
                           double[] s,
                           double[] a,
                           int lda)
        
     ..

  Purpose
  =======

  DLASR applies a sequence of plane rotations to a real matrix A,
  from either the left or the right.
  
  When SIDE = 'L', the transformation takes the form
  
     A := P*A
  
  and when SIDE = 'R', the transformation takes the form
  
     A := A*P**T
  
  where P is an orthogonal matrix consisting of a sequence of z plane
  rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
  and P**T is the transpose of P.
  
  When DIRECT = 'F' (Forward sequence), then
  
     P = P(z-1) * ... * P(2) * P(1)
  
  and when DIRECT = 'B' (Backward sequence), then
  
     P = P(1) * P(2) * ... * P(z-1)
  
  where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

  
     R(k) = (  c(k)  s(k) )
          = ( -s(k)  c(k) ).
  
  When PIVOT = 'V' (Variable pivot), the rotation is performed
  for the plane (k,k+1), i.e., P(k) has the form
  
     P(k) = (  1                                            )
            (       ...                                     )
            (              1                                )
            (                   c(k)  s(k)                  )
            (                  -s(k)  c(k)                  )
            (                                1              )
            (                                     ...       )
            (                                            1  )
  
  where R(k) appears as a rank-2 modification to the identity matrix in
  rows and columns k and k+1.
  
  When PIVOT = 'T' (Top pivot), the rotation is performed for the
  plane (1,k+1), so P(k) has the form
  
     P(k) = (  c(k)                    s(k)                 )
            (         1                                     )
            (              ...                              )
            (                     1                         )
            ( -s(k)                    c(k)                 )
            (                                 1             )
            (                                      ...      )
            (                                             1 )
  
  where R(k) appears in rows and columns 1 and k+1.
  
  Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
  performed for the plane (k,z), giving P(k) the form
  
     P(k) = ( 1                                             )
            (      ...                                      )
            (             1                                 )
            (                  c(k)                    s(k) )
            (                         1                     )
            (                              ...              )
            (                                     1         )
            (                 -s(k)                    c(k) )
  
  where R(k) appears in rows and columns k and z.  The rotations are
  performed without ever forming P(k) explicitly.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          Specifies whether the plane rotation matrix P is applied to
          A on the left or the right.
          = 'L':  Left, compute A := P*A
          = 'R':  Right, compute A:= A*P**T

  PIVOT   (input) CHARACTER*1
          Specifies the plane for which P(k) is a plane rotation
          matrix.
          = 'V':  Variable pivot, the plane (k,k+1)
          = 'T':  Top pivot, the plane (1,k+1)
          = 'B':  Bottom pivot, the plane (k,z)

  DIRECT  (input) CHARACTER*1
          Specifies whether P is a forward or backward sequence of
          plane rotations.
          = 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
          = 'B':  Backward, P = P(1)*P(2)*...*P(z-1)

  M       (input) INTEGER
          The number of rows of the matrix A.  If m <= 1, an immediate

          return is effected.

  N       (input) INTEGER
          The number of columns of the matrix A.  If n <= 1, an
          immediate return is effected.

  C       (input) DOUBLE PRECISION array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The cosines c(k) of the plane rotations.

  S       (input) DOUBLE PRECISION array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The sines s(k) of the plane rotations.  The 2-by-2 plane
          rotation part of the matrix P(k), R(k), has the form
          R(k) = (  c(k)  s(k) )
                 ( -s(k)  c(k) ).

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          The M-by-N matrix A.  On exit, A is overwritten by P*A if
          SIDE = 'R' or by A*P**T if SIDE = 'L'.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        pivot -
        direct -
        m -
        n -
        c -
        s -
        a -
        lda -

      • dlasrt

        public abstract void dlasrt(java.lang.String id,
                            int n,
                            double[] d,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Sort the numbers in D in increasing order (if ID = 'I') or
  in decreasing order (if ID = 'D' ).

  Use Quick Sort, reverting to Insertion sort on arrays of
  size <= 20. Dimension of STACK limits N to about 2**32.

  Arguments
  =========

  ID      (input) CHARACTER*1
          = 'I': sort D in increasing order;
          = 'D': sort D in decreasing order.

  N       (input) INTEGER
          The length of the array D.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the array to be sorted.
          On exit, D has been sorted into increasing order
          (D(1) <= ... <= D(N) ) or into decreasing order
          (D(1) >= ... >= D(N) ), depending on ID.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        id -
        n -
        d -
        info -

      • dlassq

        public abstract void dlassq(int n,
                            double[] x,
                            int incx,
                            org.netlib.util.doubleW scale,
                            org.netlib.util.doubleW sumsq)
        
     ..

  Purpose
  =======

  DLASSQ  returns the values  scl  and  smsq  such that

     ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,


  where  x( i ) = X( 1 + ( i - 1 )*INCX ). The value of  sumsq  is
  assumed to be non-negative and  scl  returns the value

     scl = max( scale, abs( x( i ) ) ).

  scale and sumsq must be supplied in SCALE and SUMSQ and
  scl and smsq are overwritten on SCALE and SUMSQ respectively.

  The routine makes only one pass through the vector x.

  Arguments
  =========

  N       (input) INTEGER
          The number of elements to be used from the vector X.

  X       (input) DOUBLE PRECISION array, dimension (N)
          The vector for which a scaled sum of squares is computed.
             x( i )  = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.

  INCX    (input) INTEGER
          The increment between successive values of the vector X.
          INCX > 0.

  SCALE   (input/output) DOUBLE PRECISION
          On entry, the value  scale  in the equation above.
          On exit, SCALE is overwritten with  scl , the scaling factor

          for the sum of squares.

  SUMSQ   (input/output) DOUBLE PRECISION
          On entry, the value  sumsq  in the equation above.
          On exit, SUMSQ is overwritten with  smsq , the basic sum of
          squares from which  scl  has been factored out.

 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        scale -
        sumsq -

      • dlasv2

        public abstract void dlasv2(double f,
                            double g,
                            double h,
                            org.netlib.util.doubleW ssmin,
                            org.netlib.util.doubleW ssmax,
                            org.netlib.util.doubleW snr,
                            org.netlib.util.doubleW csr,
                            org.netlib.util.doubleW snl,
                            org.netlib.util.doubleW csl)
        
     ..

  Purpose
  =======

  DLASV2 computes the singular value decomposition of a 2-by-2
  triangular matrix
     [  F   G  ]
     [  0   H  ].
  On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
  smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and

  right singular vectors for abs(SSMAX), giving the decomposition

     [ CSL  SNL ] [  F   G  ] [ CSR -SNR ]  =  [ SSMAX   0   ]
     [-SNL  CSL ] [  0   H  ] [ SNR  CSR ]     [  0    SSMIN ].

  Arguments
  =========

  F       (input) DOUBLE PRECISION
          The (1,1) element of the 2-by-2 matrix.

  G       (input) DOUBLE PRECISION
          The (1,2) element of the 2-by-2 matrix.

  H       (input) DOUBLE PRECISION
          The (2,2) element of the 2-by-2 matrix.

  SSMIN   (output) DOUBLE PRECISION
          abs(SSMIN) is the smaller singular value.

  SSMAX   (output) DOUBLE PRECISION
          abs(SSMAX) is the larger singular value.

  SNL     (output) DOUBLE PRECISION
  CSL     (output) DOUBLE PRECISION
          The vector (CSL, SNL) is a unit left singular vector for the

          singular value abs(SSMAX).

  SNR     (output) DOUBLE PRECISION
  CSR     (output) DOUBLE PRECISION
          The vector (CSR, SNR) is a unit right singular vector for the
          singular value abs(SSMAX).

  Further Details
  ===============

  Any input parameter may be aliased with any output parameter.

  Barring over/underflow and assuming a guard digit in subtraction, all
  output quantities are correct to within a few units in the last
  place (ulps).

  In IEEE arithmetic, the code works correctly if one matrix element is
  infinite.

  Overflow will not occur unless the largest singular value itself
  overflows or is within a few ulps of overflow. (On machines with
  partial overflow, like the Cray, overflow may occur if the largest
  singular value is within a factor of 2 of overflow.)

  Underflow is harmless if underflow is gradual. Otherwise, results
  may correspond to a matrix modified by perturbations of size near
  the underflow threshold.

 =====================================================================


     .. Parameters ..
        Parameters:
        f -
        g -
        h -
        ssmin -
        ssmax -
        snr -
        csr -
        snl -
        csl -

      • dlaswp

        public abstract void dlaswp(int n,
                            double[] a,
                            int lda,
                            int k1,
                            int k2,
                            int[] ipiv,
                            int incx)
        
     ..

  Purpose
  =======

  DLASWP performs a series of row interchanges on the matrix A.
  One row interchange is initiated for each of rows K1 through K2 of A.

  Arguments
  =========

  N       (input) INTEGER
          The number of columns of the matrix A.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the matrix of column dimension N to which the row
          interchanges will be applied.
          On exit, the permuted matrix.

  LDA     (input) INTEGER
          The leading dimension of the array A.

  K1      (input) INTEGER
          The first element of IPIV for which a row interchange will
          be done.

  K2      (input) INTEGER
          The last element of IPIV for which a row interchange will
          be done.

  IPIV    (input) INTEGER array, dimension (K2*abs(INCX))
          The vector of pivot indices.  Only the elements in positions

          K1 through K2 of IPIV are accessed.
          IPIV(K) = L implies rows K and L are to be interchanged.

  INCX    (input) INTEGER
          The increment between successive values of IPIV.  If IPIV
          is negative, the pivots are applied in reverse order.

  Further Details
  ===============

  Modified by
   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 =====================================================================


     .. Local Scalars ..
        Parameters:
        n -
        a -
        lda -
        k1 -
        k2 -
        ipiv -
        incx -

      • dlasy2

        public abstract void dlasy2(boolean ltranl,
                            boolean ltranr,
                            int isgn,
                            int n1,
                            int n2,
                            double[] tl,
                            int ldtl,
                            double[] tr,
                            int ldtr,
                            double[] b,
                            int ldb,
                            org.netlib.util.doubleW scale,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW xnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in

         op(TL)*X + ISGN*X*op(TR) = SCALE*B,

  where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or

  -1.  op(T) = T or T', where T' denotes the transpose of T.

  Arguments
  =========

  LTRANL  (input) LOGICAL
          On entry, LTRANL specifies the op(TL):
             = .FALSE., op(TL) = TL,
             = .TRUE., op(TL) = TL'.

  LTRANR  (input) LOGICAL
          On entry, LTRANR specifies the op(TR):
            = .FALSE., op(TR) = TR,
            = .TRUE., op(TR) = TR'.

  ISGN    (input) INTEGER
          On entry, ISGN specifies the sign of the equation
          as described before. ISGN may only be 1 or -1.

  N1      (input) INTEGER
          On entry, N1 specifies the order of matrix TL.
          N1 may only be 0, 1 or 2.

  N2      (input) INTEGER
          On entry, N2 specifies the order of matrix TR.
          N2 may only be 0, 1 or 2.

  TL      (input) DOUBLE PRECISION array, dimension (LDTL,2)
          On entry, TL contains an N1 by N1 matrix.

  LDTL    (input) INTEGER
          The leading dimension of the matrix TL. LDTL >= max(1,N1).

  TR      (input) DOUBLE PRECISION array, dimension (LDTR,2)
          On entry, TR contains an N2 by N2 matrix.

  LDTR    (input) INTEGER
          The leading dimension of the matrix TR. LDTR >= max(1,N2).

  B       (input) DOUBLE PRECISION array, dimension (LDB,2)
          On entry, the N1 by N2 matrix B contains the right-hand
          side of the equation.

  LDB     (input) INTEGER
          The leading dimension of the matrix B. LDB >= max(1,N1).

  SCALE   (output) DOUBLE PRECISION
          On exit, SCALE contains the scale factor. SCALE is chosen
          less than or equal to 1 to prevent the solution overflowing.


  X       (output) DOUBLE PRECISION array, dimension (LDX,2)
          On exit, X contains the N1 by N2 solution.

  LDX     (input) INTEGER
          The leading dimension of the matrix X. LDX >= max(1,N1).

  XNORM   (output) DOUBLE PRECISION
          On exit, XNORM is the infinity-norm of the solution.

  INFO    (output) INTEGER
          On exit, INFO is set to
             0: successful exit.
             1: TL and TR have too close eigenvalues, so TL or
                TR is perturbed to get a nonsingular equation.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltranl -
        ltranr -
        isgn -
        n1 -
        n2 -
        tl -
        ldtl -
        tr -
        ldtr -
        b -
        ldb -
        scale -
        x -
        ldx -
        xnorm -
        info -

      • dlasyf

        public abstract void dlasyf(java.lang.String uplo,
                            int n,
                            int nb,
                            org.netlib.util.intW kb,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] w,
                            int ldw,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLASYF computes a partial factorization of a real symmetric matrix A

  using the Bunch-Kaufman diagonal pivoting method. The partial
  factorization has the form:

  A  =  ( I  U12 ) ( A11  0  ) (  I    0   )  if UPLO = 'U', or:
        ( 0  U22 ) (  0   D  ) ( U12' U22' )

  A  =  ( L11  0 ) (  D   0  ) ( L11' L21' )  if UPLO = 'L'
        ( L21  I ) (  0  A22 ) (  0    I   )

  where the order of D is at most NB. The actual order is returned in
  the argument KB, and is either NB or NB-1, or N if N <= NB.

  DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
  (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
  A22 (if UPLO = 'L').

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NB      (input) INTEGER
          The maximum number of columns of the matrix A that should be

          factored.  NB should be at least 2 to allow for 2-by-2 pivot

          blocks.

  KB      (output) INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, A contains details of the partial factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If UPLO = 'U', only the last KB elements of IPIV are set;
          if UPLO = 'L', only the first KB elements are set.

          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  W       (workspace) DOUBLE PRECISION array, dimension (LDW,NB)

  LDW     (input) INTEGER
          The leading dimension of the array W.  LDW >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nb -
        kb -
        a -
        lda -
        ipiv -
        w -
        ldw -
        info -

      • dlatbs

        public abstract void dlatbs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] x,
                            org.netlib.util.doubleW scale,
                            double[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLATBS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow, where A is an upper or lower
  triangular band matrix.  Here A' denotes the transpose of A, x and b

  are n-element vectors, and s is a scaling factor, usually less than
  or equal to 1, chosen so that the components of x will be less than
  the overflow threshold.  If the unscaled problem will not cause
  overflow, the Level 2 BLAS routine DTBSV is called.  If the matrix A

  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
  non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  X       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, DTBSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        kd -
        ab -
        ldab -
        x -
        scale -
        cnorm -
        info -

      • dlatdf

        public abstract void dlatdf(int ijob,
                            int n,
                            double[] z,
                            int ldz,
                            double[] rhs,
                            org.netlib.util.doubleW rdsum,
                            org.netlib.util.doubleW rdscal,
                            int[] ipiv,
                            int[] jpiv)
        
     ..

  Purpose
  =======

  DLATDF uses the LU factorization of the n-by-n matrix Z computed by
  DGETC2 and computes a contribution to the reciprocal Dif-estimate
  by solving Z * x = b for x, and choosing the r.h.s. b such that
  the norm of x is as large as possible. On entry RHS = b holds the
  contribution from earlier solved sub-systems, and on return RHS = x.


  The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
  where P and Q are permutation matrices. L is lower triangular with
  unit diagonal elements and U is upper triangular.

  Arguments
  =========

  IJOB    (input) INTEGER
          IJOB = 2: First compute an approximative null-vector e
              of Z using DGECON, e is normalized and solve for
              Zx = +-e - f with the sign giving the greater value
              of 2-norm(x). About 5 times as expensive as Default.
          IJOB .ne. 2: Local look ahead strategy where all entries of
              the r.h.s. b is choosen as either +1 or -1 (Default).

  N       (input) INTEGER
          The number of columns of the matrix Z.

  Z       (input) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, the LU part of the factorization of the n-by-n
          matrix Z computed by DGETC2:  Z = P * L * U * Q

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDA >= max(1, N).

  RHS     (input/output) DOUBLE PRECISION array, dimension N.
          On entry, RHS contains contributions from other subsystems.
          On exit, RHS contains the solution of the subsystem with
          entries acoording to the value of IJOB (see above).

  RDSUM   (input/output) DOUBLE PRECISION
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by DTGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.

  RDSCAL  (input/output) DOUBLE PRECISION
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when DTGSY2 is called by
                DTGSYL.

  IPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  This routine is a further developed implementation of algorithm
  BSOLVE in [1] using complete pivoting in the LU factorization.

  [1] Bo Kagstrom and Lars Westin,
      Generalized Schur Methods with Condition Estimators for
      Solving the Generalized Sylvester Equation, IEEE Transactions
      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

  [2] Peter Poromaa,
      On Efficient and Robust Estimators for the Separation
      between two Regular Matrix Pairs with Applications in
      Condition Estimation. Report IMINF-95.05, Departement of
      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.


  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        n -
        z -
        ldz -
        rhs -
        rdsum -
        rdscal -
        ipiv -
        jpiv -

      • dlatps

        public abstract void dlatps(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            double[] ap,
                            double[] x,
                            org.netlib.util.doubleW scale,
                            double[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLATPS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow, where A is an upper or lower
  triangular matrix stored in packed form.  Here A' denotes the
  transpose of A, x and b are n-element vectors, and s is a scaling
  factor, usually less than or equal to 1, chosen so that the
  components of x will be less than the overflow threshold.  If the
  unscaled problem will not cause overflow, the Level 2 BLAS routine
  DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
  then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  X       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, DTPSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        ap -
        x -
        scale -
        cnorm -
        info -

      • dlatrd

        public abstract void dlatrd(java.lang.String uplo,
                            int n,
                            int nb,
                            double[] a,
                            int lda,
                            double[] e,
                            double[] tau,
                            double[] w,
                            int ldw)
        
     ..

  Purpose
  =======

  DLATRD reduces NB rows and columns of a real symmetric matrix A to
  symmetric tridiagonal form by an orthogonal similarity
  transformation Q' * A * Q, and returns the matrices V and W which are
  needed to apply the transformation to the unreduced part of A.

  If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
  matrix, of which the upper triangle is supplied;
  if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
  matrix, of which the lower triangle is supplied.

  This is an auxiliary routine called by DSYTRD.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U': Upper triangular
          = 'L': Lower triangular

  N       (input) INTEGER
          The order of the matrix A.

  NB      (input) INTEGER
          The number of rows and columns to be reduced.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit:
          if UPLO = 'U', the last NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the orthogonal matrix Q as a

            product of elementary reflectors;
          if UPLO = 'L', the first NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements below the diagonal
            with the array TAU, represent the  orthogonal matrix Q as a
            product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= (1,N).

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          elements of the last NB columns of the reduced matrix;
          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
          the first NB columns of the reduced matrix.

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors, stored in
          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.

          See Further Details.

  W       (output) DOUBLE PRECISION array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.

  LDW     (input) INTEGER
          The leading dimension of the array W. LDW >= max(1,N).

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n) H(n-1) . . . H(n-nb+1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),

  and tau in TAU(i-1).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),

  and tau in TAU(i).

  The elements of the vectors v together form the n-by-nb matrix V
  which is needed, with W, to apply the transformation to the unreduced
  part of the matrix, using a symmetric rank-2k update of the form:
  A := A - V*W' - W*V'.

  The contents of A on exit are illustrated by the following examples
  with n = 5 and nb = 2:

  if UPLO = 'U':                       if UPLO = 'L':

    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )

  where d denotes a diagonal element of the reduced matrix, a denotes
  an element of the original matrix that is unchanged, and vi denotes
  an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nb -
        a -
        lda -
        e -
        tau -
        w -
        ldw -

      • dlatrs

        public abstract void dlatrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            double[] a,
                            int lda,
                            double[] x,
                            org.netlib.util.doubleW scale,
                            double[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLATRS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow.  Here A is an upper or lower
  triangular matrix, A' denotes the transpose of A, x and b are
  n-element vectors, and s is a scaling factor, usually less than
  or equal to 1, chosen so that the components of x will be less than
  the overflow threshold.  If the unscaled problem will not cause
  overflow, the Level 2 BLAS routine DTRSV is called.  If the matrix A

  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
  non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading n by n
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading n by n lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max (1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, DTRSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        a -
        lda -
        x -
        scale -
        cnorm -
        info -

      • dlatrz

        public abstract void dlatrz(int m,
                            int n,
                            int l,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work)
        
     ..

  Purpose
  =======

  DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
  [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
  of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
  matrix and, R and A1 are M-by-M upper triangular matrices.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing the
          meaningful part of the Householder vectors. N-M >= L >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements N-L+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (M)
          The scalar factors of the elementary reflectors.

  WORK    (workspace) DOUBLE PRECISION array, dimension (M)

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an l element vector. tau and z( k )
  are chosen to annihilate the elements of the kth row of A2.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A2, such that the elements of z( k ) are
  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A1.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        l -
        a -
        lda -
        tau -
        work -

      • dlatzm

        public abstract void dlatzm(java.lang.String side,
                            int m,
                            int n,
                            double[] v,
                            int incv,
                            double tau,
                            double[] c1,
                            double[] c2,
                            int Ldc,
                            double[] work)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DORMRZ.

  DLATZM applies a Householder matrix generated by DTZRQF to a matrix.


  Let P = I - tau*u*u',   u = ( 1 ),
                              ( v )
  where v is an (m-1) vector if SIDE = 'L', or a (n-1) vector if
  SIDE = 'R'.

  If SIDE equals 'L', let
         C = [ C1 ] 1
             [ C2 ] m-1
               n
  Then C is overwritten by P*C.

  If SIDE equals 'R', let
         C = [ C1, C2 ] m
                1  n-1
  Then C is overwritten by C*P.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form P * C
          = 'R': form C * P

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) DOUBLE PRECISION array, dimension
                  (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of P. V is not used
          if TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0

  TAU     (input) DOUBLE PRECISION
          The value tau in the representation of P.

  C1      (input/output) DOUBLE PRECISION array, dimension
                         (LDC,N) if SIDE = 'L'
                         (M,1)   if SIDE = 'R'
          On entry, the n-vector C1 if SIDE = 'L', or the m-vector C1
          if SIDE = 'R'.

          On exit, the first row of P*C if SIDE = 'L', or the first
          column of C*P if SIDE = 'R'.

  C2      (input/output) DOUBLE PRECISION array, dimension
                         (LDC, N)   if SIDE = 'L'
                         (LDC, N-1) if SIDE = 'R'
          On entry, the (m - 1) x n matrix C2 if SIDE = 'L', or the
          m x (n - 1) matrix C2 if SIDE = 'R'.

          On exit, rows 2:m of P*C if SIDE = 'L', or columns 2:m of C*P
          if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the arrays C1 and C2. LDC >= (1,M).


  WORK    (workspace) DOUBLE PRECISION array, dimension
                      (N) if SIDE = 'L'
                      (M) if SIDE = 'R'

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        incv -
        tau -
        c1 -
        c2 -
        Ldc -
        work -

      • dlauu2

        public abstract void dlauu2(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAUU2 computes the product U * U' or L' * L, where the triangular
  factor U or L is stored in the upper or lower triangular part of
  the array A.

  If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
  overwriting the factor U in A.
  If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
  overwriting the factor L in A.

  This is the unblocked form of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the triangular factor U or L.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U';
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L' * L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • dlauum

        public abstract void dlauum(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DLAUUM computes the product U * U' or L' * L, where the triangular
  factor U or L is stored in the upper or lower triangular part of
  the array A.

  If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
  overwriting the factor U in A.
  If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
  overwriting the factor L in A.

  This is the blocked form of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the triangular factor U or L.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U';
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L' * L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • dlazq3

        public void dlazq3(int i0,
                   org.netlib.util.intW n0,
                   double[] z,
                   int pp,
                   org.netlib.util.doubleW dmin,
                   org.netlib.util.doubleW sigma,
                   org.netlib.util.doubleW desig,
                   org.netlib.util.doubleW qmax,
                   org.netlib.util.intW nfail,
                   org.netlib.util.intW iter,
                   org.netlib.util.intW ndiv,
                   boolean ieee,
                   org.netlib.util.intW ttype,
                   org.netlib.util.doubleW dmin1,
                   org.netlib.util.doubleW dmin2,
                   org.netlib.util.doubleW dn,
                   org.netlib.util.doubleW dn1,
                   org.netlib.util.doubleW dn2,
                   org.netlib.util.doubleW tau)
        
     ..

  Purpose
  =======

  DLAZQ3 checks for deflation, computes a shift (TAU) and calls dqds.
  In case of failure it changes shifts, and tries again until output
  is positive.

  Arguments
  =========

  I0     (input) INTEGER
         First index.

  N0     (input) INTEGER
         Last index.

  Z      (input) DOUBLE PRECISION array, dimension ( 4*N )
         Z holds the qd array.

  PP     (input) INTEGER
         PP=0 for ping, PP=1 for pong.

  DMIN   (output) DOUBLE PRECISION
         Minimum value of d.

  SIGMA  (output) DOUBLE PRECISION
         Sum of shifts used in current segment.

  DESIG  (input/output) DOUBLE PRECISION
         Lower order part of SIGMA

  QMAX   (input) DOUBLE PRECISION
         Maximum value of q.

  NFAIL  (output) INTEGER
         Number of times shift was too big.

  ITER   (output) INTEGER
         Number of iterations.

  NDIV   (output) INTEGER
         Number of divisions.

  IEEE   (input) LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).

  TTYPE  (input/output) INTEGER
         Shift type.  TTYPE is passed as an argument in order to save
         its value between calls to DLAZQ3

  DMIN1  (input/output) REAL
  DMIN2  (input/output) REAL
  DN     (input/output) REAL
  DN1    (input/output) REAL
  DN2    (input/output) REAL
  TAU    (input/output) REAL
         These are passed as arguments in order to save their values
         between calls to DLAZQ3

  This is a thread safe version of DLASQ3, which passes TTYPE, DMIN1,
  DMIN2, DN, DN1. DN2 and TAU through the argument list in place of
  declaring them in a SAVE statment.

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        sigma -
        desig -
        qmax -
        nfail -
        iter -
        ndiv -
        ieee -
        ttype -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -

      • dlazq4

        public void dlazq4(int i0,
                   int n0,
                   double[] z,
                   int pp,
                   int n0in,
                   double dmin,
                   double dmin1,
                   double dmin2,
                   double dn,
                   double dn1,
                   double dn2,
                   org.netlib.util.doubleW tau,
                   org.netlib.util.intW ttype,
                   org.netlib.util.doubleW g)
        
     ..

  Purpose
  =======

  DLAZQ4 computes an approximation TAU to the smallest eigenvalue 
  using values of d from the previous transform.

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  N0IN  (input) INTEGER
        The value of N0 at start of EIGTEST.

  DMIN  (input) DOUBLE PRECISION
        Minimum value of d.

  DMIN1 (input) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

  DMIN2 (input) DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (input) DOUBLE PRECISION
        d(N)

  DN1   (input) DOUBLE PRECISION
        d(N-1)

  DN2   (input) DOUBLE PRECISION
        d(N-2)

  TAU   (output) DOUBLE PRECISION
        This is the shift.

  TTYPE (output) INTEGER
        Shift type.

  G     (input/output) DOUBLE PRECISION
        G is passed as an argument in order to save its value between
        calls to DLAZQ4

  Further Details
  ===============
  CNST1 = 9/16

  This is a thread safe version of DLASQ4, which passes G through the
  argument list in place of declaring G in a SAVE statment.

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        n0in -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -
        ttype -
        g -

      • dopgtr

        public abstract void dopgtr(java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] tau,
                            double[] q,
                            int ldq,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DOPGTR generates a real orthogonal matrix Q which is defined as the
  product of n-1 elementary reflectors H(i) of order n, as returned by

  DSPTRD using packed storage:

  if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),

  if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to DSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to DSPTRD.

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The vectors which define the elementary reflectors, as
          returned by DSPTRD.

  TAU     (input) DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSPTRD.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          The N-by-N orthogonal matrix Q.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (N-1)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        tau -
        q -
        ldq -
        work -
        info -

      • dopmtr

        public abstract void dopmtr(java.lang.String side,
                            java.lang.String uplo,
                            java.lang.String trans,
                            int m,
                            int n,
                            double[] ap,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DOPMTR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  nq-1 elementary reflectors, as returned by DSPTRD using packed
  storage:

  if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);

  if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to DSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to DSPTRD.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension
                               (M*(M+1)/2) if SIDE = 'L'
                               (N*(N+1)/2) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DSPTRD.  AP is modified by the routine but
          restored on exit.

  TAU     (input) DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L'
                                     or (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSPTRD.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L'
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        uplo -
        trans -
        m -
        n -
        ap -
        tau -
        c -
        Ldc -
        work -
        info -

      • dorg2l

        public abstract void dorg2l(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORG2L generates an m by n real matrix Q with orthonormal columns,
  which is defined as the last n columns of a product of k elementary
  reflectors of order m

        Q  =  H(k) . . . H(2) H(1)

  as returned by DGEQLF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGEQLF in the last k columns of its array
          argument A.
          On exit, the m by n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • dorg2r

        public abstract void dorg2r(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORG2R generates an m by n real matrix Q with orthonormal columns,
  which is defined as the first n columns of a product of k elementary

  reflectors of order m

        Q  =  H(1) H(2) . . . H(k)

  as returned by DGEQRF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGEQRF in the first k columns of its array
          argument A.
          On exit, the m-by-n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • dorgbr

        public abstract void dorgbr(java.lang.String vect,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGBR generates one of the real orthogonal matrices Q or P**T
  determined by DGEBRD when reducing a real matrix A to bidiagonal
  form: A = Q * B * P**T.  Q and P**T are defined as products of
  elementary reflectors H(i) or G(i) respectively.

  If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
  is of order M:
  if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
  columns of Q, where m >= n >= k;
  if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
  M-by-M matrix.

  If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
  is of order N:
  if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m

  rows of P**T, where n >= m >= k;
  if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
  an N-by-N matrix.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          Specifies whether the matrix Q or the matrix P**T is
          required, as defined in the transformation applied by DGEBRD:
          = 'Q':  generate Q;
          = 'P':  generate P**T.

  M       (input) INTEGER
          The number of rows of the matrix Q or P**T to be returned.
          M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q or P**T to be returned.
          N >= 0.
          If VECT = 'Q', M >= N >= min(M,K);
          if VECT = 'P', N >= M >= min(N,K).

  K       (input) INTEGER
          If VECT = 'Q', the number of columns in the original M-by-K
          matrix reduced by DGEBRD.
          If VECT = 'P', the number of rows in the original K-by-N
          matrix reduced by DGEBRD.
          K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DGEBRD.
          On exit, the M-by-N matrix Q or P**T.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension
                                (min(M,K)) if VECT = 'Q'
                                (min(N,K)) if VECT = 'P'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i), which determines Q or P**T, as
          returned by DGEBRD in its array argument TAUQ or TAUP.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,min(M,N)).
          For optimum performance LWORK >= min(M,N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorghr

        public abstract void dorghr(int n,
                            int ilo,
                            int ihi,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGHR generates a real orthogonal matrix Q which is defined as the
  product of IHI-ILO elementary reflectors of order N, as returned by
  DGEHRD:

  Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI must have the same values as in the previous call
          of DGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DGEHRD.
          On exit, the N-by-N orthogonal matrix Q.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEHRD.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= IHI-ILO.
          For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorgl2

        public abstract void dorgl2(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGL2 generates an m by n real matrix Q with orthonormal rows,
  which is defined as the first m rows of a product of k elementary
  reflectors of order n

        Q  =  H(k) . . . H(2) H(1)

  as returned by DGELQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines

          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by DGELQF in the first k rows of its array argument A.
          On exit, the m-by-n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • dorglq

        public abstract void dorglq(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
  which is defined as the first M rows of a product of K elementary
  reflectors of order N

        Q  =  H(k) . . . H(2) H(1)

  as returned by DGELQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines

          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by DGELQF in the first k rows of its array argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorgql

        public abstract void dorgql(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGQL generates an M-by-N real matrix Q with orthonormal columns,
  which is defined as the last N columns of a product of K elementary
  reflectors of order M

        Q  =  H(k) . . . H(2) H(1)

  as returned by DGEQLF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGEQLF in the last k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorgqr

        public abstract void dorgqr(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGQR generates an M-by-N real matrix Q with orthonormal columns,
  which is defined as the first N columns of a product of K elementary

  reflectors of order M

        Q  =  H(1) H(2) . . . H(k)

  as returned by DGEQRF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGEQRF in the first k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorgr2

        public abstract void dorgr2(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGR2 generates an m by n real matrix Q with orthonormal rows,
  which is defined as the last m rows of a product of k elementary
  reflectors of order n

        Q  =  H(1) H(2) . . . H(k)

  as returned by DGERQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGERQF in the last k rows of its array argument
          A.
          On exit, the m by n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • dorgrq

        public abstract void dorgrq(int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
  which is defined as the last M rows of a product of K elementary
  reflectors of order N

        Q  =  H(1) H(2) . . . H(k)

  as returned by DGERQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by DGERQF in the last k rows of its array argument
          A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorgtr

        public abstract void dorgtr(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORGTR generates a real orthogonal matrix Q which is defined as the
  product of n-1 elementary reflectors of order N, as returned by
  DSYTRD:

  if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),

  if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from DSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from DSYTRD.

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DSYTRD.
          On exit, the N-by-N orthogonal matrix Q.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSYTRD.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N-1).
          For optimum performance LWORK >= (N-1)*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dorm2l

        public abstract void dorm2l(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORM2L overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • dorm2r

        public abstract void dorm2r(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORM2R overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • dormbr

        public abstract void dormbr(java.lang.String vect,
                            java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
  with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
  with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      P * C          C * P
  TRANS = 'T':      P**T * C       C * P**T

  Here Q and P**T are the orthogonal matrices determined by DGEBRD when
  reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and

  P**T are defined as products of elementary reflectors H(i) and G(i)
  respectively.

  Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
  order of the orthogonal matrix Q or P**T that is applied.

  If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
  if nq >= k, Q = H(1) H(2) . . . H(k);
  if nq < k, Q = H(1) H(2) . . . H(nq-1).

  If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
  if k < nq, P = G(1) G(2) . . . G(k);
  if k >= nq, P = G(1) G(2) . . . G(nq-1).

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'Q': apply Q or Q**T;
          = 'P': apply P or P**T.

  SIDE    (input) CHARACTER*1
          = 'L': apply Q, Q**T, P or P**T from the Left;
          = 'R': apply Q, Q**T, P or P**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q  or P;
          = 'T':  Transpose, apply Q**T or P**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          If VECT = 'Q', the number of columns in the original
          matrix reduced by DGEBRD.
          If VECT = 'P', the number of rows in the original
          matrix reduced by DGEBRD.
          K >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                                (LDA,min(nq,K)) if VECT = 'Q'
                                (LDA,nq)        if VECT = 'P'
          The vectors which define the elementary reflectors H(i) and
          G(i), whose products determine the matrices Q and P, as
          returned by DGEBRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If VECT = 'Q', LDA >= max(1,nq);
          if VECT = 'P', LDA >= max(1,min(nq,K)).

  TAU     (input) DOUBLE PRECISION array, dimension (min(nq,K))
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i) which determines Q or P, as returned
          by DGEBRD in the array argument TAUQ or TAUP.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
          or P*C or P**T*C or C*P or C*P**T.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        vect -
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormhr

        public abstract void dormhr(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int ilo,
                            int ihi,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMHR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  IHI-ILO elementary reflectors, as returned by DGEHRD:

  Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI must have the same values as in the previous call
          of DGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
          ILO = 1 and IHI = 0, if M = 0;
          if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
          ILO = 1 and IHI = 0, if N = 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DGEHRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

  TAU     (input) DOUBLE PRECISION array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEHRD.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        trans -
        m -
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dorml2

        public abstract void dorml2(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORML2 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • dormlq

        public abstract void dormlq(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMLQ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormql

        public abstract void dormql(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMQL overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormqr

        public abstract void dormqr(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMQR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormr2

        public abstract void dormr2(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMR2 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • dormr3

        public abstract void dormr3(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            int l,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMR3 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DTZRZF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        l -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • dormrq

        public abstract void dormrq(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMRQ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormrz

        public abstract void dormrz(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            int l,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMRZ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          DTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DTZRZF.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        l -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dormtr

        public abstract void dormtr(java.lang.String side,
                            java.lang.String uplo,
                            java.lang.String trans,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] c,
                            int Ldc,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DORMTR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  nq-1 elementary reflectors, as returned by DSYTRD:

  if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);

  if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from DSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from DSYTRD.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  A       (input) DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DSYTRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

  TAU     (input) DOUBLE PRECISION array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSYTRD.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        uplo -
        trans -
        m -
        n -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • dpbcon

        public abstract void dpbcon(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite band matrix using the
  Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A, stored in the

          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).


  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  ANORM   (input) DOUBLE PRECISION
          The 1-norm (or infinity-norm) of the symmetric band matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dpbequ

        public abstract void dpbequ(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] s,
                            org.netlib.util.doubleW scond,
                            org.netlib.util.doubleW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite band matrix A and reduce its condition
  number (with respect to the two-norm).  S contains the scale factors,
  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
  choice of S puts the condition number of B within a factor N of the
  smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular of A is stored;
          = 'L':  Lower triangular of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column

          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB     (input) INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

  S       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) DOUBLE PRECISION
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        s -
        scond -
        amax -
        info -

      • dpbrfs

        public abstract void dpbrfs(java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] afb,
                            int ldafb,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and banded, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column

          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A as computed by

          DPBTRF, in the same storage format as A (see AB).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPBTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dpbstf

        public abstract void dpbstf(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBSTF computes a split Cholesky factorization of a real
  symmetric positive definite band matrix A.

  This routine is designed to be used in conjunction with DSBGST.

  The factorization has the form  A = S**T*S  where S is a band matrix

  of the same bandwidth as A and the following structure:

    S = ( U    )
        ( M  L )

  where U is upper triangular of order m = (n+kd)/2, and L is lower
  triangular of order n-m.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first kd+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the factor S from the split Cholesky
          factorization A = S**T*S. See Further Details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the factorization could not be completed,
               because the updated element a(i,i) was negative; the
               matrix A is not positive definite.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 7, KD = 2:

  S = ( s11  s12  s13                     )
      (      s22  s23  s24                )
      (           s33  s34                )
      (                s44                )
      (           s53  s54  s55           )
      (                s64  s65  s66      )
      (                     s75  s76  s77 )

  If UPLO = 'U', the array AB holds:

  on entry:                          on exit:

    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75

   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76

  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77


  If UPLO = 'L', the array AB holds:

  on entry:                          on exit:

  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • dpbsv

        public abstract void dpbsv(java.lang.String uplo,
                           int n,
                           int kd,
                           int nrhs,
                           double[] ab,
                           int ldab,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite band matrix and X
  and B are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T * U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular band matrix, and L is a lower
  triangular band matrix, with the same number of superdiagonals or
  subdiagonals as A.  The factored form of A is then used to solve the

  system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
          See below for further details.

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • dpbsvx

        public abstract void dpbsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] afb,
                            int ldafb,
                            org.netlib.util.StringW equed,
                            double[] s,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite band matrix and X
  and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T * U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular band matrix, and L is a lower
     triangular band matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFB contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  AB and AFB will not
                  be modified.
          = 'N':  The matrix A will be copied to AFB and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFB and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right-hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array, except

          if FACT = 'F' and EQUED = 'Y', then A must contain the
          equilibrated matrix diag(S)*A*diag(S).  The j-th column of A

          is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
          See below for further details.

          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  LDAB    (input) INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

  AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the band matrix
          A, in the same storage format as A (see AB).  If EQUED = 'Y',
          then AFB is the factored form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

          If FACT = 'E', then AFB is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the equilibrated
          matrix A (see the description of A for the form of the
          equilibrated matrix).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) DOUBLE PRECISION array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45  a46
                         a55  a56
     (aij=conjg(aji))         a66

  Band storage of the upper triangle of A:

    *   a13  a24  a35  a46
   a12  a23  a34  a45  a56
     a11  a22  a33  a44  a55  a66

  Similarly, if UPLO = 'L' the format of A is as follows:

     a11  a22  a33  a44  a55  a66
     a21  a32  a43  a54  a65   *
     a31  a42  a53  a64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dpbtf2

        public abstract void dpbtf2(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBTF2 computes the Cholesky factorization of a real symmetric
  positive definite band matrix A.

  The factorization has the form
     A = U' * U ,  if UPLO = 'U', or
     A = L  * L',  if UPLO = 'L',
  where U is an upper triangular matrix, U' is the transpose of U, and

  L is lower triangular.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U'*U or A = L*L' of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • dpbtrf

        public abstract void dpbtrf(java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBTRF computes the Cholesky factorization of a real symmetric
  positive definite band matrix A.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  Contributed by
  Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • dpbtrs

        public abstract void dpbtrs(java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPBTRS solves a system of linear equations A*X = B with a symmetric
  positive definite band matrix A using the Cholesky factorization
  A = U**T*U or A = L*L**T computed by DPBTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A, stored in the

          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).


  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • dpocon

        public abstract void dpocon(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite matrix using the
  Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by DPOTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ANORM   (input) DOUBLE PRECISION
          The 1-norm (or infinity-norm) of the symmetric matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dpoequ

        public abstract void dpoequ(int n,
                            double[] a,
                            int lda,
                            double[] s,
                            org.netlib.util.doubleW scond,
                            org.netlib.util.doubleW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite matrix A and reduce its condition number

  (with respect to the two-norm).  S contains the scale factors,
  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
  choice of S puts the condition number of B within a factor N of the
  smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The N-by-N symmetric positive definite matrix whose scaling
          factors are to be computed.  Only the diagonal elements of A

          are referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  S       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) DOUBLE PRECISION
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        s -
        scond -
        amax -
        info -

      • dporfs

        public abstract void dporfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPORFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite,
  and provides error bounds and backward error estimates for the
  solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) DOUBLE PRECISION array, dimension (LDAF,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by DPOTRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPOTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dposv

        public abstract void dposv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix and X and B
  are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T* U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is a lower triangular
  matrix.  The factored form of A is then used to solve the system of
  equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • dposvx

        public abstract void dposvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            org.netlib.util.StringW equed,
                            double[] s,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix and X and B
  are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T* U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is a lower triangular

     matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AF contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  A and AF will not
                  be modified.
          = 'N':  The matrix A will be copied to AF and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AF and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A, except if FACT = 'F' and
          EQUED = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.  A is not modified if
          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)

          If FACT = 'F', then AF is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, in the same storage
          format as A.  If EQUED .ne. 'N', then AF is the factored form
          of the equilibrated matrix diag(S)*A*diag(S).

          If FACT = 'N', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the original
          matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the equilibrated
          matrix A (see the description of A for the form of the
          equilibrated matrix).

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) DOUBLE PRECISION array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dpotf2

        public abstract void dpotf2(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOTF2 computes the Cholesky factorization of a real symmetric
  positive definite matrix A.

  The factorization has the form
     A = U' * U ,  if UPLO = 'U', or
     A = L  * L',  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U'*U  or A = L*L'.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • dpotrf

        public abstract void dpotrf(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOTRF computes the Cholesky factorization of a real symmetric
  positive definite matrix A.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  This is the block version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • dpotri

        public abstract void dpotri(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOTRI computes the inverse of a real symmetric positive definite
  matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
  computed by DPOTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, as computed by
          DPOTRF.
          On exit, the upper or lower triangle of the (symmetric)
          inverse of A, overwriting the input factor U or L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is

                zero, and the inverse could not be computed.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • dpotrs

        public abstract void dpotrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPOTRS solves a system of linear equations A*X = B with a symmetric
  positive definite matrix A using the Cholesky factorization
  A = U**T*U or A = L*L**T computed by DPOTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by DPOTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • dppcon

        public abstract void dppcon(java.lang.String uplo,
                            int n,
                            double[] ap,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite packed matrix using
  the Cholesky factorization A = U**T*U or A = L*L**T computed by
  DPPTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

  ANORM   (input) DOUBLE PRECISION
          The 1-norm (or infinity-norm) of the symmetric matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dppequ

        public abstract void dppequ(java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] s,
                            org.netlib.util.doubleW scond,
                            org.netlib.util.doubleW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite matrix A in packed storage and reduce
  its condition number (with respect to the two-norm).  S contains the

  scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix

  B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
  This choice of S puts the condition number of B within a factor N of

  the smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  S       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) DOUBLE PRECISION
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        s -
        scond -
        amax -
        info -

      • dpprfs

        public abstract void dpprfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] afp,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and packed, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
          packed columnwise in a linear array in the same format as A
          (see AP).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPPTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        afp -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dppsv

        public abstract void dppsv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           double[] ap,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix stored in
  packed format and X and B are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T* U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is a lower triangular
  matrix.  The factored form of A is then used to solve the system of
  equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, in the same storage
          format as A.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • dppsvx

        public abstract void dppsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] afp,
                            org.netlib.util.StringW equed,
                            double[] s,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix stored in
  packed format and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T* U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is a lower triangular

     matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFP contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  AP and AFP will not
                  be modified.
          = 'N':  The matrix A will be copied to AFP and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFP and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array, except if FACT = 'F'

          and EQUED = 'Y', then A must contain the equilibrated matrix

          diag(S)*A*diag(S).  The j-th column of A is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.  A is not modified if
          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  AFP     (input or output) DOUBLE PRECISION array, dimension
                            (N*(N+1)/2)
          If FACT = 'F', then AFP is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L', in the same storage
          format as A.  If EQUED .ne. 'N', then AFP is the factored
          form of the equilibrated matrix A.

          If FACT = 'N', then AFP is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L' of the original matrix A.


          If FACT = 'E', then AFP is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L' of the equilibrated
          matrix A (see the description of AP for the form of the
          equilibrated matrix).

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) DOUBLE PRECISION array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        ap -
        afp -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dpptrf

        public abstract void dpptrf(java.lang.String uplo,
                            int n,
                            double[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPTRF computes the Cholesky factorization of a real symmetric
  positive definite matrix A stored in packed format.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T, in the same

          storage format as A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  Further Details
  ======= =======

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        info -

      • dpptri

        public abstract void dpptri(java.lang.String uplo,
                            int n,
                            double[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPTRI computes the inverse of a real symmetric positive definite
  matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
  computed by DPPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor is stored in AP;
          = 'L':  Lower triangular factor is stored in AP.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, packed columnwise as

          a linear array.  The j-th column of U or L is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

          On exit, the upper or lower triangle of the (symmetric)
          inverse of A, overwriting the input factor U or L.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is

                zero, and the inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        info -

      • dpptrs

        public abstract void dpptrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPPTRS solves a system of linear equations A*X = B with a symmetric
  positive definite matrix A in packed storage using the Cholesky
  factorization A = U**T*U or A = L*L**T computed by DPPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • dptcon

        public abstract void dptcon(int n,
                            double[] d,
                            double[] e,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTCON computes the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite tridiagonal matrix
  using the factorization A = L*D*L**T or A = U**T*D*U computed by
  DPTTRF.

  Norm(inv(A)) is computed by a direct method, and the reciprocal of
  the condition number is computed as
               RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          factorization of A, as computed by DPTTRF.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) off-diagonal elements of the unit bidiagonal factor
          U or L from the factorization of A,  as computed by DPTTRF.

  ANORM   (input) DOUBLE PRECISION
          The 1-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
          1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The method used is described in Nicholas J. Higham, "Efficient
  Algorithms for Computing the Condition Number of a Tridiagonal
  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        anorm -
        rcond -
        work -
        info -

      • dpteqr

        public abstract void dpteqr(java.lang.String compz,
                            int n,
                            double[] d,
                            double[] e,
                            double[] z,
                            int ldz,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
  symmetric positive definite tridiagonal matrix by first factoring the
  matrix using DPTTRF, and then calling DBDSQR to compute the singular

  values of the bidiagonal factor.

  This routine computes the eigenvalues of the positive definite
  tridiagonal matrix to high relative accuracy.  This means that if the
  eigenvalues range over many orders of magnitude in size, then the
  small eigenvalues and corresponding eigenvectors will be computed
  more accurately than, for example, with the standard QR method.

  The eigenvectors of a full or band symmetric positive definite matrix
  can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to
  reduce this matrix to tridiagonal form. (The reduction to tridiagonal
  form, however, may preclude the possibility of obtaining high
  relative accuracy in the small eigenvalues of the original matrix, if
  these eigenvalues range over many orders of magnitude.)

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvectors of original symmetric
                  matrix also.  Array Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal
          matrix.
          On normal exit, D contains the eigenvalues, in descending
          order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix used in the
          reduction to tridiagonal form.
          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the

          original symmetric matrix;
          if COMPZ = 'I', the orthonormal eigenvectors of the
          tridiagonal matrix.
          If INFO > 0 on exit, Z contains the eigenvectors associated
          with only the stored eigenvalues.
          If  COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          COMPZ = 'V' or 'I', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, and i is:
                <= N  the Cholesky factorization of the matrix could
                      not be performed because the i-th principal minor
                      was not positive definite.
                > N   the SVD algorithm failed to converge;
                      if INFO = N+i, i off-diagonal elements of the
                      bidiagonal factor did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • dptrfs

        public abstract void dptrfs(int n,
                            int nrhs,
                            double[] d,
                            double[] e,
                            double[] df,
                            double[] ef,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and tridiagonal, and provides error bounds and backward error
  estimates for the solution.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

  DF      (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          factorization computed by DPTTRF.

  EF      (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the factorization computed by DPTTRF.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPTTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        df -
        ef -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        info -

      • dptsv

        public abstract void dptsv(int n,
                           int nrhs,
                           double[] d,
                           double[] e,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTSV computes the solution to a real system of linear equations
  A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
  matrix, and X and B are N-by-NRHS matrices.

  A is factored as A = L*D*L**T, and the factored form of A is then
  used to solve the system of equations.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the factorization A = L*D*L**T.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L**T factorization of
          A.  (E can also be regarded as the superdiagonal of the unit

          bidiagonal factor U from the U**T*D*U factorization of A.)

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the solution has not been
                computed.  The factorization has not been completed
                unless i = N.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        info -

      • dptsvx

        public abstract void dptsvx(java.lang.String fact,
                            int n,
                            int nrhs,
                            double[] d,
                            double[] e,
                            double[] df,
                            double[] ef,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTSVX uses the factorization A = L*D*L**T to compute the solution
  to a real system of linear equations A*X = B, where A is an N-by-N
  symmetric positive definite tridiagonal matrix and X and B are
  N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
     is a unit lower bidiagonal matrix and D is diagonal.  The
     factorization can also be regarded as having the form
     A = U**T*D*U.

  2. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, DF and EF contain the factored form of A.
                  D, E, DF, and EF will not be modified.
          = 'N':  The matrix A will be copied to DF and EF and
                  factored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

  DF      (input or output) DOUBLE PRECISION array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.
          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.

  EF      (input or output) DOUBLE PRECISION array, dimension (N-1)
          If FACT = 'F', then EF is an input argument and on entry
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.
          If FACT = 'N', then EF is an output argument and on exit
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The reciprocal condition number of the matrix A.  If RCOND
          is less than the machine precision (in particular, if
          RCOND = 0), the matrix is singular to working precision.
          This condition is indicated by a return code of INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in any
          element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        n -
        nrhs -
        d -
        e -
        df -
        ef -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        info -

      • dpttrf

        public abstract void dpttrf(int n,
                            double[] d,
                            double[] e,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTTRF computes the L*D*L' factorization of a real symmetric
  positive definite tridiagonal matrix A.  The factorization may also
  be regarded as having the form A = U'*D*U.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the L*D*L' factorization of A.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L' factorization of A.

          E can also be regarded as the superdiagonal of the unit
          bidiagonal factor U from the U'*D*U factorization of A.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite; if k < N, the factorization could not
               be completed, while if k = N, the factorization was
               completed, but D(N) <= 0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • dpttrs

        public abstract void dpttrs(int n,
                            int nrhs,
                            double[] d,
                            double[] e,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DPTTRS solves a tridiagonal system of the form
     A * X = B
  using the L*D*L' factorization of A computed by DPTTRF.  D is a
  diagonal matrix specified in the vector D, L is a unit bidiagonal
  matrix whose subdiagonal is specified in the vector E, and X and B
  are N by NRHS matrices.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          L*D*L' factorization of A.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the L*D*L' factorization of A.  E can also be regarded
          as the superdiagonal of the unit bidiagonal factor U from the
          factorization A = U'*D*U.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side vectors B for the system of
          linear equations.
          On exit, the solution vectors, X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        info -

      • dptts2

        public abstract void dptts2(int n,
                            int nrhs,
                            double[] d,
                            double[] e,
                            double[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  DPTTS2 solves a tridiagonal system of the form
     A * X = B
  using the L*D*L' factorization of A computed by DPTTRF.  D is a
  diagonal matrix specified in the vector D, L is a unit bidiagonal
  matrix whose subdiagonal is specified in the vector E, and X and B
  are N by NRHS matrices.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          L*D*L' factorization of A.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the L*D*L' factorization of A.  E can also be regarded
          as the superdiagonal of the unit bidiagonal factor U from the
          factorization A = U'*D*U.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side vectors B for the system of
          linear equations.
          On exit, the solution vectors, X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -

      • drscl

        public abstract void drscl(int n,
                           double sa,
                           double[] sx,
                           int incx)
        
     ..

  Purpose
  =======

  DRSCL multiplies an n-element real vector x by the real scalar 1/a.
  This is done without overflow or underflow as long as
  the final result x/a does not overflow or underflow.

  Arguments
  =========

  N       (input) INTEGER
          The number of components of the vector x.

  SA      (input) DOUBLE PRECISION
          The scalar a which is used to divide each component of x.
          SA must be >= 0, or the subroutine will divide by zero.

  SX      (input/output) DOUBLE PRECISION array, dimension
                         (1+(N-1)*abs(INCX))
          The n-element vector x.

  INCX    (input) INTEGER
          The increment between successive values of the vector SX.
          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n


 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        sa -
        sx -
        incx -

      • dsbev

        public abstract void dsbev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           int kd,
                           double[] ab,
                           int ldab,
                           double[] w,
                           double[] z,
                           int ldz,
                           double[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBEV computes all the eigenvalues and, optionally, eigenvectors of
  a real symmetric band matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (max(1,3*N-2))


  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        kd -
        ab -
        ldab -
        w -
        z -
        ldz -
        work -
        info -

      • dsbevd

        public abstract void dsbevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBEVD computes all the eigenvalues and, optionally, eigenvectors of

  a real symmetric band matrix A. If eigenvectors are desired, it uses

  a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          IF N <= 1,                LWORK must be at least 1.
          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
          If JOBZ  = 'V' and N > 2, LWORK must be at least
                         ( 1 + 5*N + 2*N**2 ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array LIWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        kd -
        ab -
        ldab -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsbevx

        public abstract void dsbevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] q,
                            int ldq,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric band matrix A.  Eigenvalues and eigenvectors can

  be selected by specifying either a range of values or a range of
  indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found;
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found;
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                         reduction to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  If JOBZ = 'V', then
          LDQ >= max(1,N).

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing AB to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (7*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        kd -
        ab -
        ldab -
        q -
        ldq -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dsbgst

        public abstract void dsbgst(java.lang.String vect,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            double[] ab,
                            int ldab,
                            double[] bb,
                            int ldbb,
                            double[] x,
                            int ldx,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBGST reduces a real symmetric-definite banded generalized
  eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
  such that C has the same bandwidth as A.

  B must have been previously factorized as S**T*S by DPBSTF, using a
  split Cholesky factorization. A is overwritten by C = X**T*A*X, where
  X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
  bandwidth of A.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.


  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the transformed matrix X**T*A*X, stored in the same

          format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input) DOUBLE PRECISION array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of

          B, as returned by DPBSTF, stored in the first KB+1 rows of
          the array.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  X       (output) DOUBLE PRECISION array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X.
          If VECT = 'N', the array X is not referenced.

  LDX     (input) INTEGER
          The leading dimension of the array X.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        x -
        ldx -
        work -
        info -

      • dsbgv

        public abstract void dsbgv(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           int ka,
                           int kb,
                           double[] ab,
                           int ldab,
                           double[] bb,
                           int ldbb,
                           double[] w,
                           double[] z,
                           int ldz,
                           double[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBGV computes all the eigenvalues, and optionally, the eigenvectors

  of a real generalized symmetric-definite banded eigenproblem, of
  the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
  and banded, and B is also positive definite.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KB >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i). The eigenvectors are
          normalized so that Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= N.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobz -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        w -
        z -
        ldz -
        work -
        info -

      • dsbgvd

        public abstract void dsbgvd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            double[] ab,
                            int ldab,
                            double[] bb,
                            int ldbb,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite banded eigenproblem, of the

  form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and

  banded, and B is also positive definite.  If eigenvectors are
  desired, it uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 3*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.


  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsbgvx

        public abstract void dsbgvx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            double[] ab,
                            int ldab,
                            double[] bb,
                            int ldbb,
                            double[] q,
                            int ldq,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite banded eigenproblem, of
  the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric

  and banded, and B is also positive definite.  Eigenvalues and
  eigenvectors can be selected by specifying either all eigenvalues,
  a range of values or a range of indices for the desired eigenvalues.


  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
          If JOBZ = 'V', the n-by-n matrix used in the reduction of
          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
          and consequently C to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  If JOBZ = 'N',
          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (7*N)

  IWORK   (workspace/output) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (M)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvalues that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0 : successful exit
          < 0 : if INFO = -i, the i-th argument had an illegal value
          <= N: if INFO = i, then i eigenvectors failed to converge.
                  Their indices are stored in IFAIL.
          > N : DPBSTF returned an error code; i.e.,
                if INFO = N + i, for 1 <= i <= N, then the leading
                minor of order i of B is not positive definite.
                The factorization of B could not be completed and
                no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        q -
        ldq -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dsbtrd

        public abstract void dsbtrd(java.lang.String vect,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            double[] d,
                            double[] e,
                            double[] q,
                            int ldq,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSBTRD reduces a real symmetric band matrix A to symmetric
  tridiagonal form T by an orthogonal similarity transformation:
  Q**T * A * Q = T.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'N':  do not form Q;
          = 'V':  form Q;
          = 'U':  update a matrix X, by forming X*Q.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by

          values generated during the reduction.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if VECT = 'U', then Q must contain an N-by-N
          matrix X; if VECT = 'N' or 'V', then Q need not be set.

          On exit:
          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
          if VECT = 'U', Q contains the product X*Q;
          if VECT = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Modified by Linda Kaufman, Bell Labs.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        uplo -
        n -
        kd -
        ab -
        ldab -
        d -
        e -
        q -
        ldq -
        work -
        info -

      • dsgesv

        public void dsgesv(int n,
                   int nrhs,
                   double[] a,
                   int lda,
                   int[] ipiv,
                   double[] b,
                   int ldb,
                   double[] x,
                   int ldx,
                   double[] work,
                   float[] swork,
                   org.netlib.util.intW iter,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSGESV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

  DSGESV first attempts to factorize the matrix in SINGLE PRECISION 
  and use this factorization within an iterative refinement procedure t
  produce a solution with DOUBLE PRECISION normwise backward error
  quality (see below). If the approach fails the method switches to a
  DOUBLE PRECISION factorization and solve.

  The iterative refinement is not going to be a winning strategy if
  the ratio SINGLE PRECISION performance over DOUBLE PRECISION performa
  is too small. A reasonable strategy should take the number of right-h
  sides and the size of the matrix into account. This might be done wit
  call to ILAENV in the future. Up to now, we always try iterative refi

  The iterative refinement process is stopped if
      ITER > ITERMAX
  or for all the RHS we have:
      RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX 
  where
      o ITER is the number of the current iteration in the iterative
        refinement process
      o RNRM is the infinity-norm of the residual
      o XNRM is the infinity-norm of the solution
      o ANRM is the infinity-operator-norm of the matrix A
      o EPS is the machine epsilon returned by DLAMCH('Epsilon')
  The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively

  Arguments
  =========

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input or input/ouptut) DOUBLE PRECISION array,
          dimension (LDA,N)
          On entry, the N-by-N coefficient matrix A.
          On exit, if iterative refinement has been successfully used
          (INFO.EQ.0 and ITER.GE.0, see description below), then A is
          unchanged, if double precision factorization has been used
          (INFO.EQ.0 and ITER.LT.0, see description below), then the
          array A contains the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).
          Corresponds either to the single precision factorization 
          (if INFO.EQ.0 and ITER.GE.0) or the double precision 
          factorization (if INFO.EQ.0 and ITER.LT.0).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS matrix of right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (N*NRHS)
          This array is used to hold the residual vectors.

  SWORK   (workspace) REAL array, dimension (N*(N+NRHS))
          This array is used to use the single precision matrix and the
          right-hand sides or solutions in single precision.

  ITER    (output) INTEGER
          < 0: iterative refinement has failed, double precision
               factorization has been performed
               -1 : taking into account machine parameters, N, NRHS, it
                    is a priori not worth working in SINGLE PRECISION
               -2 : overflow of an entry when moving from double to
                    SINGLE PRECISION
               -3 : failure of SGETRF
               -31: stop the iterative refinement after the 30th
                    iterations
          > 0: iterative refinement has been sucessfully used.
               Returns the number of iterations

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) computed in DOUBLE PRECISION is
                exactly zero.  The factorization has been completed,
                but the factor U is exactly singular, so the solution
                could not be computed.

  =========

     .. Parameters ..
        Parameters:
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        work -
        swork -
        iter -
        info -

      • dspcon

        public abstract void dspcon(java.lang.String uplo,
                            int n,
                            double[] ap,
                            int[] ipiv,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric packed matrix A using the factorization
  A = U*D*U**T or A = L*D*L**T computed by DSPTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSPTRF, stored as a
          packed triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

  ANORM   (input) DOUBLE PRECISION
          The 1-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  IWORK    (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dspev

        public abstract void dspev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           double[] ap,
                           double[] w,
                           double[] z,
                           int ldz,
                           double[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A in packed storage.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ap -
        w -
        z -
        ldz -
        work -
        info -

      • dspevd

        public abstract void dspevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPEVD computes all the eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A in packed storage. If eigenvectors are
  desired, it uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the required LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK must be at least 1.
          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
          If JOBZ = 'V' and N > 1, LWORK must be at least
                                                 1 + 6*N + N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ap -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dspevx

        public abstract void dspevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            double[] ap,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A in packed storage.  Eigenvalues/vectors

  can be selected by specifying either a range of values or a range of

  indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found;
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found;
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing AP to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the selected eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (8*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        ap -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dspgst

        public abstract void dspgst(int itype,
                            java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] bp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPGST reduces a real symmetric-definite generalized eigenproblem
  to standard form, using packed storage.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

  B must have been previously factorized as U**T*U or L*L**T by DPPTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
          = 2 or 3: compute U*A*U**T or L**T*A*L.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**T*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**T.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  BP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor from the Cholesky factorization of B,
          stored in the same format as A, as returned by DPPTRF.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        ap -
        bp -
        info -

      • dspgv

        public abstract void dspgv(int itype,
                           java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           double[] ap,
                           double[] bp,
                           double[] w,
                           double[] z,
                           int ldz,
                           double[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPGV computes all the eigenvalues and, optionally, the eigenvectors

  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
  Here A and B are assumed to be symmetric, stored in packed format,
  and B is also positive definite.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension
                            (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPPTRF or DSPEV returned an error code:
             <= N:  if INFO = i, DSPEV failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero.
             > N:   if INFO = n + i, for 1 <= i <= n, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        ap -
        bp -
        w -
        z -
        ldz -
        work -
        info -

      • dspgvd

        public abstract void dspgvd(int itype,
                            java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] bp,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
  B are assumed to be symmetric, stored in packed format, and B is also
  positive definite.
  If eigenvectors are desired, it uses a divide and conquer algorithm.


  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the required LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPPTRF or DSPEVD returned an error code:
             <= N:  if INFO = i, DSPEVD failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        ap -
        bp -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dspgvx

        public abstract void dspgvx(int itype,
                            java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] bp,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
  and B are assumed to be symmetric, stored in packed storage, and B
  is also positive definite.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of indices
  for the desired eigenvalues.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A and B are stored;
          = 'L':  Lower triangle of A and B are stored.

  N       (input) INTEGER
          The order of the matrix pencil (A,B).  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'N', then Z is not referenced.
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.

          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (8*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPPTRF or DSPEVX returned an error code:
             <= N:  if INFO = i, DSPEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 =====================================================================


     .. Local Scalars ..
        Parameters:
        itype -
        jobz -
        range -
        uplo -
        n -
        ap -
        bp -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dsprfs

        public abstract void dsprfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] afp,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric indefinite
  and packed, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


  AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The factored form of the matrix A.  AFP contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by DSPTRF, stored as a packed
          triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DSPTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        afp -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dspsv

        public abstract void dspsv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           double[] ap,
                           int[] ipiv,
                           double[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric matrix stored in packed format and X
  and B are N-by-NRHS matrices.

  The diagonal pivoting method is used to factor A as
     A = U * D * U**T,  if UPLO = 'U', or
     A = L * D * L**T,  if UPLO = 'L',
  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, D is symmetric and block diagonal with 1-by-1
  and 2-by-2 diagonal blocks.  The factored form of A is then used to
  solve the system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D, as

          determined by DSPTRF.  If IPIV(k) > 0, then rows and columns

          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
          then rows and columns k-1 and -IPIV(k) were interchanged and

          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
          diagonal block.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is

                exactly singular, so the solution could not be
                computed.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        ipiv -
        b -
        ldb -
        info -

      • dspsvx

        public abstract void dspsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] afp,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
  A = L*D*L**T to compute the solution to a real system of linear
  equations A * X = B, where A is an N-by-N symmetric matrix stored
  in packed format and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A as
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)

     triangular matrices and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

  2. If some D(i,i)=0, so that D is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, AFP and IPIV contain the factored form of
                  A.  AP, AFP and IPIV will not be modified.
          = 'N':  The matrix A will be copied to AFP and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          See below for further details.

  AFP     (input or output) DOUBLE PRECISION array, dimension
                            (N*(N+1)/2)
          If FACT = 'F', then AFP is an input argument and on entry
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

          If FACT = 'N', then AFP is an output argument and on exit
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains details of the interchanges and the block structure

          of D, as determined by DSPTRF.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains details of the interchanges and the block structure

          of D, as determined by DSPTRF.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  D(i,i) is exactly zero.  The factorization
                       has been completed but the factor D is exactly
                       singular, so the solution and error bounds could
                       not be computed. RCOND = 0 is returned.
                = N+1: D is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        ap -
        afp -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dsptrd

        public abstract void dsptrd(java.lang.String uplo,
                            int n,
                            double[] ap,
                            double[] d,
                            double[] e,
                            double[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPTRD reduces a real symmetric matrix A stored in packed form to
  symmetric tridiagonal form T by an orthogonal similarity
  transformation: Q**T * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  overwriting A(i+2:n,i), and tau is stored in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        d -
        e -
        tau -
        info -

      • dsptrf

        public abstract void dsptrf(java.lang.String uplo,
                            int n,
                            double[] ap,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPTRF computes the factorization of a real symmetric matrix A stored
  in packed format using the Bunch-Kaufman diagonal pivoting method:

     A = U*D*U**T  or  A = L*D*L**T

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with
  1-by-1 and 2-by-2 diagonal blocks.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L, stored as a packed triangular
          matrix overwriting A (see below for further details).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it

               is used to solve a system of equations.

  Further Details
  ===============

  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        info -

      • dsptri

        public abstract void dsptri(java.lang.String uplo,
                            int n,
                            double[] ap,
                            int[] ipiv,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPTRI computes the inverse of a real symmetric indefinite matrix
  A in packed storage using the factorization A = U*D*U**T or
  A = L*D*L**T computed by DSPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by DSPTRF,
          stored as a packed triangular matrix.

          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix, stored as a packed triangular matrix. The j-th column
          of inv(A) is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
          if UPLO = 'L',
             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

               inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        work -
        info -

      • dsptrs

        public abstract void dsptrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] ap,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSPTRS solves a system of linear equations A*X = B with a real
  symmetric matrix A stored in packed format using the factorization
  A = U*D*U**T or A = L*D*L**T computed by DSPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSPTRF, stored as a
          packed triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        ipiv -
        b -
        ldb -
        info -

      • dstebz

        public abstract void dstebz(java.lang.String range,
                            java.lang.String order,
                            int n,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            double[] d,
                            double[] e,
                            org.netlib.util.intW m,
                            org.netlib.util.intW nsplit,
                            double[] w,
                            int[] iblock,
                            int[] isplit,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEBZ computes the eigenvalues of a symmetric tridiagonal
  matrix T.  The user may ask for all eigenvalues, all eigenvalues
  in the half-open interval (VL, VU], or the IL-th through IU-th
  eigenvalues.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  RANGE   (input) CHARACTER*1
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  ORDER   (input) CHARACTER*1
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and

                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal

          to VL, or greater than VU, will not be returned.  VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute tolerance for the eigenvalues.  An eigenvalue
          (or cluster) is considered to be located if it has been
          determined to lie in an interval whose width is ABSTOL or
          less.  If ABSTOL is less than or equal to zero, then ULP*|T|

          will be used, where |T| means the 1-norm of T.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.


  M       (output) INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

  NSPLIT  (output) INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

  W       (output) DOUBLE PRECISION array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalues.  (DSTEBZ may use the remaining N-M elements as
          workspace.)

  IBLOCK  (output) INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (DSTEBZ may use the remaining N-M elements as
          workspace.)

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to ISPLIT(1),

          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the

                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick

                                out eigenvalues IL:IU.  In some cases,

                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  Internal Parameters
  ===================

  RELFAC  DOUBLE PRECISION, default = 2.0e0
          The relative tolerance.  An interval (a,b] lies within
          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
          where "ulp" is the machine precision (distance from 1 to
          the next larger floating point number.)

  FUDGE   DOUBLE PRECISION, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        order -
        n -
        vl -
        vu -
        il -
        iu -
        abstol -
        d -
        e -
        m -
        nsplit -
        w -
        iblock -
        isplit -
        work -
        iwork -
        info -

      • dstedc

        public abstract void dstedc(java.lang.String compz,
                            int n,
                            double[] d,
                            double[] e,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
  symmetric tridiagonal matrix using the divide and conquer method.
  The eigenvectors of a full or band real symmetric matrix can also be

  found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
  matrix to tridiagonal form.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.  See DLAED3 for details.

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.
          = 'V':  Compute eigenvectors of original dense symmetric
                  matrix also.  On entry, Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.

  N       (input) INTEGER
          The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If  COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If eigenvectors are desired, then LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 3*N + 2*N*lg N + 3*N**2 ),
                         where lg( N ) = smallest integer k such
                         that 2**k >= N.
          If COMPZ = 'I' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LWORK need
          only be max(1,2*(N-1)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LIWORK must be at least
                         ( 6 + 6*N + 5*N*lg N ).
          If COMPZ = 'I' and N > 1 then LIWORK must be at least
                         ( 3 + 5*N ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LIWORK
          need only be 1.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dstegr

        public abstract void dstegr(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            double[] d,
                            double[] e,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            int[] isuppz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEGR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T. Any such unreduced matrix h
  a well defined set of pairwise different real eigenvalues, the corres
  real eigenvectors are pairwise orthogonal.

  The spectrum may be computed either completely or partially by specif
  either an interval (VL,VU] or a range of indices IL:IU for the desire
  eigenvalues.

  DSTEGR is a compatability wrapper around the improved DSTEMR routine.
  See DSTEMR for further details.

  One important change is that the ABSTOL parameter no longer provides 
  benefit and hence is no longer used.

  Note : DSTEGR and DSTEMR work only on machines which follow
  IEEE-754 floating-point standard in their handling of infinities and

  NaNs.  Normal execution may create these exceptiona values and hence

  may abort due to a floating point exception in environments which
  do not conform to the IEEE-754 standard.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

  E       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          Unused.  Was the absolute error tolerance for the
          eigenvalues/eigenvectors in previous versions.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigen
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in DLARRE,
                if INFO = 2X, internal error in DLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by DLARRE or
                DLARRV, respectively.

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, LBNL/NERSC, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dstein

        public abstract void dstein(int n,
                            double[] d,
                            double[] e,
                            int m,
                            double[] w,
                            int[] iblock,
                            int[] isplit,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEIN computes the eigenvectors of a real symmetric tridiagonal
  matrix T corresponding to specified eigenvalues, using inverse
  iteration.

  The maximum number of iterations allowed for each eigenvector is
  specified by an internal parameter MAXITS (currently set to 5).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input) DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix
          T, in elements 1 to N-1.

  M       (input) INTEGER
          The number of eigenvectors to be found.  0 <= M <= N.

  W       (input) DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the eigenvalues for
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block.  ( The output array
          W from DSTEBZ with ORDER = 'B' is expected here. )

  IBLOCK  (input) INTEGER array, dimension (N)
          The submatrix indices associated with the corresponding
          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
          the first submatrix from the top, =2 if W(i) belongs to
          the second submatrix, etc.  ( The output array IBLOCK
          from DSTEBZ is expected here. )

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.
          ( The output array ISPLIT from DSTEBZ is expected here. )

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, M)
          The computed eigenvectors.  The eigenvector associated
          with the eigenvalue W(i) is stored in the i-th column of
          Z.  Any vector which fails to converge is set to its current

          iterate after MAXITS iterations.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (5*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  IFAIL   (output) INTEGER array, dimension (M)
          On normal exit, all elements of IFAIL are zero.
          If one or more eigenvectors fail to converge after
          MAXITS iterations, then their indices are stored in
          array IFAIL.

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, then i eigenvectors failed to converge
               in MAXITS iterations.  Their indices are stored in
               array IFAIL.

  Internal Parameters
  ===================

  MAXITS  INTEGER, default = 5
          The maximum number of iterations performed.

  EXTRA   INTEGER, default = 2
          The number of iterations performed after norm growth
          criterion is satisfied, should be at least 1.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        m -
        w -
        iblock -
        isplit -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dstemr

        public void dstemr(java.lang.String jobz,
                   java.lang.String range,
                   int n,
                   double[] d,
                   double[] e,
                   double vl,
                   double vu,
                   int il,
                   int iu,
                   org.netlib.util.intW m,
                   double[] w,
                   double[] z,
                   int ldz,
                   int nzc,
                   int[] isuppz,
                   org.netlib.util.booleanW tryrac,
                   double[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEMR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T. Any such unreduced matrix h
  a well defined set of pairwise different real eigenvalues, the corres
  real eigenvectors are pairwise orthogonal.

  The spectrum may be computed either completely or partially by specif
  either an interval (VL,VU] or a range of indices IL:IU for the desire
  eigenvalues.

  Depending on the number of desired eigenvalues, these are computed ei
  by bisection or the dqds algorithm. Numerically orthogonal eigenvecto
  computed by the use of various suitable L D L^T factorizations near c
  of close eigenvalues (referred to as RRRs, Relatively Robust
  Representations). An informal sketch of the algorithm follows.

  For each unreduced block (submatrix) of T,
     (a) Compute T - sigma I  = L D L^T, so that L and D
         define all the wanted eigenvalues to high relative accuracy.
         This means that small relative changes in the entries of D and
         cause only small relative changes in the eigenvalues and
         eigenvectors. The standard (unfactored) representation of the

         tridiagonal matrix T does not have this property in general.
     (b) Compute the eigenvalues to suitable accuracy.
         If the eigenvectors are desired, the algorithm attains full
         accuracy of the computed eigenvalues only right before
         the corresponding vectors have to be computed, see steps c) an
     (c) For each cluster of close eigenvalues, select a new
         shift close to the cluster, find a new factorization, and refi
         the shifted eigenvalues to suitable accuracy.
     (d) For each eigenvalue with a large enough relative separation co
         the corresponding eigenvector by forming a rank revealing twis
         factorization. Go back to (c) for any clusters that remain.

  For more details, see:
  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representat
    to compute orthogonal eigenvectors of symmetric tridiagonal matrice
    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors an
    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, V
    2004.  Also LAPACK Working Note 154.
  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
    tridiagonal eigenvalue/eigenvector problem",
    Computer Science Division Technical Report No. UCB/CSD-97-971,
    UC Berkeley, May 1997.

  Notes:
  1.DSTEMR works only on machines which follow IEEE-754
  floating-point standard in their handling of infinities and NaNs.
  This permits the use of efficient inner loops avoiding a check for
  zero divisors.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

  E       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and can be computed with a workspace

          query by setting NZC = -1, see below.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

  NZC     (input) INTEGER
          The number of eigenvectors to be held in the array Z.
          If RANGE = 'A', then NZC >= max(1,N).
          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,
          If RANGE = 'I', then NZC >= IU-IL+1.
          If NZC = -1, then a workspace query is assumed; the
          routine calculates the number of columns of the array Z that

          are needed to hold the eigenvectors.
          This value is returned as the first entry of the Z array, and
          no error message related to NZC is issued by XERBLA.

  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigen
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

  TRYRAC  (input/output) LOGICAL
          If TRYRAC.EQ..TRUE., indicates that the code should check whe
          the tridiagonal matrix defines its eigenvalues to high relati
          accuracy.  If so, the code uses relative-accuracy preserving

          algorithms that might be (a bit) slower depending on the matr
          If the matrix does not define its eigenvalues to high relativ
          accuracy, the code can uses possibly faster algorithms.
          If TRYRAC.EQ..FALSE., the code is not required to guarantee
          relatively accurate eigenvalues and can use the fastest possi
          techniques.
          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
          does not define its eigenvalues to high relative accuracy.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in DLARRE,
                if INFO = 2X, internal error in DLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by DLARRE or
                DLARRV, respectively.


  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        m -
        w -
        z -
        ldz -
        nzc -
        isuppz -
        tryrac -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsteqr

        public abstract void dsteqr(java.lang.String compz,
                            int n,
                            double[] d,
                            double[] e,
                            double[] z,
                            int ldz,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
  symmetric tridiagonal matrix using the implicit QL or QR method.
  The eigenvectors of a full or band symmetric matrix can also be found
  if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to

  tridiagonal form.

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvalues and eigenvectors of the original

                  symmetric matrix.  On entry, Z must contain the
                  orthogonal matrix used to reduce the original matrix

                  to tridiagonal form.
          = 'I':  Compute eigenvalues and eigenvectors of the
                  tridiagonal matrix.  Z is initialized to the identity
                  matrix.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if  COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          eigenvectors are desired, then  LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2))

          If COMPZ = 'N', then WORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm has failed to find all the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero; on exit, D
                and E contain the elements of a symmetric tridiagonal
                matrix which is orthogonally similar to the original
                matrix.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • dsterf

        public abstract void dsterf(int n,
                            double[] d,
                            double[] e,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
  using the Pal-Walker-Kahan variant of the QL or QR algorithm.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix.

          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm failed to find all of the eigenvalues in

                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • dstev

        public abstract void dstev(java.lang.String jobz,
                           int n,
                           double[] d,
                           double[] e,
                           double[] z,
                           int ldz,
                           double[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEV computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric tridiagonal matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2))

          If JOBZ = 'N', WORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • dstevd

        public abstract void dstevd(java.lang.String jobz,
                            int n,
                            double[] d,
                            double[] e,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEVD computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric tridiagonal matrix. If eigenvectors are desired, it
  uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LIWORK must be at least 3+5*N.


          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dstevr

        public abstract void dstevr(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            double[] d,
                            double[] e,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            int[] isuppz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEVR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T.  Eigenvalues and
  eigenvectors can be selected by specifying either a range of values
  or a range of indices for the desired eigenvalues.

  Whenever possible, DSTEVR calls DSTEMR to compute the
  eigenspectrum using Relatively Robust Representations.  DSTEMR
  computes eigenvalues by the dqds algorithm, while orthogonal
  eigenvectors are computed from various "good" L D L^T representations
  (also known as Relatively Robust Representations). Gram-Schmidt
  orthogonalization is avoided as far as possible. More specifically,
  the various steps of the algorithm are as follows. For the i-th
  unreduced block of T,
     (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
          is a relatively robust representation,
     (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
         relative accuracy by the dqds algorithm,
     (c) If there is a cluster of close eigenvalues, "choose" sigma_i
         close to the cluster, and go to step (a),
     (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
         compute the corresponding eigenvector by forming a
         rank-revealing twisted factorization.
  The desired accuracy of the output can be specified by the input
  parameter ABSTOL.

  For more details, see "A new O(n^2) algorithm for the symmetric
  tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
  Computer Science Division Technical Report No. UCB//CSD-97-971,
  UC Berkeley, May 1997.


  Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
  on machines which conform to the ieee-754 floating point standard.
  DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
  when partial spectrum requests are made.

  Normal execution of DSTEMR may create NaNs and infinities and
  hence may abort due to a floating point exception in environments
  which do not handle NaNs and infinities in the ieee standard default

  manner.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.
 For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
 DSTEIN are called

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, D may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A in elements 1 to N-1 of E.
          On exit, E may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
          eigenvalues are computed to high relative accuracy when
          possible in future releases.  The current code does not
          make any guarantees about high relative accuracy, but
          future releases will. See J. Barlow and J. Demmel,
          "Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices", LAPACK Working Note #7, for a discussion

          of which matrices define their eigenvalues to high relative
          accuracy.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ).
 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal (and
          minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,20*N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal (and
          minimal) LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  Internal error

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of
       California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dstevx

        public abstract void dstevx(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            double[] d,
                            double[] e,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSTEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix A.  Eigenvalues and
  eigenvectors can be selected by specifying either a range of values
  or a range of indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, D may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A in elements 1 to N-1 of E.
          On exit, E may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less
          than or equal to zero, then  EPS*|T|  will be used in
          its place, where |T| is the 1-norm of the tridiagonal
          matrix.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge (INFO > 0), then that
          column of Z contains the latest approximation to the
          eigenvector, and the index of the eigenvector is returned
          in IFAIL.  If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) DOUBLE PRECISION array, dimension (5*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • dsycon

        public abstract void dsycon(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double anorm,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric matrix A using the factorization
  A = U*D*U**T or A = L*D*L**T computed by DSYTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSYTRF.

  ANORM   (input) DOUBLE PRECISION
          The 1-norm of the original matrix A.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  IWORK    (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • dsyev

        public abstract void dsyev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           double[] a,
                           int lda,
                           double[] w,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYEV computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          orthonormal eigenvectors of the matrix A.
          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

          or the upper triangle (if UPLO='U') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,3*N-1).
          For optimal efficiency, LWORK >= (NB+2)*N,
          where NB is the blocksize for DSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        a -
        lda -
        w -
        work -
        lwork -
        info -

      • dsyevd

        public abstract void dsyevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] w,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A. If eigenvectors are desired, it uses a
  divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Because of large use of BLAS of level 3, DSYEVD needs N**2 more
  workspace than DSYEVX.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          orthonormal eigenvectors of the matrix A.
          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

          or the upper triangle (if UPLO='U') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK must be at least 1.
          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
          If JOBZ = 'V' and N > 1, LWORK must be at least
                                                1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If N <= 1,                LIWORK must be at least 1.
          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
                to converge; i off-diagonal elements of an intermediate
                tridiagonal form did not converge to zero;
                if INFO = i and JOBZ = 'V', then the algorithm failed
                to compute an eigenvalue while working on the submatrix
                lying in rows and columns INFO/(N+1) through
                mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  Modified description of INFO. Sven, 16 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        a -
        lda -
        w -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsyevr

        public abstract void dsyevr(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            int[] isuppz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYEVR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of
  indices for the desired eigenvalues.

  DSYEVR first reduces the matrix A to tridiagonal form T with a call
  to DSYTRD.  Then, whenever possible, DSYEVR calls DSTEMR to compute
  the eigenspectrum using Relatively Robust Representations.  DSTEMR
  computes eigenvalues by the dqds algorithm, while orthogonal
  eigenvectors are computed from various "good" L D L^T representations
  (also known as Relatively Robust Representations). Gram-Schmidt
  orthogonalization is avoided as far as possible. More specifically,
  the various steps of the algorithm are as follows.

  For each unreduced block (submatrix) of T,
     (a) Compute T - sigma I  = L D L^T, so that L and D
         define all the wanted eigenvalues to high relative accuracy.
         This means that small relative changes in the entries of D and
         cause only small relative changes in the eigenvalues and
         eigenvectors. The standard (unfactored) representation of the

         tridiagonal matrix T does not have this property in general.
     (b) Compute the eigenvalues to suitable accuracy.
         If the eigenvectors are desired, the algorithm attains full
         accuracy of the computed eigenvalues only right before
         the corresponding vectors have to be computed, see steps c) an
     (c) For each cluster of close eigenvalues, select a new
         shift close to the cluster, find a new factorization, and refi
         the shifted eigenvalues to suitable accuracy.
     (d) For each eigenvalue with a large enough relative separation co
         the corresponding eigenvector by forming a rank revealing twis
         factorization. Go back to (c) for any clusters that remain.

  The desired accuracy of the output can be specified by the input
  parameter ABSTOL.

  For more details, see DSTEMR's documentation and:
  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representat
    to compute orthogonal eigenvectors of symmetric tridiagonal matrice
    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors an
    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, V
    2004.  Also LAPACK Working Note 154.
  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
    tridiagonal eigenvalue/eigenvector problem",
    Computer Science Division Technical Report No. UCB/CSD-97-971,
    UC Berkeley, May 1997.


  Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
  on machines which conform to the ieee-754 floating point standard.
  DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and
  when partial spectrum requests are made.

  Normal execution of DSTEMR may create NaNs and infinities and
  hence may abort due to a floating point exception in environments
  which do not handle NaNs and infinities in the ieee standard default

  manner.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.
 For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
 DSTEIN are called

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
          eigenvalues are computed to high relative accuracy when
          possible in future releases.  The current code does not
          make any guarantees about high relative accuracy, but
          future releases will. See J. Barlow and J. Demmel,
          "Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices", LAPACK Working Note #7, for a discussion

          of which matrices define their eigenvalues to high relative
          accuracy.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ).
 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,26*N).
          For optimal efficiency, LWORK >= (NB+6)*N,
          where NB is the max of the blocksize for DSYTRD and DORMTR
          returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  Internal error

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of
       California at Berkeley, USA
     Jason Riedy, Computer Science Division, University of
       California at Berkeley, USA

 =====================================================================


     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsyevx

        public abstract void dsyevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of indices
  for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= 1, when N <= 1;
          otherwise 8*N.
          For optimal efficiency, LWORK >= (NB+3)*N,
          where NB is the max of the blocksize for DSYTRD and DORMTR
          returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

 =====================================================================


     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        ifail -
        info -

      • dsygs2

        public abstract void dsygs2(int itype,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYGS2 reduces a real symmetric-definite generalized eigenproblem
  to standard form.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U')*A*inv(U) or inv(L)*A*inv(L')

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U` or L'*A*L.

  B must have been previously factorized as U'*U or L*L' by DPOTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U')*A*inv(U) or inv(L)*A*inv(L');
          = 2 or 3: compute U*A*U' or L'*A*L.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored, and how B has been factorized.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by DPOTRF.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        info -

      • dsygst

        public abstract void dsygst(int itype,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYGST reduces a real symmetric-definite generalized eigenproblem
  to standard form.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

  B must have been previously factorized as U**T*U or L*L**T by DPOTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
          = 2 or 3: compute U*A*U**T or L**T*A*L.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**T*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**T.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by DPOTRF.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        info -

      • dsygv

        public abstract void dsygv(int itype,
                           java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           double[] a,
                           int lda,
                           double[] b,
                           int ldb,
                           double[] w,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYGV computes all the eigenvalues, and optionally, the eigenvectors

  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
  Here A and B are assumed to be symmetric and B is also
  positive definite.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          matrix Z of eigenvectors.  The eigenvectors are normalized
          as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')

          or the lower triangle (if UPLO='L') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the symmetric positive definite matrix B.
          If UPLO = 'U', the leading N-by-N upper triangular part of B

          contains the upper triangular part of the matrix B.
          If UPLO = 'L', the leading N-by-N lower triangular part of B

          contains the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,3*N-1).
          For optimal efficiency, LWORK >= (NB+2)*N,
          where NB is the blocksize for DSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPOTRF or DSYEV returned an error code:
             <= N:  if INFO = i, DSYEV failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        w -
        work -
        lwork -
        info -

      • dsygvd

        public abstract void dsygvd(int itype,
                            java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] w,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
  B are assumed to be symmetric and B is also positive definite.
  If eigenvectors are desired, it uses a divide and conquer algorithm.


  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          matrix Z of eigenvectors.  The eigenvectors are normalized
          as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')

          or the lower triangle (if UPLO='L') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the symmetric matrix B.  If UPLO = 'U', the
          leading N-by-N upper triangular part of B contains the
          upper triangular part of the matrix B.  If UPLO = 'L',
          the leading N-by-N lower triangular part of B contains
          the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  W       (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If N <= 1,                LIWORK >= 1.
          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPOTRF or DSYEVD returned an error code:
             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
                    failed to converge; i off-diagonal elements of an
                    intermediate tridiagonal form did not converge to
                    zero;
                    if INFO = i and JOBZ = 'V', then the algorithm
                    failed to compute an eigenvalue while working on
                    the submatrix lying in rows and columns INFO/(N+1)

                    through mod(INFO,N+1);
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  Modified so that no backsubstitution is performed if DSYEVD fails to

  converge (NEIG in old code could be greater than N causing out of
  bounds reference to A - reported by Ralf Meyer).  Also corrected the

  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        w -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dsygvx

        public abstract void dsygvx(int itype,
                            java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double vl,
                            double vu,
                            int il,
                            int iu,
                            double abstol,
                            org.netlib.util.intW m,
                            double[] w,
                            double[] z,
                            int ldz,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
  and B are assumed to be symmetric and B is also positive definite.
  Eigenvalues and eigenvectors can be selected by specifying either a
  range of values or a range of indices for the desired eigenvalues.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A and B are stored;
          = 'L':  Lower triangle of A and B are stored.

  N       (input) INTEGER
          The order of the matrix pencil (A,B).  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix B.  If UPLO = 'U', the
          leading N-by-N upper triangular part of B contains the
          upper triangular part of the matrix B.  If UPLO = 'L',
          the leading N-by-N lower triangular part of B contains
          the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) DOUBLE PRECISION
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) DOUBLE PRECISION array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'N', then Z is not referenced.
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.

          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,8*N).
          For optimal efficiency, LWORK >= (NB+3)*N,
          where NB is the blocksize for DSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPOTRF or DSYEVX returned an error code:
             <= N:  if INFO = i, DSYEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 =====================================================================


     .. Parameters ..
        Parameters:
        itype -
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        ifail -
        info -

      • dsyrfs

        public abstract void dsyrfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric indefinite, and
  provides error bounds and backward error estimates for the solution.


  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) DOUBLE PRECISION array, dimension (LDAF,N)
          The factored form of the matrix A.  AF contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by DSYTRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSYTRF.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DSYTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dsysv

        public abstract void dsysv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           double[] a,
                           int lda,
                           int[] ipiv,
                           double[] b,
                           int ldb,
                           double[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
  matrices.

  The diagonal pivoting method is used to factor A as
     A = U * D * U**T,  if UPLO = 'U', or
     A = L * D * L**T,  if UPLO = 'L',
  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with
  1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
  used to solve the system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the block diagonal matrix D and the
          multipliers used to obtain the factor U or L from the
          factorization A = U*D*U**T or A = L*D*L**T as computed by
          DSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D, as

          determined by DSYTRF.  If IPIV(k) > 0, then rows and columns

          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
          then rows and columns k-1 and -IPIV(k) were interchanged and

          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
          diagonal block.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >= 1, and for best performance
          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
          DSYTRF.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, so the solution could not be computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        work -
        lwork -
        info -

      • dsysvx

        public abstract void dsysvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] af,
                            int ldaf,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            org.netlib.util.doubleW rcond,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYSVX uses the diagonal pivoting factorization to compute the
  solution to a real system of linear equations A * X = B,
  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
  matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A.
     The form of the factorization is
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)

     triangular matrices, and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

  2. If some D(i,i)=0, so that D is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, AF and IPIV contain the factored form of
                  A.  AF and IPIV will not be modified.
          = 'N':  The matrix A will be copied to AF and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)

          If FACT = 'F', then AF is an input argument and on entry
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.

          If FACT = 'N', then AF is an output argument and on exit
          returns the block diagonal matrix D and the multipliers used

          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains details of the interchanges and the block structure

          of D, as determined by DSYTRF.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains details of the interchanges and the block structure

          of D, as determined by DSYTRF.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >= max(1,3*N), and for best
          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
          NB is the optimal blocksize for DSYTRF.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, and i is
                <= N:  D(i,i) is exactly zero.  The factorization
                       has been completed but the factor D is exactly
                       singular, so the solution and error bounds could
                       not be computed. RCOND = 0 is returned.
                = N+1: D is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        lwork -
        iwork -
        info -

      • dsytd2

        public abstract void dsytd2(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] d,
                            double[] e,
                            double[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
  form T by an orthogonal similarity transformation: Q' * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),

  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi

  denotes an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        d -
        e -
        tau -
        info -

      • dsytf2

        public abstract void dsytf2(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTF2 computes the factorization of a real symmetric matrix A using

  the Bunch-Kaufman diagonal pivoting method:

     A = U*D*U'  or  A = L*D*L'

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, U' is the transpose of U, and D is symmetric and
  block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details).


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it

               is used to solve a system of equations.

  Further Details
  ===============

  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l.204 and l.372
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J. Lewis, Boeing Computer Services Company
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        info -

      • dsytrd

        public abstract void dsytrd(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            double[] d,
                            double[] e,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTRD reduces a real symmetric matrix A to real symmetric
  tridiagonal form T by an orthogonal similarity transformation:
  Q**T * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= 1.
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),

  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi

  denotes an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        d -
        e -
        tau -
        work -
        lwork -
        info -

      • dsytrf

        public abstract void dsytrf(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTRF computes the factorization of a real symmetric matrix A using

  the Bunch-Kaufman diagonal pivoting method.  The form of the
  factorization is

     A = U*D*U**T  or  A = L*D*L**T

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with
  1-by-1 and 2-by-2 diagonal blocks.

  This is the blocked version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details).


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is

                exactly singular, and division by zero will occur if it
                is used to solve a system of equations.

  Further Details
  ===============

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        work -
        lwork -
        info -

      • dsytri

        public abstract void dsytri(java.lang.String uplo,
                            int n,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTRI computes the inverse of a real symmetric indefinite matrix
  A using the factorization A = U*D*U**T or A = L*D*L**T computed by
  DSYTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by DSYTRF.

          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix.  If UPLO = 'U', the upper triangular part of the
          inverse is formed and the part of A below the diagonal is not
          referenced; if UPLO = 'L' the lower triangular part of the
          inverse is formed and the part of A above the diagonal is
          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSYTRF.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

               inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        work -
        info -

      • dsytrs

        public abstract void dsytrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            int[] ipiv,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DSYTRS solves a system of linear equations A*X = B with a real
  symmetric matrix A using the factorization A = U*D*U**T or
  A = L*D*L**T computed by DSYTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSYTRF.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • dtbcon

        public abstract void dtbcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            int kd,
                            double[] ab,
                            int ldab,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTBCON estimates the reciprocal of the condition number of a
  triangular band matrix A, in either the 1-norm or the infinity-norm.


  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        kd -
        ab -
        ldab -
        rcond -
        work -
        iwork -
        info -

      • dtbrfs

        public abstract void dtbrfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int kd,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTBRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular band
  coefficient matrix.

  The solution matrix X must be computed by DTBTRS or some other
  means before entering this routine.  DTBRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dtbtrs

        public abstract void dtbtrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int kd,
                            int nrhs,
                            double[] ab,
                            int ldab,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTBTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular band matrix of order N, and B is an
  N-by NRHS matrix.  A check is made to verify that A is nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • dtgevc

        public abstract void dtgevc(java.lang.String side,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            double[] s,
                            int lds,
                            double[] p,
                            int ldp,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..


  Purpose
  =======

  DTGEVC computes some or all of the right and/or left eigenvectors of

  a pair of real matrices (S,P), where S is a quasi-triangular matrix
  and P is upper triangular.  Matrix pairs of this type are produced by
  the generalized Schur factorization of a matrix pair (A,B):

     A = Q*S*Z**T,  B = Q*P*Z**T

  as computed by DGGHRD + DHGEQZ.

  The right eigenvector x and the left eigenvector y of (S,P)
  corresponding to an eigenvalue w are defined by:
  
     S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
  
  where y**H denotes the conjugate tranpose of y.
  The eigenvalues are not input to this routine, but are computed
  directly from the diagonal blocks of S and P.
  
  This routine returns the matrices X and/or Y of right and left
  eigenvectors of (S,P), or the products Z*X and/or Q*Y,
  where Z and Q are input matrices.
  If Q and Z are the orthogonal factors from the generalized Schur
  factorization of a matrix pair (A,B), then Z*X and Q*Y
  are the matrices of right and left eigenvectors of (A,B).
 
  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

  HOWMNY  (input) CHARACTER*1
          = 'A': compute all right and/or left eigenvectors;
          = 'B': compute all right and/or left eigenvectors,
                 backtransformed by the matrices in VR and/or VL;
          = 'S': compute selected right and/or left eigenvectors,
                 specified by the logical array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY='S', SELECT specifies the eigenvectors to be
          computed.  If w(j) is a real eigenvalue, the corresponding
          real eigenvector is computed if SELECT(j) is .TRUE..
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector
          is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
          and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
          set to .FALSE..
          Not referenced if HOWMNY = 'A' or 'B'.

  N       (input) INTEGER
          The order of the matrices S and P.  N >= 0.

  S       (input) DOUBLE PRECISION array, dimension (LDS,N)
          The upper quasi-triangular matrix S from a generalized Schur

          factorization, as computed by DHGEQZ.

  LDS     (input) INTEGER
          The leading dimension of array S.  LDS >= max(1,N).

  P       (input) DOUBLE PRECISION array, dimension (LDP,N)
          The upper triangular matrix P from a generalized Schur
          factorization, as computed by DHGEQZ.
          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
          of S must be in positive diagonal form.

  LDP     (input) INTEGER
          The leading dimension of array P.  LDP >= max(1,N).

  VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of left Schur vectors returned by DHGEQZ).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);

          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by

                      SELECT, stored consecutively in the columns of
                      VL, in the same order as their eigenvalues.

          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part.

          Not referenced if SIDE = 'R'.

  LDVL    (input) INTEGER
          The leading dimension of array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.

  VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Z (usually the orthogonal matrix Z
          of right Schur vectors returned by DHGEQZ).

          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
          if HOWMNY = 'B' or 'b', the matrix Z*X;
          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                      specified by SELECT, stored consecutively in the

                      columns of VR, in the same order as their
                      eigenvalues.

          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part.
          
          Not referenced if SIDE = 'L'.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B', LDVR >= N.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
          is set to N.  Each selected real eigenvector occupies one
          column and each selected complex eigenvector occupies two
          columns.

  WORK    (workspace) DOUBLE PRECISION array, dimension (6*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex

                eigenvalue.

  Further Details
  ===============

  Allocation of workspace:
  ---------- -- ---------

     WORK( j ) = 1-norm of j-th column of A, above the diagonal
     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
     WORK( 2*N+1:3*N ) = real part of eigenvector
     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

  Rowwise vs. columnwise solution methods:
  ------- --  ---------- -------- -------

  Finding a generalized eigenvector consists basically of solving the
  singular triangular system

   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

  Consider finding the i-th right eigenvector (assume all eigenvalues
  are real). The equation to be solved is:
       n                   i
  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
      k=j                 k=j

  where  C = (A - w B)  (The components v(i+1:n) are 0.)

  The "rowwise" method is:

  (1)  v(i) := 1
  for j = i-1,. . .,1:
                          i
      (2) compute  s = - sum C(j,k) v(k)   and
                        k=j+1

      (3) v(j) := s / C(j,j)

  Step 2 is sometimes called the "dot product" step, since it is an
  inner product between the j-th row and the portion of the eigenvector
  that has been computed so far.

  The "columnwise" method consists basically in doing the sums
  for all the rows in parallel.  As each v(j) is computed, the
  contribution of v(j) times the j-th column of C is added to the
  partial sums.  Since FORTRAN arrays are stored columnwise, this has
  the advantage that at each step, the elements of C that are accessed

  are adjacent to one another, whereas with the rowwise method, the
  elements accessed at a step are spaced LDS (and LDP) words apart.

  When finding left eigenvectors, the matrix in question is the
  transpose of the one in storage, so the rowwise method then
  actually accesses columns of A and B at each step, and so is the
  preferred method.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        howmny -
        select -
        n -
        s -
        lds -
        p -
        ldp -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        info -

      • dtgex2

        public abstract void dtgex2(boolean wantq,
                            boolean wantz,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] q,
                            int ldq,
                            double[] z,
                            int ldz,
                            int j1,
                            int n1,
                            int n2,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
  of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
  (A, B) by an orthogonal equivalence transformation.

  (A, B) must be in generalized real Schur canonical form (as returned

  by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
  diagonal blocks. B is upper triangular.

  Optionally, the matrices Q and Z of generalized Schur vectors are
  updated.

         Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
         Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'


  Arguments
  =========

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A      (input/output) DOUBLE PRECISION arrays, dimensions (LDA,N)
          On entry, the matrix A in the pair (A, B).
          On exit, the updated matrix A.

  LDA     (input)  INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B      (input/output) DOUBLE PRECISION arrays, dimensions (LDB,N)
          On entry, the matrix B in the pair (A, B).
          On exit, the updated matrix B.

  LDB     (input)  INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
          On exit, the updated matrix Q.
          Not referenced if WANTQ = .FALSE..

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
          On exit, the updated matrix Z.
          Not referenced if WANTZ = .FALSE..

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.

  J1      (input) INTEGER
          The index to the first block (A11, B11). 1 <= J1 <= N.

  N1      (input) INTEGER
          The order of the first block (A11, B11). N1 = 0, 1 or 2.

  N2      (input) INTEGER
          The order of the second block (A22, B22). N2 = 0, 1 or 2.

  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)).

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )

  INFO    (output) INTEGER
            =0: Successful exit
            >0: If INFO = 1, the transformed matrix (A, B) would be
                too far from generalized Schur form; the blocks are
                not swapped and (A, B) and (Q, Z) are unchanged.
                The problem of swapping is too ill-conditioned.
            <0: If INFO = -16: LWORK is too small. Appropriate value
                for LWORK is returned in WORK(1).

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  In the current code both weak and strong stability tests are
  performed. The user can omit the strong stability test by changing
  the internal logical parameter WANDS to .FALSE.. See ref. [2] for
  details.

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  =====================================================================
  Replaced various illegal calls to DCOPY by calls to DLASET, or by DO

  loops. Sven Hammarling, 1/5/02.

     .. Parameters ..
        Parameters:
        wantq -
        wantz -
        n -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        j1 -
        n1 -
        n2 -
        work -
        lwork -
        info -

      • dtgexc

        public abstract void dtgexc(boolean wantq,
                            boolean wantz,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] q,
                            int ldq,
                            double[] z,
                            int ldz,
                            org.netlib.util.intW ifst,
                            org.netlib.util.intW ilst,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGEXC reorders the generalized real Schur decomposition of a real
  matrix pair (A,B) using an orthogonal equivalence transformation

                 (A, B) = Q * (A, B) * Z',

  so that the diagonal block of (A, B) with row index IFST is moved
  to row ILST.

  (A, B) must be in generalized real Schur canonical form (as returned

  by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
  diagonal blocks. B is upper triangular.

  Optionally, the matrices Q and Z of generalized Schur vectors are
  updated.

         Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
         Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'


  Arguments
  =========

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the matrix A in generalized real Schur canonical
          form.
          On exit, the updated matrix A, again in generalized
          real Schur canonical form.

  LDA     (input)  INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the matrix B in generalized real Schur canonical
          form (A,B).
          On exit, the updated matrix B, again in generalized
          real Schur canonical form (A,B).

  LDB     (input)  INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
          On exit, the updated matrix Q.
          If WANTQ = .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
          On exit, the updated matrix Z.
          If WANTZ = .FALSE., Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.

  IFST    (input/output) INTEGER
  ILST    (input/output) INTEGER
          Specify the reordering of the diagonal blocks of (A, B).
          The block with row index IFST is moved to row ILST, by a
          sequence of swapping between adjacent blocks.
          On exit, if IFST pointed on entry to the second row of
          a 2-by-2 block, it is changed to point to the first row;
          ILST always points to the first row of the block in its
          final position (which may differ from its input value by
          +1 or -1). 1 <= IFST, ILST <= N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
           =0:  successful exit.
           <0:  if INFO = -i, the i-th argument had an illegal value.
           =1:  The transformed matrix pair (A, B) would be too far
                from generalized Schur form; the problem is ill-
                conditioned. (A, B) may have been partially reordered,

                and ILST points to the first row of the current
                position of the block being moved.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  =====================================================================

     .. Parameters ..
        Parameters:
        wantq -
        wantz -
        n -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        ifst -
        ilst -
        work -
        lwork -
        info -

      • dtgsen

        public abstract void dtgsen(int ijob,
                            boolean wantq,
                            boolean wantz,
                            boolean[] select,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] alphar,
                            double[] alphai,
                            double[] beta,
                            double[] q,
                            int ldq,
                            double[] z,
                            int ldz,
                            org.netlib.util.intW m,
                            org.netlib.util.doubleW pl,
                            org.netlib.util.doubleW pr,
                            double[] dif,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGSEN reorders the generalized real Schur decomposition of a real
  matrix pair (A, B) (in terms of an orthonormal equivalence trans-
  formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
  appears in the leading diagonal blocks of the upper quasi-triangular

  matrix A and the upper triangular B. The leading columns of Q and
  Z form orthonormal bases of the corresponding left and right eigen-
  spaces (deflating subspaces). (A, B) must be in generalized real
  Schur canonical form (as returned by DGGES), i.e. A is block upper
  triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
  triangular.

  DTGSEN also computes the generalized eigenvalues

              w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

  of the reordered matrix pair (A, B).

  Optionally, DTGSEN computes the estimates of reciprocal condition
  numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
  (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
  between the matrix pairs (A11, B11) and (A22,B22) that correspond to

  the selected cluster and the eigenvalues outside the cluster, resp.,

  and norms of "projections" onto left and right eigenspaces w.r.t.
  the selected cluster in the (1,1)-block.

  Arguments
  =========

  IJOB    (input) INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  SELECT  (input) LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster.
          To select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension(LDA,N)
          On entry, the upper quasi-triangular matrix A, with (A, B) in
          generalized real Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension(LDB,N)
          On entry, the upper triangular matrix B, with (A, B) in
          generalized real Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real generalized Schur form of (A,B) were further reduced
          to triangular form using complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= 1;
          and if WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.

  M       (output) INTEGER
          The dimension of the specified pair of left and right eigen-

          spaces (deflating subspaces). 0 <= M <= N.

  PL      (output) DOUBLE PRECISION
  PR      (output) DOUBLE PRECISION
          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal of the norm of "projections" onto left and right
          eigenspaces with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3, PL and PR are not referenced.

  DIF     (output) DOUBLE PRECISION array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.

          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.

  WORK    (workspace/output) DOUBLE PRECISION array,
          dimension (MAX(1,LWORK)) 
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >=  4*N+16.
          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          IF IJOB = 0, IWORK is not referenced.  Otherwise,
          on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.

  Further Details
  ===============

  DTGSEN first collects the selected eigenvalues by computing
  orthogonal U and W that move them to the top left corner of (A, B).
  In other words, the selected eigenvalues are the eigenvalues of
  (A11, B11) in:

                U'*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

  where N = n1+n2 and U' means the transpose of U. The first n1 columns
  of U and W span the specified pair of left and right eigenspaces
  (deflating subspaces) of (A, B).

  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
  reordered generalized real Schur form of (C, D) is given by

           (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',

  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.

  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.

  The Difu and Difl are defined as:

       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix

       Zu = [ kron(In2, A11)  -kron(A22', In1) ]
            [ kron(In2, B11)  -kron(B22', In1) ].

  Here, Inx is the identity matrix of size nx and A22' is the
  transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.

  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is

       EPS * norm((A, B)) / DIF(2),

  where EPS is the machine precision.

  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.

  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where

       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2

  and (L, R) is the solution to the generalized Sylvester equation

       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12

  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / PL.

  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is

   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).


  An approximate bound on x can be computed from DIF(1:2), PL and PR.

  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as

   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))

   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))


  See LAPACK User's Guide section 4.11 or the following references
  for more information.

  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see DLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
  (IJOB = 2 will be used)). See DTGSYL for more details.

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  References
  ==========

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
      1996.

  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        wantq -
        wantz -
        select -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        q -
        ldq -
        z -
        ldz -
        m -
        pl -
        pr -
        dif -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dtgsja

        public abstract void dtgsja(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int p,
                            int n,
                            int k,
                            int l,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double tola,
                            double tolb,
                            double[] alpha,
                            double[] beta,
                            double[] u,
                            int ldu,
                            double[] v,
                            int ldv,
                            double[] q,
                            int ldq,
                            double[] work,
                            org.netlib.util.intW ncycle,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGSJA computes the generalized singular value decomposition (GSVD)
  of two real upper triangular (or trapezoidal) matrices A and B.

  On entry, it is assumed that matrices A and B have the following
  forms, which may be obtained by the preprocessing subroutine DGGSVP
  from a general M-by-N matrix A and P-by-N matrix B:

               N-K-L  K    L
     A =    K ( 0    A12  A13 ) if M-K-L >= 0;
            L ( 0     0   A23 )
        M-K-L ( 0     0    0  )

             N-K-L  K    L
     A =  K ( 0    A12  A13 ) if M-K-L < 0;
        M-K ( 0     0   A23 )

             N-K-L  K    L
     B =  L ( 0     0   B13 )
        P-L ( 0     0    0  )

  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
  otherwise A23 is (M-K)-by-L upper trapezoidal.

  On exit,

              U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),

  where U, V and Q are orthogonal matrices, Z' denotes the transpose
  of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
  ``diagonal'' matrices, which are of the following structures:

  If M-K-L >= 0,

                      K  L
         D1 =     K ( I  0 )
                  L ( 0  C )
              M-K-L ( 0  0 )

                    K  L
         D2 = L   ( 0  S )
              P-L ( 0  0 )

                 N-K-L  K    L
    ( 0 R ) = K (  0   R11  R12 ) K
              L (  0    0   R22 ) L

  where

    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
    S = diag( BETA(K+1),  ... , BETA(K+L) ),
    C**2 + S**2 = I.

    R is stored in A(1:K+L,N-K-L+1:N) on exit.

  If M-K-L < 0,

                 K M-K K+L-M
      D1 =   K ( I  0    0   )
           M-K ( 0  C    0   )

                   K M-K K+L-M
      D2 =   M-K ( 0  S    0   )
           K+L-M ( 0  0    I   )
             P-L ( 0  0    0   )

                 N-K-L  K   M-K  K+L-M
 ( 0 R ) =    K ( 0    R11  R12  R13  )
            M-K ( 0     0   R22  R23  )
          K+L-M ( 0     0    0   R33  )

  where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.

  R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored

      (  0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.

  The computation of the orthogonal transformation matrices U, V or Q
  is optional.  These matrices may either be formed explicitly, or they
  may be postmultiplied into input matrices U1, V1, or Q1.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  U must contain an orthogonal matrix U1 on entry, and

                  the product U1*U is returned;
          = 'I':  U is initialized to the unit matrix, and the
                  orthogonal matrix U is returned;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  V must contain an orthogonal matrix V1 on entry, and

                  the product V1*V is returned;
          = 'I':  V is initialized to the unit matrix, and the
                  orthogonal matrix V is returned;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and

                  the product Q1*Q is returned;
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal matrix Q is returned;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  K       (input) INTEGER
  L       (input) INTEGER
          K and L specify the subblocks in the input matrices A and B:

          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
          of A and B, whose GSVD is going to be computed by DTGSJA.
          See Further details.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
          matrix R or part of R.  See Purpose for details.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
          a part of R.  See Purpose for details.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TOLA    (input) DOUBLE PRECISION
  TOLB    (input) DOUBLE PRECISION
          TOLA and TOLB are the convergence criteria for the Jacobi-
          Kogbetliantz iteration procedure. Generally, they are the
          same as used in the preprocessing step, say
              TOLA = max(M,N)*norm(A)*MAZHEPS,
              TOLB = max(P,N)*norm(B)*MAZHEPS.

  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = diag(C),
            BETA(K+1:K+L)  = diag(S),
          or if M-K-L < 0,
            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
          Furthermore, if K+L < N,
            ALPHA(K+L+1:N) = 0 and
            BETA(K+L+1:N)  = 0.

  U       (input/output) DOUBLE PRECISION array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually

          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBU = 'I', U contains the orthogonal matrix U;
          if JOBU = 'U', U contains the product U1*U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (input/output) DOUBLE PRECISION array, dimension (LDV,P)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually

          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBV = 'I', V contains the orthogonal matrix V;
          if JOBV = 'V', V contains the product V1*V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually

          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBQ = 'I', Q contains the orthogonal matrix Q;
          if JOBQ = 'Q', Q contains the product Q1*Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

  NCYCLE  (output) INTEGER
          The number of cycles required for convergence.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the procedure does not converge after MAXIT cycles.

  Internal Parameters
  ===================

  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure

          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1.

  Further Details
  ===============

  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
  matrix B13 to the form:

           U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

  where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
  of Z.  C1 and S1 are diagonal matrices satisfying

                C1**2 + S1**2 = I,

  and R1 is an L-by-L nonsingular upper triangular matrix.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        p -
        n -
        k -
        l -
        a -
        lda -
        b -
        ldb -
        tola -
        tolb -
        alpha -
        beta -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        work -
        ncycle -
        info -

      • dtgsna

        public abstract void dtgsna(java.lang.String job,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            double[] s,
                            double[] dif,
                            int mm,
                            org.netlib.util.intW m,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGSNA estimates reciprocal condition numbers for specified
  eigenvalues and/or eigenvectors of a matrix pair (A, B) in
  generalized real Schur canonical form (or of any matrix pair
  (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where
  Z' denotes the transpose of Z.

  (A, B) must be in generalized real Schur form (as returned by DGGES),
  i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
  blocks. B is upper triangular.


  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (DIF):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (DIF);
          = 'B': for both eigenvalues and eigenvectors (S and DIF).

  HOWMNY  (input) CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the eigenpair corresponding to a real eigenvalue w(j),
          SELECT(j) must be set to .TRUE.. To select condition numbers

          corresponding to a complex conjugate pair of eigenvalues w(j)
          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be

          set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

  N       (input) INTEGER
          The order of the square matrix pair (A, B). N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The upper quasi-triangular matrix A in the pair (A,B).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
          The upper triangular matrix B in the pair (A,B).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  VL      (input) DOUBLE PRECISION array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns of VL, as returned by DTGEVC.
          If JOB = 'V', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL. LDVL >= 1.
          If JOB = 'E' or 'B', LDVL >= N.

  VR      (input) DOUBLE PRECISION array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns ov VR, as returned by DTGEVC.
          If JOB = 'V', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR. LDVR >= 1.
          If JOB = 'E' or 'B', LDVR >= N.

  S       (output) DOUBLE PRECISION array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the

          selected eigenvalues, stored in consecutive elements of the
          array. For a complex conjugate pair of eigenvalues two
          consecutive elements of S are set to the same value. Thus
          S(j), DIF(j), and the j-th columns of VL and VR all
          correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.

  DIF     (output) DOUBLE PRECISION array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array. For a complex eigenvector two
          consecutive elements of DIF are set to the same value. If
          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
          is set to 0; this can only occur when the true value would be
          very small anyway.
          If JOB = 'E', DIF is not referenced.

  MM      (input) INTEGER
          The number of elements in the arrays S and DIF. MM >= M.

  M       (output) INTEGER
          The number of elements of the arrays S and DIF used to store

          the specified condition numbers; for each selected real
          eigenvalue one element is used, and for each selected complex
          conjugate pair of eigenvalues, two elements are used.
          If HOWMNY = 'A', M is set to N.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N + 6)
          If JOB = 'E', IWORK is not referenced.

  INFO    (output) INTEGER
          =0: Successful exit
          <0: If INFO = -i, the i-th argument had an illegal value


  Further Details
  ===============

  The reciprocal of the condition number of a generalized eigenvalue
  w = (a, b) is defined as

       S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))

  where u and v are the left and right eigenvectors of (A, B)
  corresponding to w; |z| denotes the absolute value of the complex
  number, and norm(u) denotes the 2-norm of the vector u.
  The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)
  of the matrix pair (A, B). If both a and b equal zero, then (A B) is

  singular and S(I) = -1 is returned.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(A, B) / S(I)

  where EPS is the machine precision.

  The reciprocal of the condition number DIF(i) of right eigenvector u

  and left eigenvector v corresponding to the generalized eigenvalue w

  is defined as follows:

  a) If the i-th eigenvalue w = (a,b) is real

     Suppose U and V are orthogonal transformations such that

                U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                                        ( 0  S22 ),( 0 T22 )  n-1
                                          1  n-1     1 n-1

     Then the reciprocal condition number DIF(i) is

                Difl((a, b), (S22, T22)) = sigma-min( Zl ),

     where sigma-min(Zl) denotes the smallest singular value of the
     2(n-1)-by-2(n-1) matrix

         Zl = [ kron(a, In-1)  -kron(1, S22) ]
              [ kron(b, In-1)  -kron(1, T22) ] .

     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
     Kronecker product between the matrices X and Y.

     Note that if the default method for computing DIF(i) is wanted
     (see DLATDF), then the parameter DIFDRI (see below) should be
     changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
     See DTGSYL for more details.

  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

     Suppose U and V are orthogonal transformations such that

                U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                                       ( 0    S22 ),( 0    T22) n-2
                                         2    n-2     2    n-2

     and (S11, T11) corresponds to the complex conjugate eigenvalue
     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
     that

         U1'*S11*V1 = ( s11 s12 )   and U1'*T11*V1 = ( t11 t12 )
                      (  0  s22 )                    (  0  t22 )

     where the generalized eigenvalues w = s11/t11 and
     conjg(w) = s22/t22.

     Then the reciprocal condition number DIF(i) is bounded by

         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
     Z1 is the complex 2-by-2 matrix

              Z1 =  [ s11  -s22 ]
                    [ t11  -t22 ],

     This is done by computing (using real arithmetic) the
     roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
     where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
     the determinant of X.

     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an

     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

              Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]
                   [ kron(T11', In-2)  -kron(I2, T22) ]

     Note that if the default method for computing DIF is wanted (see
     DLATDF), then the parameter DIFDRI (see below) should be changed
     from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
     for more details.

  For each eigenvalue/vector specified by SELECT, DIF stores a
  Frobenius norm-based estimate of Difl.

  An approximate error bound for the i-th computed eigenvector VL(i) or
  VR(i) is given by

             EPS * norm(A, B) / DIF(i).

  See ref. [2-3] for more details and further references.

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  References
  ==========

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
      No 1, 1996.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        howmny -
        select -
        n -
        a -
        lda -
        b -
        ldb -
        vl -
        ldvl -
        vr -
        ldvr -
        s -
        dif -
        mm -
        m -
        work -
        lwork -
        iwork -
        info -

      • dtgsy2

        public abstract void dtgsy2(java.lang.String trans,
                            int ijob,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] c,
                            int Ldc,
                            double[] d,
                            int ldd,
                            double[] e,
                            int lde,
                            double[] f,
                            int ldf,
                            org.netlib.util.doubleW scale,
                            org.netlib.util.doubleW rdsum,
                            org.netlib.util.doubleW rdscal,
                            int[] iwork,
                            org.netlib.util.intW pq,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGSY2 solves the generalized Sylvester equation:

              A * R - L * B = scale * C                (1)
              D * R - L * E = scale * F,

  using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,

  (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
  N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
  must be in generalized Schur canonical form, i.e. A, B are upper
  quasi triangular and D, E are upper triangular. The solution (R, L)
  overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
  chosen to avoid overflow.

  In matrix notation solving equation (1) corresponds to solve
  Z*x = scale*b, where Z is defined as

         Z = [ kron(In, A)  -kron(B', Im) ]             (2)
             [ kron(In, D)  -kron(E', Im) ],

  Ik is the identity matrix of size k and X' is the transpose of X.
  kron(X, Y) is the Kronecker product between the matrices X and Y.
  In the process of solving (1), we solve a number of such systems
  where Dim(In), Dim(In) = 1 or 2.

  If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
  which is equivalent to solve for R and L in

              A' * R  + D' * L   = scale *  C           (3)
              R  * B' + L  * E'  = scale * -F

  This case is used to compute an estimate of Dif[(A, D), (B, E)] =
  sigma_min(Z) using reverse communicaton with DLACON.

  DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
  of an upper bound on the separation between to matrix pairs. Then
  the input (A, D), (B, E) are sub-pencils of the matrix pair in
  DTGSYL. See DTGSYL for details.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N', solve the generalized Sylvester equation (1).
          = 'T': solve the 'transposed' system (3).

  IJOB    (input) INTEGER
          Specifies what kind of functionality to be performed.
          = 0: solve (1) only.
          = 1: A contribution from this subsystem to a Frobenius
               norm-based estimate of the separation between two matrix
               pairs is computed. (look ahead strategy is used).
          = 2: A contribution from this subsystem to a Frobenius
               norm-based estimate of the separation between two matrix
               pairs is computed. (DGECON on sub-systems is used.)
          Not referenced if TRANS = 'T'.

  M       (input) INTEGER
          On entry, M specifies the order of A and D, and the row
          dimension of C, F, R and L.

  N       (input) INTEGER
          On entry, N specifies the order of B and E, and the column
          dimension of C, F, R and L.

  A       (input) DOUBLE PRECISION array, dimension (LDA, M)
          On entry, A contains an upper quasi triangular matrix.

  LDA     (input) INTEGER
          The leading dimension of the matrix A. LDA >= max(1, M).

  B       (input) DOUBLE PRECISION array, dimension (LDB, N)
          On entry, B contains an upper quasi triangular matrix.

  LDB     (input) INTEGER
          The leading dimension of the matrix B. LDB >= max(1, N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC, N)
          On entry, C contains the right-hand-side of the first matrix

          equation in (1).
          On exit, if IJOB = 0, C has been overwritten by the
          solution R.

  LDC     (input) INTEGER
          The leading dimension of the matrix C. LDC >= max(1, M).

  D       (input) DOUBLE PRECISION array, dimension (LDD, M)
          On entry, D contains an upper triangular matrix.

  LDD     (input) INTEGER
          The leading dimension of the matrix D. LDD >= max(1, M).

  E       (input) DOUBLE PRECISION array, dimension (LDE, N)
          On entry, E contains an upper triangular matrix.

  LDE     (input) INTEGER
          The leading dimension of the matrix E. LDE >= max(1, N).

  F       (input/output) DOUBLE PRECISION array, dimension (LDF, N)
          On entry, F contains the right-hand-side of the second matrix
          equation in (1).
          On exit, if IJOB = 0, F has been overwritten by the
          solution L.

  LDF     (input) INTEGER
          The leading dimension of the matrix F. LDF >= max(1, M).

  SCALE   (output) DOUBLE PRECISION
          On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
          R and L (C and F on entry) will hold the solutions to a
          slightly perturbed system but the input matrices A, B, D and

          E have not been changed. If SCALE = 0, R and L will hold the

          solutions to the homogeneous system with C = F = 0. Normally,
          SCALE = 1.

  RDSUM   (input/output) DOUBLE PRECISION
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by DTGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.

  RDSCAL  (input/output) DOUBLE PRECISION
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when DTGSY2 is called by
                DTGSYL.

  IWORK   (workspace) INTEGER array, dimension (M+N+2)

  PQ      (output) INTEGER
          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
          8-by-8) solved by this routine.

  INFO    (output) INTEGER
          On exit, if INFO is set to
            =0: Successful exit
            <0: If INFO = -i, the i-th argument had an illegal value.
            >0: The matrix pairs (A, D) and (B, E) have common or very

                close eigenvalues.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================
  Replaced various illegal calls to DCOPY by calls to DLASET.
  Sven Hammarling, 27/5/02.

     .. Parameters ..
        Parameters:
        trans -
        ijob -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        d -
        ldd -
        e -
        lde -
        f -
        ldf -
        scale -
        rdsum -
        rdscal -
        iwork -
        pq -
        info -

      • dtgsyl

        public abstract void dtgsyl(java.lang.String trans,
                            int ijob,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] c,
                            int Ldc,
                            double[] d,
                            int ldd,
                            double[] e,
                            int lde,
                            double[] f,
                            int ldf,
                            org.netlib.util.doubleW scale,
                            org.netlib.util.doubleW dif,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTGSYL solves the generalized Sylvester equation:

              A * R - L * B = scale * C                 (1)
              D * R - L * E = scale * F

  where R and L are unknown m-by-n matrices, (A, D), (B, E) and
  (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
  respectively, with real entries. (A, D) and (B, E) must be in
  generalized (real) Schur canonical form, i.e. A, B are upper quasi
  triangular and D, E are upper triangular.

  The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
  scaling factor chosen to avoid overflow.

  In matrix notation (1) is equivalent to solve  Zx = scale b, where
  Z is defined as

             Z = [ kron(In, A)  -kron(B', Im) ]         (2)
                 [ kron(In, D)  -kron(E', Im) ].

  Here Ik is the identity matrix of size k and X' is the transpose of
  X. kron(X, Y) is the Kronecker product between the matrices X and Y.


  If TRANS = 'T', DTGSYL solves the transposed system Z'*y = scale*b,
  which is equivalent to solve for R and L in

              A' * R  + D' * L   = scale *  C           (3)
              R  * B' + L  * E'  = scale * (-F)

  This case (TRANS = 'T') is used to compute an one-norm-based estimate
  of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
  and (B,E), using DLACON.

  If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
  of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the

  reciprocal of the smallest singular value of Z. See [1-2] for more
  information.

  This is a level 3 BLAS algorithm.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N', solve the generalized Sylvester equation (1).
          = 'T', solve the 'transposed' system (3).

  IJOB    (input) INTEGER
          Specifies what kind of functionality to be performed.
           =0: solve (1) only.
           =1: The functionality of 0 and 3.
           =2: The functionality of 0 and 4.
           =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
               (look ahead strategy IJOB  = 1 is used).
           =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
               ( DGECON on sub-systems is used ).
          Not referenced if TRANS = 'T'.

  M       (input) INTEGER
          The order of the matrices A and D, and the row dimension of
          the matrices C, F, R and L.

  N       (input) INTEGER
          The order of the matrices B and E, and the column dimension
          of the matrices C, F, R and L.

  A       (input) DOUBLE PRECISION array, dimension (LDA, M)
          The upper quasi triangular matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1, M).

  B       (input) DOUBLE PRECISION array, dimension (LDB, N)
          The upper quasi triangular matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1, N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC, N)
          On entry, C contains the right-hand-side of the first matrix

          equation in (1) or (3).
          On exit, if IJOB = 0, 1 or 2, C has been overwritten by
          the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,

          the solution achieved during the computation of the
          Dif-estimate.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1, M).

  D       (input) DOUBLE PRECISION array, dimension (LDD, M)
          The upper triangular matrix D.

  LDD     (input) INTEGER
          The leading dimension of the array D. LDD >= max(1, M).

  E       (input) DOUBLE PRECISION array, dimension (LDE, N)
          The upper triangular matrix E.

  LDE     (input) INTEGER
          The leading dimension of the array E. LDE >= max(1, N).

  F       (input/output) DOUBLE PRECISION array, dimension (LDF, N)
          On entry, F contains the right-hand-side of the second matrix
          equation in (1) or (3).
          On exit, if IJOB = 0, 1 or 2, F has been overwritten by
          the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,

          the solution achieved during the computation of the
          Dif-estimate.

  LDF     (input) INTEGER
          The leading dimension of the array F. LDF >= max(1, M).

  DIF     (output) DOUBLE PRECISION
          On exit DIF is the reciprocal of a lower bound of the
          reciprocal of the Dif-function, i.e. DIF is an upper bound of
          Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
          IF IJOB = 0 or TRANS = 'T', DIF is not touched.

  SCALE   (output) DOUBLE PRECISION
          On exit SCALE is the scaling factor in (1) or (3).
          If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,

          to a slightly perturbed system but the input matrices A, B, D
          and E have not been changed. If SCALE = 0, C and F hold the
          solutions R and L, respectively, to the homogeneous system
          with C = F = 0. Normally, SCALE = 1.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK > = 1.
          If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (M+N+6)

  INFO    (output) INTEGER
            =0: successful exit
            <0: If INFO = -i, the i-th argument had an illegal value.
            >0: (A, D) and (B, E) have common or close eigenvalues.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
      No 1, 1996.

  [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
      Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
      Appl., 15(4):1045-1060, 1994

  [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
      Condition Estimators for Solving the Generalized Sylvester
      Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
      July 1989, pp 745-751.

  =====================================================================
  Replaced various illegal calls to DCOPY by calls to DLASET.
  Sven Hammarling, 1/5/02.

     .. Parameters ..
        Parameters:
        trans -
        ijob -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        d -
        ldd -
        e -
        lde -
        f -
        ldf -
        scale -
        dif -
        work -
        lwork -
        iwork -
        info -

      • dtpcon

        public abstract void dtpcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            double[] ap,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTPCON estimates the reciprocal of the condition number of a packed
  triangular matrix A, in either the 1-norm or the infinity-norm.

  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        ap -
        rcond -
        work -
        iwork -
        info -

      • dtprfs

        public abstract void dtprfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTPRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular packed
  coefficient matrix.

  The solution matrix X must be computed by DTPTRS or some other
  means before entering this routine.  DTPRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        ap -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dtptri

        public abstract void dtptri(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            double[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTPTRI computes the inverse of a real upper or lower triangular
  matrix A stored in packed format.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)

          On entry, the upper or lower triangular matrix A, stored
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.
          On exit, the (triangular) inverse of the original matrix, in

          the same packed storage format.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, A(i,i) is exactly zero.  The triangular
                matrix is singular and its inverse can not be computed.

  Further Details
  ===============

  A triangular matrix A can be transferred to packed storage using one

  of the following program segments:

  UPLO = 'U':                      UPLO = 'L':

        JC = 1                           JC = 1
        DO 2 J = 1, N                    DO 2 J = 1, N
           DO 1 I = 1, J                    DO 1 I = J, N
              AP(JC+I-1) = A(I,J)              AP(JC+I-J) = A(I,J)
      1    CONTINUE                    1    CONTINUE
           JC = JC + J                      JC = JC + N - J + 1
      2 CONTINUE                       2 CONTINUE

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        ap -
        info -

      • dtptrs

        public abstract void dtptrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            double[] ap,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTPTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular matrix of order N stored in packed format,
  and B is an N-by-NRHS matrix.  A check is made to verify that A is
  nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • dtrcon

        public abstract void dtrcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.doubleW rcond,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRCON estimates the reciprocal of the condition number of a
  triangular matrix A, in either the 1-norm or the infinity-norm.

  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        a -
        lda -
        rcond -
        work -
        iwork -
        info -

      • dtrevc

        public abstract void dtrevc(java.lang.String side,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            double[] t,
                            int ldt,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTREVC computes some or all of the right and/or left eigenvectors of

  a real upper quasi-triangular matrix T.
  Matrices of this type are produced by the Schur factorization of
  a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
  
  The right eigenvector x and the left eigenvector y of T corresponding
  to an eigenvalue w are defined by:
  
     T*x = w*x,     (y**H)*T = w*(y**H)
  
  where y**H denotes the conjugate transpose of y.
  The eigenvalues are not input to this routine, but are read directly

  from the diagonal blocks of T.
  
  This routine returns the matrices X and/or Y of right and left
  eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
  input matrix.  If Q is the orthogonal factor that reduces a matrix
  A to Schur form T, then Q*X and Q*Y are the matrices of right and
  left eigenvectors of A.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.

  HOWMNY  (input) CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed by the matrices in VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.

  SELECT  (input/output) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          If w(j) is a real eigenvalue, the corresponding real
          eigenvector is computed if SELECT(j) is .TRUE..
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector is

          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to

          .FALSE..
          Not referenced if HOWMNY = 'A' or 'B'.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input) DOUBLE PRECISION array, dimension (LDT,N)
          The upper quasi-triangular matrix T in Schur canonical form.


  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns

                           of VL, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part.
          Not referenced if SIDE = 'R'.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.

  VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns

                           of VR, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part.
          Not referenced if SIDE = 'L'.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B', LDVR >= N.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.
          If HOWMNY = 'A' or 'B', M is set to N.
          Each selected real eigenvector occupies one column and each
          selected complex eigenvector occupies two columns.

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        howmny -
        select -
        n -
        t -
        ldt -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        info -

      • dtrexc

        public abstract void dtrexc(java.lang.String compq,
                            int n,
                            double[] t,
                            int ldt,
                            double[] q,
                            int ldq,
                            org.netlib.util.intW ifst,
                            org.netlib.util.intW ilst,
                            double[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTREXC reorders the real Schur factorization of a real matrix
  A = Q*T*Q**T, so that the diagonal block of T with row index IFST is

  moved to row ILST.

  The real Schur form T is reordered by an orthogonal similarity
  transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
  is updated by postmultiplying it with Z.

  T must be in Schur canonical form (as returned by DHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  COMPQ   (input) CHARACTER*1
          = 'V':  update the matrix Q of Schur vectors;
          = 'N':  do not update Q.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          Schur canonical form.
          On exit, the reordered upper quasi-triangular matrix, again
          in Schur canonical form.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          orthogonal transformation matrix Z which reorders T.
          If COMPQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max(1,N).

  IFST    (input/output) INTEGER
  ILST    (input/output) INTEGER
          Specify the reordering of the diagonal blocks of T.
          The block with row index IFST is moved to row ILST, by a
          sequence of transpositions between adjacent blocks.
          On exit, if IFST pointed on entry to the second row of a
          2-by-2 block, it is changed to point to the first row; ILST
          always points to the first row of the block in its final
          position (which may differ from its input value by +1 or -1).
          1 <= IFST <= N; 1 <= ILST <= N.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          = 1:  two adjacent blocks were too close to swap (the problem
                is very ill-conditioned); T may have been partially
                reordered, and ILST points to the first row of the
                current position of the block being moved.

  =====================================================================

     .. Parameters ..
        Parameters:
        compq -
        n -
        t -
        ldt -
        q -
        ldq -
        ifst -
        ilst -
        work -
        info -

      • dtrrfs

        public abstract void dtrrfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] x,
                            int ldx,
                            double[] ferr,
                            double[] berr,
                            double[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular
  coefficient matrix.

  The solution matrix X must be computed by DTRTRS or some other
  means before entering this routine.  DTRRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • dtrsen

        public abstract void dtrsen(java.lang.String job,
                            java.lang.String compq,
                            boolean[] select,
                            int n,
                            double[] t,
                            int ldt,
                            double[] q,
                            int ldq,
                            double[] wr,
                            double[] wi,
                            org.netlib.util.intW m,
                            org.netlib.util.doubleW s,
                            org.netlib.util.doubleW sep,
                            double[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRSEN reorders the real Schur factorization of a real matrix
  A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
  the leading diagonal blocks of the upper quasi-triangular matrix T,
  and the leading columns of Q form an orthonormal basis of the
  corresponding right invariant subspace.

  Optionally the routine computes the reciprocal condition numbers of
  the cluster of eigenvalues and/or the invariant subspace.

  T must be in Schur canonical form (as returned by DHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elemnts equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (S) or the invariant subspace (SEP):
          = 'N': none;
          = 'E': for eigenvalues only (S);
          = 'V': for invariant subspace only (SEP);
          = 'B': for both eigenvalues and invariant subspace (S and
                 SEP).

  COMPQ   (input) CHARACTER*1
          = 'V': update the matrix Q of Schur vectors;
          = 'N': do not update Q.

  SELECT  (input) LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To

          select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, T is overwritten by the reordered matrix T, again in
          Schur canonical form, with the selected eigenvalues in the
          leading diagonal blocks.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          orthogonal transformation matrix which reorders T; the
          leading M columns of Q form an orthonormal basis for the
          specified invariant subspace.
          If COMPQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          The real and imaginary parts, respectively, of the reordered

          eigenvalues of T. The eigenvalues are stored in the same
          order as on the diagonal of T, with WR(i) = T(i,i) and, if
          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
          sufficiently ill-conditioned, then its value may differ
          significantly from its value before reordering.

  M       (output) INTEGER
          The dimension of the specified invariant subspace.
          0 < = M <= N.

  S       (output) DOUBLE PRECISION
          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
          condition number for the selected cluster of eigenvalues.
          S cannot underestimate the true reciprocal condition number
          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
          If JOB = 'N' or 'V', S is not referenced.

  SEP     (output) DOUBLE PRECISION
          If JOB = 'V' or 'B', SEP is the estimated reciprocal
          condition number of the specified invariant subspace. If
          M = 0 or N, SEP = norm(T).
          If JOB = 'N' or 'E', SEP is not referenced.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If JOB = 'N', LWORK >= max(1,N);
          if JOB = 'E', LWORK >= max(1,M*(N-M));
          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOB = 'N' or 'E', LIWORK >= 1;
          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1: reordering of T failed because some eigenvalues are too

               close to separate (the problem is very ill-conditioned);
               T may have been partially reordered, and WR and WI
               contain the eigenvalues in the same order as in T; S and
               SEP (if requested) are set to zero.

  Further Details
  ===============

  DTRSEN first collects the selected eigenvalues by computing an
  orthogonal transformation Z to move them to the top left corner of T.
  In other words, the selected eigenvalues are the eigenvalues of T11
  in:

                Z'*T*Z = ( T11 T12 ) n1
                         (  0  T22 ) n2
                            n1  n2

  where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
  of Z span the specified invariant subspace of T.

  If T has been obtained from the real Schur factorization of a matrix

  A = Q*T*Q', then the reordered real Schur factorization of A is given
  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
  the corresponding invariant subspace of A.

  The reciprocal condition number of the average of the eigenvalues of

  T11 may be returned in S. S lies between 0 (very badly conditioned)
  and 1 (very well conditioned). It is computed as follows. First we
  compute R so that

                         P = ( I  R ) n1
                             ( 0  0 ) n2
                               n1 n2

  is the projector on the invariant subspace associated with T11.
  R is the solution of the Sylvester equation:

                        T11*R - R*T22 = T12.

  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
  the two-norm of M. Then S is computed as the lower bound

                      (1 + F-norm(R)**2)**(-1/2)

  on the reciprocal of 2-norm(P), the true reciprocal condition number.
  S cannot underestimate 1 / 2-norm(P) by more than a factor of
  sqrt(N).

  An approximate error bound for the computed average of the
  eigenvalues of T11 is

                         EPS * norm(T) / S

  where EPS is the machine precision.

  The reciprocal condition number of the right invariant subspace
  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.

  SEP is defined as the separation of T11 and T22:

                     sep( T11, T22 ) = sigma-min( C )

  where sigma-min(C) is the smallest singular value of the
  n1*n2-by-n1*n2 matrix

     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
  product. We estimate sigma-min(C) by the reciprocal of an estimate of
  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

  When SEP is small, small changes in T can cause large changes in
  the invariant subspace. An approximate bound on the maximum angular
  error in the computed right invariant subspace is

                      EPS * norm(T) / SEP

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        compq -
        select -
        n -
        t -
        ldt -
        q -
        ldq -
        wr -
        wi -
        m -
        s -
        sep -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • dtrsna

        public abstract void dtrsna(java.lang.String job,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            double[] t,
                            int ldt,
                            double[] vl,
                            int ldvl,
                            double[] vr,
                            int ldvr,
                            double[] s,
                            double[] sep,
                            int mm,
                            org.netlib.util.intW m,
                            double[] work,
                            int ldwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRSNA estimates reciprocal condition numbers for specified
  eigenvalues and/or right eigenvectors of a real upper
  quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
  orthogonal).

  T must be in Schur canonical form (as returned by DHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (SEP):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (SEP);
          = 'B': for both eigenvalues and eigenvectors (S and SEP).

  HOWMNY  (input) CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the eigenpair corresponding to a real eigenvalue w(j),
          SELECT(j) must be set to .TRUE.. To select condition numbers

          corresponding to a complex conjugate pair of eigenvalues w(j)
          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be

          set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input) DOUBLE PRECISION array, dimension (LDT,N)
          The upper quasi-triangular matrix T, in Schur canonical form.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  VL      (input) DOUBLE PRECISION array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
          (or of any Q*T*Q**T with Q orthogonal), corresponding to the

          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VL, as returned by
          DHSEIN or DTREVC.
          If JOB = 'V', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.
          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.

  VR      (input) DOUBLE PRECISION array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of T

          (or of any Q*T*Q**T with Q orthogonal), corresponding to the

          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VR, as returned by
          DHSEIN or DTREVC.
          If JOB = 'V', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.
          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

  S       (output) DOUBLE PRECISION array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the

          selected eigenvalues, stored in consecutive elements of the
          array. For a complex conjugate pair of eigenvalues two
          consecutive elements of S are set to the same value. Thus
          S(j), SEP(j), and the j-th columns of VL and VR all
          correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.

  SEP     (output) DOUBLE PRECISION array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array. For a complex eigenvector two
          consecutive elements of SEP are set to the same value. If
          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
          is set to 0; this can only occur when the true value would be
          very small anyway.
          If JOB = 'E', SEP is not referenced.

  MM      (input) INTEGER
          The number of elements in the arrays S (if JOB = 'E' or 'B')

           and/or SEP (if JOB = 'V' or 'B'). MM >= M.

  M       (output) INTEGER
          The number of elements of the arrays S and/or SEP actually
          used to store the estimated condition numbers.
          If HOWMNY = 'A', M is set to N.

  WORK    (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+6)
          If JOB = 'E', WORK is not referenced.

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

  IWORK   (workspace) INTEGER array, dimension (2*(N-1))
          If JOB = 'E', IWORK is not referenced.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The reciprocal of the condition number of an eigenvalue lambda is
  defined as

          S(lambda) = |v'*u| / (norm(u)*norm(v))

  where u and v are the right and left eigenvectors of T corresponding

  to lambda; v' denotes the conjugate-transpose of v, and norm(u)
  denotes the Euclidean norm. These reciprocal condition numbers always
  lie between zero (very badly conditioned) and one (very well
  conditioned). If n = 1, S(lambda) is defined to be 1.

  An approximate error bound for a computed eigenvalue W(i) is given by

                      EPS * norm(T) / S(i)

  where EPS is the machine precision.

  The reciprocal of the condition number of the right eigenvector u
  corresponding to lambda is defined as follows. Suppose

              T = ( lambda  c  )
                  (   0    T22 )

  Then the reciprocal condition number is

          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

  where sigma-min denotes the smallest singular value. We approximate
  the smallest singular value by the reciprocal of an estimate of the
  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
  defined to be abs(T(1,1)).

  An approximate error bound for a computed right eigenvector VR(i)
  is given by

                      EPS * norm(T) / SEP(i)

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        howmny -
        select -
        n -
        t -
        ldt -
        vl -
        ldvl -
        vr -
        ldvr -
        s -
        sep -
        mm -
        m -
        work -
        ldwork -
        iwork -
        info -

      • dtrsyl

        public abstract void dtrsyl(java.lang.String trana,
                            java.lang.String tranb,
                            int isgn,
                            int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            double[] c,
                            int Ldc,
                            org.netlib.util.doubleW scale,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRSYL solves the real Sylvester matrix equation:

     op(A)*X + X*op(B) = scale*C or
     op(A)*X - X*op(B) = scale*C,

  where op(A) = A or A**T, and  A and B are both upper quasi-
  triangular. A is M-by-M and B is N-by-N; the right hand side C and
  the solution X are M-by-N; and scale is an output scale factor, set
  <= 1 to avoid overflow in X.

  A and B must be in Schur canonical form (as returned by DHSEQR), that
  is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
  each 2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  TRANA   (input) CHARACTER*1
          Specifies the option op(A):
          = 'N': op(A) = A    (No transpose)
          = 'T': op(A) = A**T (Transpose)
          = 'C': op(A) = A**H (Conjugate transpose = Transpose)

  TRANB   (input) CHARACTER*1
          Specifies the option op(B):
          = 'N': op(B) = B    (No transpose)
          = 'T': op(B) = B**T (Transpose)
          = 'C': op(B) = B**H (Conjugate transpose = Transpose)

  ISGN    (input) INTEGER
          Specifies the sign in the equation:
          = +1: solve op(A)*X + X*op(B) = scale*C
          = -1: solve op(A)*X - X*op(B) = scale*C

  M       (input) INTEGER
          The order of the matrix A, and the number of rows in the
          matrices X and C. M >= 0.

  N       (input) INTEGER
          The order of the matrix B, and the number of columns in the
          matrices X and C. N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,M)
          The upper quasi-triangular matrix A, in Schur canonical form.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
          The upper quasi-triangular matrix B, in Schur canonical form.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N right hand side matrix C.
          On exit, C is overwritten by the solution matrix X.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M)

  SCALE   (output) DOUBLE PRECISION
          The scale factor, scale, set <= 1 to avoid overflow in X.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1: A and B have common or very close eigenvalues; perturbed
               values were used to solve the equation (but the matrices
               A and B are unchanged).

  =====================================================================

     .. Parameters ..
        Parameters:
        trana -
        tranb -
        isgn -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        scale -
        info -

      • dtrti2

        public abstract void dtrti2(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRTI2 computes the inverse of a real upper or lower triangular
  matrix.

  This is the Level 2 BLAS version of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading n by n upper triangular part of the array A contains

          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of the array A contains

          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.

          On exit, the (triangular) inverse of the original matrix, in

          the same storage format.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        a -
        lda -
        info -

      • dtrtri

        public abstract void dtrtri(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            double[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRTRI computes the inverse of a real upper or lower triangular
  matrix A.

  This is the Level 3 BLAS version of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains

          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains

          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.
          On exit, the (triangular) inverse of the original matrix, in

          the same storage format.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can not be computed.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        a -
        lda -
        info -

      • dtrtrs

        public abstract void dtrtrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            double[] a,
                            int lda,
                            double[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTRTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular matrix of order N, and B is an N-by-NRHS
  matrix.  A check is made to verify that A is nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the i-th diagonal element of A is zero,
               indicating that the matrix is singular and the solutions
               X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • dtzrqf

        public abstract void dtzrqf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine DTZRZF.

  DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
  to upper triangular form by means of orthogonal transformations.

  The upper trapezoidal matrix A is factored as

     A = ( R  0 ) * Z,

  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
  triangular matrix.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= M.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements M+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (M)
          The scalar factors of the elementary reflectors.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an ( n - m ) element vector.
  tau and z( k ) are chosen to annihilate the elements of the kth row
  of X.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A, such that the elements of z( k ) are
  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        info -

      • dtzrzf

        public abstract void dtzrzf(int m,
                            int n,
                            double[] a,
                            int lda,
                            double[] tau,
                            double[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
  to upper triangular form by means of orthogonal transformations.

  The upper trapezoidal matrix A is factored as

     A = ( R  0 ) * Z,

  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
  triangular matrix.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= M.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements M+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) DOUBLE PRECISION array, dimension (M)
          The scalar factors of the elementary reflectors.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,L
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an ( n - m ) element vector.
  tau and z( k ) are chosen to annihilate the elements of the kth row
  of X.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A, such that the elements of z( k ) are
  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • ieeeck

        public abstract int ieeeck(int ispec,
                           float zero,
                           float one)
        
     ..

  Purpose
  =======

  IEEECK is called from the ILAENV to verify that Infinity and
  possibly NaN arithmetic is safe (i.e. will not trap).

  Arguments
  =========

  ISPEC   (input) INTEGER
          Specifies whether to test just for inifinity arithmetic
          or whether to test for infinity and NaN arithmetic.
          = 0: Verify infinity arithmetic only.
          = 1: Verify infinity and NaN arithmetic.

  ZERO    (input) REAL
          Must contain the value 0.0
          This is passed to prevent the compiler from optimizing
          away this code.

  ONE     (input) REAL
          Must contain the value 1.0
          This is passed to prevent the compiler from optimizing
          away this code.

  RETURN VALUE:  INTEGER
          = 0:  Arithmetic failed to produce the correct answers
          = 1:  Arithmetic produced the correct answers

     .. Local Scalars ..
        Parameters:
        ispec -
        zero -
        one -

      • ilaenv

        public abstract int ilaenv(int ispec,
                           java.lang.String name,
                           java.lang.String opts,
                           int n1,
                           int n2,
                           int n3,
                           int n4)
        
     ..

  Purpose
  =======

  ILAENV is called from the LAPACK routines to choose problem-dependent
  parameters for the local environment.  See ISPEC for a description of
  the parameters.

  ILAENV returns an INTEGER
  if ILAENV >= 0: ILAENV returns the value of the parameter specified b
  if ILAENV < 0:  if ILAENV = -k, the k-th argument had an illegal valu

  This version provides a set of parameters which should give good,
  but not optimal, performance on many of the currently available
  computers.  Users are encouraged to modify this subroutine to set
  the tuning parameters for their particular machine using the option
  and problem size information in the arguments.

  This routine will not function correctly if it is converted to all
  lower case.  Converting it to all upper case is allowed.

  Arguments
  =========

  ISPEC   (input) INTEGER
          Specifies the parameter to be returned as the value of
          ILAENV.
          = 1: the optimal blocksize; if this value is 1, an unblocked

               algorithm will give the best performance.
          = 2: the minimum block size for which the block routine
               should be used; if the usable block size is less than
               this value, an unblocked routine should be used.
          = 3: the crossover point (in a block routine, for N less
               than this value, an unblocked routine should be used)
          = 4: the number of shifts, used in the nonsymmetric
               eigenvalue routines (DEPRECATED)
          = 5: the minimum column dimension for blocking to be used;
               rectangular blocks must have dimension at least k by m,

               where k is given by ILAENV(2,...) and m by ILAENV(5,...)
          = 6: the crossover point for the SVD (when reducing an m by n
               matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds

               this value, a QR factorization is used first to reduce
               the matrix to a triangular form.)
          = 7: the number of processors
          = 8: the crossover point for the multishift QR method
               for nonsymmetric eigenvalue problems (DEPRECATED)
          = 9: maximum size of the subproblems at the bottom of the
               computation tree in the divide-and-conquer algorithm
               (used by xGELSD and xGESDD)
          =10: ieee NaN arithmetic can be trusted not to trap
          =11: infinity arithmetic can be trusted not to trap
          12 <= ISPEC <= 16:
               xHSEQR or one of its subroutines,
               see IPARMQ for detailed explanation

  NAME    (input) CHARACTER*(*)
          The name of the calling subroutine, in either upper case or
          lower case.

  OPTS    (input) CHARACTER*(*)
          The character options to the subroutine NAME, concatenated
          into a single character string.  For example, UPLO = 'U',
          TRANS = 'T', and DIAG = 'N' for a triangular routine would
          be specified as OPTS = 'UTN'.

  N1      (input) INTEGER
  N2      (input) INTEGER
  N3      (input) INTEGER
  N4      (input) INTEGER
          Problem dimensions for the subroutine NAME; these may not all
          be required.

  Further Details
  ===============

  The following conventions have been used when calling ILAENV from the
  LAPACK routines:
  1)  OPTS is a concatenation of all of the character options to
      subroutine NAME, in the same order that they appear in the
      argument list for NAME, even if they are not used in determining

      the value of the parameter specified by ISPEC.
  2)  The problem dimensions N1, N2, N3, N4 are specified in the order

      that they appear in the argument list for NAME.  N1 is used
      first, N2 second, and so on, and unused problem dimensions are
      passed a value of -1.
  3)  The parameter value returned by ILAENV is checked for validity in
      the calling subroutine.  For example, ILAENV is used to retrieve

      the optimal blocksize for STRTRI as follows:

      NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
      IF( NB.LE.1 ) NB = MAX( 1, N )

  =====================================================================

     .. Local Scalars ..
        Parameters:
        ispec -
        name -
        opts -
        n1 -
        n2 -
        n3 -
        n4 -

      • ilaver

        public void ilaver(org.netlib.util.intW vers_major,
                   org.netlib.util.intW vers_minor,
                   org.netlib.util.intW vers_patch)
        
     
  -- LAPACK routine (version 3.1.1) --
     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
     January 2007
     ..

  Purpose
  =======

  This subroutine return the Lapack version

  Arguments
  =========
  VERS_MAJOR   (output) INTEGER
      return the lapack major version
  VERS_MINOR   (output) INTEGER
      return the lapack minor version from the major version
  VERS_PATCH   (output) INTEGER
      return the lapack patch version from the minor version
  =====================================================================
        Parameters:
        vers_major -
        vers_minor -
        vers_patch -

      • iparmq

        public int iparmq(int ispec,
                  java.lang.String name,
                  java.lang.String opts,
                  int n,
                  int ilo,
                  int ihi,
                  int lwork)
        

  Purpose
  =======

       This program sets problem and machine dependent parameters
       useful for xHSEQR and its subroutines. It is called whenever 
       ILAENV is called with 12 <= ISPEC <= 16

  Arguments
  =========

       ISPEC  (input) integer scalar
              ISPEC specifies which tunable parameter IPARMQ should
              return.

              ISPEC=12: (INMIN)  Matrices of order nmin or less
                        are sent directly to xLAHQR, the implicit
                        double shift QR algorithm.  NMIN must be
                        at least 11.

              ISPEC=13: (INWIN)  Size of the deflation window.
                        This is best set greater than or equal to
                        the number of simultaneous shifts NS.
                        Larger matrices benefit from larger deflation
                        windows.

              ISPEC=14: (INIBL) Determines when to stop nibbling and
                        invest in an (expensive) multi-shift QR sweep.

                        If the aggressive early deflation subroutine
                        finds LD converged eigenvalues from an order
                        NW deflation window and LD.GT.(NW*NIBBLE)/100,

                        then the next QR sweep is skipped and early
                        deflation is applied immediately to the
                        remaining active diagonal block.  Setting
                        IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
                        multi-shift QR sweep whenever early deflation
                        finds a converged eigenvalue.  Setting
                        IPARMQ(ISPEC=14) greater than or equal to 100
                        prevents TTQRE from skipping a multi-shift
                        QR sweep.

              ISPEC=15: (NSHFTS) The number of simultaneous shifts in
                        a multi-shift QR iteration.

              ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
                        following meanings.
                        0:  During the multi-shift QR sweep,
                            xLAQR5 does not accumulate reflections and

                            does not use matrix-matrix multiply to
                            update the far-from-diagonal matrix
                            entries.
                        1:  During the multi-shift QR sweep,
                            xLAQR5 and/or xLAQRaccumulates reflections 
                            matrix-matrix multiply to update the
                            far-from-diagonal matrix entries.
                        2:  During the multi-shift QR sweep.
                            xLAQR5 accumulates reflections and takes
                            advantage of 2-by-2 block structure during

                            matrix-matrix multiplies.
                        (If xTRMM is slower than xGEMM, then
                        IPARMQ(ISPEC=16)=1 may be more efficient than
                        IPARMQ(ISPEC=16)=2 despite the greater level of
                        arithmetic work implied by the latter choice.)


       NAME    (input) character string
               Name of the calling subroutine

       OPTS    (input) character string
               This is a concatenation of the string arguments to
               TTQRE.

       N       (input) integer scalar
               N is the order of the Hessenberg matrix H.

       ILO     (input) INTEGER
       IHI     (input) INTEGER
               It is assumed that H is already upper triangular
               in rows and columns 1:ILO-1 and IHI+1:N.

       LWORK   (input) integer scalar
               The amount of workspace available.

  Further Details
  ===============

       Little is known about how best to choose these parameters.
       It is possible to use different values of the parameters
       for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.

       It is probably best to choose different parameters for
       different matrices and different parameters at different
       times during the iteration, but this has not been
       implemented --- yet.


       The best choices of most of the parameters depend
       in an ill-understood way on the relative execution
       rate of xLAQR3 and xLAQR5 and on the nature of each
       particular eigenvalue problem.  Experiment may be the
       only practical way to determine which choices are most
       effective.

       Following is a list of default values supplied by IPARMQ.
       These defaults may be adjusted in order to attain better
       performance in any particular computational environment.

       IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
                        Default: 75. (Must be at least 11.)

       IPARMQ(ISPEC=13) Recommended deflation window size.
                        This depends on ILO, IHI and NS, the
                        number of simultaneous shifts returned
                        by IPARMQ(ISPEC=15).  The default for
                        (IHI-ILO+1).LE.500 is NS.  The default
                        for (IHI-ILO+1).GT.500 is 3*NS/2.

       IPARMQ(ISPEC=14) Nibble crossover point.  Default: 14.

       IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
                        a multi-shift QR iteration.

                        If IHI-ILO+1 is ...

                        greater than      ...but less    ... the
                        or equal to ...      than        default is

                                0               30       NS =   2+
                               30               60       NS =   4+
                               60              150       NS =  10
                              150              590       NS =  **
                              590             3000       NS =  64
                             3000             6000       NS = 128
                             6000             infinity   NS = 256

                    (+)  By default matrices of this order are
                         passed to the implicit double shift routine
                         xLAHQR.  See IPARMQ(ISPEC=12) above.   These
                         values of NS are used only in case of a rare
                         xLAHQR failure.

                    (**) The asterisks (**) indicate an ad-hoc
                         function increasing from 10 to 64.

       IPARMQ(ISPEC=16) Select structured matrix multiply.
                        (See ISPEC=16 above for details.)
                        Default: 3.

     ================================================================
     .. Parameters ..
        Parameters:
        ispec -
        name -
        opts -
        n -
        ilo -
        ihi -
        lwork -

      • lsamen

        public boolean lsamen(int n,
                      java.lang.String ca,
                      java.lang.String cb)
        
     ..

  Purpose
  =======

  LSAMEN  tests if the first N letters of CA are the same as the
  first N letters of CB, regardless of case.
  LSAMEN returns .TRUE. if CA and CB are equivalent except for case
  and .FALSE. otherwise.  LSAMEN also returns .FALSE. if LEN( CA )
  or LEN( CB ) is less than N.

  Arguments
  =========

  N       (input) INTEGER
          The number of characters in CA and CB to be compared.

  CA      (input) CHARACTER*(*)
  CB      (input) CHARACTER*(*)
          CA and CB specify two character strings of length at least N.
          Only the first N characters of each string will be accessed.


 =====================================================================


     .. Local Scalars ..
        Parameters:
        n -
        ca -
        cb -

      • sbdsdc

        public abstract void sbdsdc(java.lang.String uplo,
                            java.lang.String compq,
                            int n,
                            float[] d,
                            float[] e,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            float[] q,
                            int[] iq,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SBDSDC computes the singular value decomposition (SVD) of a real
  N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
  using a divide and conquer method, where S is a diagonal matrix
  with non-negative diagonal elements (the singular values of B), and
  U and VT are orthogonal matrices of left and right singular vectors,

  respectively. SBDSDC can be used to compute all singular values,
  and optionally, singular vectors or singular vectors in compact form.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.  See SLASD3 for details.

  The code currently calls SLASDQ if singular values only are desired.

  However, it can be slightly modified to compute singular values
  using the divide and conquer method.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  B is upper bidiagonal.
          = 'L':  B is lower bidiagonal.

  COMPQ   (input) CHARACTER*1
          Specifies whether singular vectors are to be computed
          as follows:
          = 'N':  Compute singular values only;
          = 'P':  Compute singular values and compute singular
                  vectors in compact form;
          = 'I':  Compute singular values and singular vectors.

  N       (input) INTEGER
          The order of the matrix B.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the elements of E contain the offdiagonal
          elements of the bidiagonal matrix whose SVD is desired.
          On exit, E has been destroyed.

  U       (output) REAL array, dimension (LDU,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, U contains the left singular vectors
             of the bidiagonal matrix.
          For other values of COMPQ, U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1.
          If singular vectors are desired, then LDU >= max( 1, N ).

  VT      (output) REAL array, dimension (LDVT,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, VT' contains the right singular
             vectors of the bidiagonal matrix.
          For other values of COMPQ, VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1.
          If singular vectors are desired, then LDVT >= max( 1, N ).

  Q       (output) REAL array, dimension (LDQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, Q contains all the REAL data in
             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, Q is not referenced.

  IQ      (output) INTEGER array, dimension (LDIQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, IQ contains all INTEGER data in
             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, IQ is not referenced.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK))
          If COMPQ = 'N' then LWORK >= (4 * N).
          If COMPQ = 'P' then LWORK >= (6 * N).
          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

  IWORK   (workspace) INTEGER array, dimension (8*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an singular value.
                The update process of divide and conquer failed.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA
  =====================================================================
  Changed dimension statement in comment describing E from (N) to
  (N-1).  Sven, 17 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        compq -
        n -
        d -
        e -
        u -
        ldu -
        vt -
        ldvt -
        q -
        iq -
        work -
        iwork -
        info -

      • sbdsqr

        public abstract void sbdsqr(java.lang.String uplo,
                            int n,
                            int ncvt,
                            int nru,
                            int ncc,
                            float[] d,
                            float[] e,
                            float[] vt,
                            int ldvt,
                            float[] u,
                            int ldu,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SBDSQR computes the singular values and, optionally, the right and/or
  left singular vectors from the singular value decomposition (SVD) of

  a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
  zero-shift QR algorithm.  The SVD of B has the form
  
     B = Q * S * P**T
  
  where S is the diagonal matrix of singular values, Q is an orthogonal
  matrix of left singular vectors, and P is an orthogonal matrix of
  right singular vectors.  If left singular vectors are requested, this
  subroutine actually returns U*Q instead of Q, and, if right singular

  vectors are requested, this subroutine returns P**T*VT instead of
  P**T, for given real input matrices U and VT.  When U and VT are the

  orthogonal matrices that reduce a general matrix A to bidiagonal
  form:  A = U*B*VT, as computed by SGEBRD, then
 
     A = (U*Q) * S * (P**T*VT)
 
  is the SVD of A.  Optionally, the subroutine may also compute Q**T*C

  for a given real input matrix C.

  See "Computing  Small Singular Values of Bidiagonal Matrices With
  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
  LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
  no. 5, pp. 873-912, Sept 1990) and
  "Accurate singular values and differential qd algorithms," by
  B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
  Department, University of California at Berkeley, July 1992
  for a detailed description of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

  N       (input) INTEGER
          The order of the matrix B.  N >= 0.

  NCVT    (input) INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

  NRU     (input) INTEGER
          The number of rows of the matrix U. NRU >= 0.

  NCC     (input) INTEGER
          The number of columns of the matrix C. NCC >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B.
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

  VT      (input/output) REAL array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * VT.
          Not referenced if NCVT = 0.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

  U       (input/output) REAL array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.
          On exit, U is overwritten by U * Q.
          Not referenced if NRU = 0.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

  C       (input/output) REAL array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * C.
          Not referenced if NCC = 0.

  LDC     (input) INTEGER
          The leading dimension of the array C.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

  WORK    (workspace) REAL array, dimension (2*N)
          if NCVT = NRU = NCC = 0, (max(1, 4*N)) otherwise

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm did not converge; D and E contain the
                elements of a bidiagonal matrix which is orthogonally
                similar to the input matrix B;  if INFO = i, i
                elements of E have not converged to zero.

  Internal Parameters
  ===================

  TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          If it is positive, TOLMUL*EPS is the desired relative
             precision in the computed singular values.
          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
             desired absolute accuracy in the computed singular
             values (corresponds to relative accuracy
             abs(TOLMUL*EPS) in the largest singular value.
          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
             between 10 (for fast convergence) and .1/EPS
             (for there to be some accuracy in the results).
          Default is to lose at either one eighth or 2 of the
             available decimal digits in each computed singular value
             (whichever is smaller).

  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the
          algorithm through its inner loop. The algorithms stops
          (and so fails to converge) if the number of passes
          through the inner loop exceeds MAXITR*N**2.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ncvt -
        nru -
        ncc -
        d -
        e -
        vt -
        ldvt -
        u -
        ldu -
        c -
        Ldc -
        work -
        info -

      • sdisna

        public abstract void sdisna(java.lang.String job,
                            int m,
                            int n,
                            float[] d,
                            float[] sep,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SDISNA computes the reciprocal condition numbers for the eigenvectors
  of a real symmetric or complex Hermitian matrix or for the left or
  right singular vectors of a general m-by-n matrix. The reciprocal
  condition number is the 'gap' between the corresponding eigenvalue or
  singular value and the nearest other one.

  The bound on the error, measured by angle in radians, in the I-th
  computed vector is given by

         SLAMCH( 'E' ) * ( ANORM / SEP( I ) )

  where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed

  to be smaller than SLAMCH( 'E' )*ANORM in order to limit the size of

  the error bound.

  SDISNA may also be used to compute error bounds for eigenvectors of
  the generalized symmetric definite eigenproblem.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies for which problem the reciprocal condition numbers

          should be computed:
          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
          = 'L':  the left singular vectors of a general matrix;
          = 'R':  the right singular vectors of a general matrix.

  M       (input) INTEGER
          The number of rows of the matrix. M >= 0.

  N       (input) INTEGER
          If JOB = 'L' or 'R', the number of columns of the matrix,
          in which case N >= 0. Ignored if JOB = 'E'.

  D       (input) REAL array, dimension (M) if JOB = 'E'
                              dimension (min(M,N)) if JOB = 'L' or 'R'

          The eigenvalues (if JOB = 'E') or singular values (if JOB =
          'L' or 'R') of the matrix, in either increasing or decreasing
          order. If singular values, they must be non-negative.

  SEP     (output) REAL array, dimension (M) if JOB = 'E'
                               dimension (min(M,N)) if JOB = 'L' or 'R'
          The reciprocal condition numbers of the vectors.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        m -
        n -
        d -
        sep -
        info -

      • sgbbrd

        public abstract void sgbbrd(java.lang.String vect,
                            int m,
                            int n,
                            int ncc,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            float[] d,
                            float[] e,
                            float[] q,
                            int ldq,
                            float[] pt,
                            int ldpt,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBBRD reduces a real general m-by-n band matrix A to upper
  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.

  The routine computes B, and optionally forms Q or P', or computes
  Q'*C for a given matrix C.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          Specifies whether or not the matrices Q and P' are to be
          formed.
          = 'N': do not form Q or P';
          = 'Q': form Q only;
          = 'P': form P' only;
          = 'B': form both.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NCC     (input) INTEGER
          The number of columns of the matrix C.  NCC >= 0.

  KL      (input) INTEGER
          The number of subdiagonals of the matrix A. KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals of the matrix A. KU >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the m-by-n band matrix A, stored in rows 1 to
          KL+KU+1. The j-th column of A is stored in the j-th column of
          the array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
          On exit, A is overwritten by values generated during the
          reduction.

  LDAB    (input) INTEGER
          The leading dimension of the array A. LDAB >= KL+KU+1.

  D       (output) REAL array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B.

  E       (output) REAL array, dimension (min(M,N)-1)
          The superdiagonal elements of the bidiagonal matrix B.

  Q       (output) REAL array, dimension (LDQ,M)
          If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
          If VECT = 'N' or 'P', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.

  PT      (output) REAL array, dimension (LDPT,N)
          If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
          If VECT = 'N' or 'Q', the array PT is not referenced.

  LDPT    (input) INTEGER
          The leading dimension of the array PT.
          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.

  C       (input/output) REAL array, dimension (LDC,NCC)
          On entry, an m-by-ncc matrix C.
          On exit, C is overwritten by Q'*C.
          C is not referenced if NCC = 0.

  LDC     (input) INTEGER
          The leading dimension of the array C.
          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.

  WORK    (workspace) REAL array, dimension (2*max(M,N))

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        m -
        n -
        ncc -
        kl -
        ku -
        ab -
        ldab -
        d -
        e -
        q -
        ldq -
        pt -
        ldpt -
        c -
        Ldc -
        work -
        info -

      • sgbcon

        public abstract void sgbcon(java.lang.String norm,
                            int n,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            int[] ipiv,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBCON estimates the reciprocal of the condition number of a real
  general band matrix A, in either the 1-norm or the infinity-norm,
  using the LU factorization computed by SGBTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input) REAL array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by SGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).

  ANORM   (input) REAL
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • sgbequ

        public abstract void sgbequ(int m,
                            int n,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            float[] r,
                            float[] c,
                            org.netlib.util.floatW rowcnd,
                            org.netlib.util.floatW colcnd,
                            org.netlib.util.floatW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBEQU computes row and column scalings intended to equilibrate an
  M-by-N band matrix A and reduce its condition number.  R returns the

  row scale factors and C the column scale factors, chosen to try to
  make the largest element in each row and column of the matrix B with

  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
  number and BIGNUM = largest safe number.  Use of these scaling
  factors is not guaranteed to reduce the condition number of A but
  works well in practice.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input) REAL array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  R       (output) REAL array, dimension (M)
          If INFO = 0, or INFO > M, R contains the row scale factors
          for A.

  C       (output) REAL array, dimension (N)
          If INFO = 0, C contains the column scale factors for A.

  ROWCND  (output) REAL
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.

  COLCND  (output) REAL
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.

  AMAX    (output) REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        info -

      • sgbrfs

        public abstract void sgbrfs(java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] afb,
                            int ldafb,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is banded, and provides
  error bounds and backward error estimates for the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The original band matrix A, stored in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  AFB     (input) REAL array, dimension (LDAFB,N)
          Details of the LU factorization of the band matrix A, as
          computed by SGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.


  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from SGBTRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SGBTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgbsv

        public abstract void sgbsv(int n,
                           int kl,
                           int ku,
                           int nrhs,
                           float[] ab,
                           int ldab,
                           int[] ipiv,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBSV computes the solution to a real system of linear equations
  A * X = B, where A is a band matrix of order N with KL subdiagonals
  and KU superdiagonals, and X and B are N-by-NRHS matrices.

  The LU decomposition with partial pivoting and row interchanges is
  used to factor A as A = L * U, where L is a product of permutation
  and unit lower triangular matrices with KL subdiagonals, and U is
  upper triangular with KL+KU superdiagonals.  The factored form of A
  is then used to solve the system of equations A * X = B.

  Arguments
  =========

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                has been completed, but the factor U is exactly
                singular, and the solution has not been computed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U because of fill-in resulting from the row interchanges.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        ipiv -
        b -
        ldb -
        info -

      • sgbsvx

        public abstract void sgbsvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] afb,
                            int ldafb,
                            int[] ipiv,
                            org.netlib.util.StringW equed,
                            float[] r,
                            float[] c,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBSVX uses the LU factorization to compute the solution to a real
  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

  where A is a band matrix of order N with KL subdiagonals and KU
  superdiagonals, and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed by this subroutine:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
     or diag(C)*B (if TRANS = 'T' or 'C').

  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
     matrix A (after equilibration if FACT = 'E') as
        A = L * U,
     where L is a product of permutation and unit lower triangular
     matrices with KL subdiagonals, and U is upper triangular with
     KL+KU superdiagonals.

  3. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
     that it solves the original system before equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFB and IPIV contain the factored form of
                  A.  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  AB, AFB, and IPIV are not modified.
          = 'N':  The matrix A will be copied to AFB and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFB and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

          If FACT = 'F' and EQUED is not 'N', then A must have been
          equilibrated by the scaling factors in R and/or C.  AB is not
          modified if FACT = 'F' or 'N', or if FACT = 'E' and
          EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  AFB     (input or output) REAL array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry
          contains details of the LU factorization of the band matrix
          A, as computed by SGBTRF.  U is stored as an upper triangular
          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
          and the multipliers used during the factorization are stored

          in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
          the factored form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit
          returns details of the LU factorization of A.

          If FACT = 'E', then AFB is an output argument and on exit
          returns details of the LU factorization of the equilibrated
          matrix A (see the description of AB for the form of the
          equilibrated matrix).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.


  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = L*U
          as computed by SGBTRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the equilibrated matrix A.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  R       (input or output) REAL array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.

  C       (input or output) REAL array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is

          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit,
          if EQUED = 'N', B is not modified;
          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
          to the original system of equations.  Note that A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) REAL array, dimension (3*N)
          On exit, WORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If WORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has been completed, but the factor U is exactly

                       singular, so the solution and error bounds
                       could not be computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the


                       value of RCOND would suggest.
  =====================================================================
  Moved setting of INFO = N+1 so INFO does not subsequently get
  overwritten.  Sven, 17 Mar 05. 
  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        ipiv -
        equed -
        r -
        c -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgbtf2

        public abstract void sgbtf2(int m,
                            int n,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBTF2 computes an LU factorization of a real m-by-n band matrix A
  using partial pivoting with row interchanges.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization

               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U, because of fill-in resulting from the row
  interchanges.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        info -

      • sgbtrf

        public abstract void sgbtrf(int m,
                            int n,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBTRF computes an LU factorization of a real m-by-n band matrix A
  using partial pivoting with row interchanges.

  This is the blocked version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization

               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

    *    *    +    +    +       *    *    *   u14  u25  u36
    *    +    +    +    +       *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked

  + need not be set on entry, but are required by the routine to store

  elements of U because of fill-in resulting from the row interchanges.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        ipiv -
        info -

      • sgbtrs

        public abstract void sgbtrs(java.lang.String trans,
                            int n,
                            int kl,
                            int ku,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGBTRS solves a system of linear equations
     A * X = B  or  A' * X = B
  with a general band matrix A using the LU factorization computed
  by SGBTRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by SGBTRF.  U is stored as an upper triangular band

          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        kl -
        ku -
        nrhs -
        ab -
        ldab -
        ipiv -
        b -
        ldb -
        info -

      • sgebak

        public abstract void sgebak(java.lang.String job,
                            java.lang.String side,
                            int n,
                            int ilo,
                            int ihi,
                            float[] scale,
                            int m,
                            float[] v,
                            int ldv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEBAK forms the right or left eigenvectors of a real general matrix

  by backward transformation on the computed eigenvectors of the
  balanced matrix output by SGEBAL.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the type of backward transformation required:
          = 'N', do nothing, return immediately;
          = 'P', do backward transformation for permutation only;
          = 'S', do backward transformation for scaling only;
          = 'B', do backward transformations for both permutation and
                 scaling.
          JOB must be the same as the argument JOB supplied to SGEBAL.


  SIDE    (input) CHARACTER*1
          = 'R':  V contains right eigenvectors;
          = 'L':  V contains left eigenvectors.

  N       (input) INTEGER
          The number of rows of the matrix V.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          The integers ILO and IHI determined by SGEBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  SCALE   (input) REAL array, dimension (N)
          Details of the permutation and scaling factors, as returned
          by SGEBAL.

  M       (input) INTEGER
          The number of columns of the matrix V.  M >= 0.

  V       (input/output) REAL array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by SHSEIN or STREVC.
          On exit, V is overwritten by the transformed eigenvectors.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        side -
        n -
        ilo -
        ihi -
        scale -
        m -
        v -
        ldv -
        info -

      • sgebal

        public abstract void sgebal(java.lang.String job,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            float[] scale,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEBAL balances a general real matrix A.  This involves, first,
  permuting A by a similarity transformation to isolate eigenvalues
  in the first 1 to ILO-1 and last IHI+1 to N elements on the
  diagonal; and second, applying a diagonal similarity transformation
  to rows and columns ILO to IHI to make the rows and columns as
  close in norm as possible.  Both steps are optional.

  Balancing may reduce the 1-norm of the matrix, and improve the
  accuracy of the computed eigenvalues and/or eigenvectors.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the operations to be performed on A:
          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0
                  for i = 1,...,N;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the input matrix A.
          On exit,  A is overwritten by the balanced matrix.
          If JOB = 'N', A is not referenced.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are set to integers such that on exit
          A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
          If JOB = 'N' or 'S', ILO = 1 and IHI = N.

  SCALE   (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied to
          A.  If P(j) is the index of the row and column interchanged
          with row and column j and D(j) is the scaling factor
          applied to row and column j, then
          SCALE(j) = P(j)    for j = 1,...,ILO-1
                   = D(j)    for j = ILO,...,IHI
                   = P(j)    for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The permutations consist of row and column interchanges which put
  the matrix in the form

             ( T1   X   Y  )
     P A P = (  0   B   Z  )
             (  0   0   T2 )

  where T1 and T2 are upper triangular matrices whose eigenvalues lie
  along the diagonal.  The column indices ILO and IHI mark the starting
  and ending columns of the submatrix B. Balancing consists of applying
  a diagonal similarity transformation inv(D) * B * D to make the
  1-norms of each row of B and its corresponding column nearly equal.
  The output matrix is

     ( T1     X*D          Y    )
     (  0  inv(D)*B*D  inv(D)*Z ).
     (  0      0           T2   )

  Information about the permutations P and the diagonal matrix D is
  returned in the vector SCALE.

  This subroutine is based on the EISPACK routine BALANC.

  Modified by Tzu-Yi Chen, Computer Science Division, University of
    California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        lda -
        ilo -
        ihi -
        scale -
        info -

      • sgebd2

        public abstract void sgebd2(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] d,
                            float[] e,
                            float[] tauq,
                            float[] taup,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEBD2 reduces a real general m by n matrix A to upper or lower
  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.

  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns in the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit,
          if m >= n, the diagonal and the first superdiagonal are
            overwritten with the upper bidiagonal matrix B; the
            elements below the diagonal, with the array TAUQ, represent
            the orthogonal matrix Q as a product of elementary
            reflectors, and the elements above the first superdiagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors;
          if m < n, the diagonal and the first subdiagonal are
            overwritten with the lower bidiagonal matrix B; the
            elements below the first subdiagonal, with the array TAUQ,

            represent the orthogonal matrix Q as a product of
            elementary reflectors, and the elements above the diagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) REAL array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).

  E       (output) REAL array, dimension (min(M,N)-1)
          The off-diagonal elements of the bidiagonal matrix B:
          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

  TAUQ    (output) REAL array dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  WORK    (workspace) REAL array, dimension (max(M,N))

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  The contents of A on exit are illustrated by the following examples:


  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi
  denotes an element of the vector defining H(i), and ui an element of

  the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        work -
        info -

      • sgebrd

        public abstract void sgebrd(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] d,
                            float[] e,
                            float[] tauq,
                            float[] taup,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEBRD reduces a general real M-by-N matrix A to upper or lower
  bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.


  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns in the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N general matrix to be reduced.
          On exit,
          if m >= n, the diagonal and the first superdiagonal are
            overwritten with the upper bidiagonal matrix B; the
            elements below the diagonal, with the array TAUQ, represent
            the orthogonal matrix Q as a product of elementary
            reflectors, and the elements above the first superdiagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors;
          if m < n, the diagonal and the first subdiagonal are
            overwritten with the lower bidiagonal matrix B; the
            elements below the first subdiagonal, with the array TAUQ,

            represent the orthogonal matrix Q as a product of
            elementary reflectors, and the elements above the diagonal,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) REAL array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).

  E       (output) REAL array, dimension (min(M,N)-1)
          The off-diagonal elements of the bidiagonal matrix B:
          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

  TAUQ    (output) REAL array dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,M,N).
          For optimum performance LWORK >= (M+N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit 
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors;
  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
  tauq is stored in TAUQ(i) and taup in TAUP(i).

  The contents of A on exit are illustrated by the following examples:


  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi
  denotes an element of the vector defining H(i), and ui an element of

  the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        work -
        lwork -
        info -

      • sgecon

        public abstract void sgecon(java.lang.String norm,
                            int n,
                            float[] a,
                            int lda,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGECON estimates the reciprocal of the condition number of a general

  real matrix A, in either the 1-norm or the infinity-norm, using
  the LU factorization computed by SGETRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The factors L and U from the factorization A = P*L*U
          as computed by SGETRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ANORM   (input) REAL
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) REAL array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        a -
        lda -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • sgeequ

        public abstract void sgeequ(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] r,
                            float[] c,
                            org.netlib.util.floatW rowcnd,
                            org.netlib.util.floatW colcnd,
                            org.netlib.util.floatW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEEQU computes row and column scalings intended to equilibrate an
  M-by-N matrix A and reduce its condition number.  R returns the row
  scale factors and C the column scale factors, chosen to try to make
  the largest element in each row and column of the matrix B with
  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
  number and BIGNUM = largest safe number.  Use of these scaling
  factors is not guaranteed to reduce the condition number of A but
  works well in practice.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The M-by-N matrix whose equilibration factors are
          to be computed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  R       (output) REAL array, dimension (M)
          If INFO = 0 or INFO > M, R contains the row scale factors
          for A.

  C       (output) REAL array, dimension (N)
          If INFO = 0,  C contains the column scale factors for A.

  ROWCND  (output) REAL
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.

  COLCND  (output) REAL
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.

  AMAX    (output) REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i,  and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        info -

      • sgees

        public void sgees(java.lang.String jobvs,
                  java.lang.String sort,
                  java.lang.Object select,
                  int n,
                  float[] a,
                  int lda,
                  org.netlib.util.intW sdim,
                  float[] wr,
                  float[] wi,
                  float[] vs,
                  int ldvs,
                  float[] work,
                  int lwork,
                  boolean[] bwork,
                  org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEES computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues, the real Schur form T, and, optionally, the matrix of
  Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).


  Optionally, it also orders the eigenvalues on the diagonal of the
  real Schur form so that selected eigenvalues are at the top left.
  The leading columns of Z then form an orthonormal basis for the
  invariant subspace corresponding to the selected eigenvalues.

  A matrix is in real Schur form if it is upper quasi-triangular with
  1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
  form
          [  a  b  ]
          [  c  a  ]

  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

  Arguments
  =========

  JOBVS   (input) CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

  SELECT  (external procedure) LOGICAL FUNCTION of two REAL arguments
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
          conjugate pair of eigenvalues is selected, then both complex

          eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO is set to N+2 (see INFO below).

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten by its real Schur form T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

  WR      (output) REAL array, dimension (N)
  WI      (output) REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues in the same order
          that they appear on the diagonal of the output Schur form T.

          Complex conjugate pairs of eigenvalues will appear
          consecutively with the eigenvalue having the positive
          imaginary part first.

  VS      (output) REAL array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur

          vectors.
          If JOBVS = 'N', VS is not referenced.

  LDVS    (input) INTEGER
          The leading dimension of the array VS.  LDVS >= 1; if
          JOBVS = 'V', LDVS >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the matrix which reduces A

                   to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem

                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvs -
        sort -
        select -
        n -
        a -
        lda -
        sdim -
        wr -
        wi -
        vs -
        ldvs -
        work -
        lwork -
        bwork -
        info -

      • sgeesx

        public void sgeesx(java.lang.String jobvs,
                   java.lang.String sort,
                   java.lang.Object select,
                   java.lang.String sense,
                   int n,
                   float[] a,
                   int lda,
                   org.netlib.util.intW sdim,
                   float[] wr,
                   float[] wi,
                   float[] vs,
                   int ldvs,
                   org.netlib.util.floatW rconde,
                   org.netlib.util.floatW rcondv,
                   float[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   boolean[] bwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEESX computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues, the real Schur form T, and, optionally, the matrix of
  Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).


  Optionally, it also orders the eigenvalues on the diagonal of the
  real Schur form so that selected eigenvalues are at the top left;
  computes a reciprocal condition number for the average of the
  selected eigenvalues (RCONDE); and computes a reciprocal condition
  number for the right invariant subspace corresponding to the
  selected eigenvalues (RCONDV).  The leading columns of Z form an
  orthonormal basis for this invariant subspace.

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
  these quantities are called s and sep respectively).

  A real matrix is in real Schur form if it is upper quasi-triangular
  with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in

  the form
            [  a  b  ]
            [  c  a  ]

  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

  Arguments
  =========

  JOBVS   (input) CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

  SELECT  (external procedure) LOGICAL FUNCTION of two REAL arguments
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a
          complex conjugate pair of eigenvalues is selected, then both

          are.  Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO may be set to N+3 (see INFO below).

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for average of selected eigenvalues only;
          = 'V': Computed for selected right invariant subspace only;
          = 'B': Computed for both.
          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the N-by-N matrix A.
          On exit, A is overwritten by its real Schur form T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

  WR      (output) REAL array, dimension (N)
  WI      (output) REAL array, dimension (N)
          WR and WI contain the real and imaginary parts, respectively,
          of the computed eigenvalues, in the same order that they
          appear on the diagonal of the output Schur form T.  Complex
          conjugate pairs of eigenvalues appear consecutively with the

          eigenvalue having the positive imaginary part first.

  VS      (output) REAL array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur

          vectors.
          If JOBVS = 'N', VS is not referenced.

  LDVS    (input) INTEGER
          The leading dimension of the array VS.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N.

  RCONDE  (output) REAL
          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
          condition number for the average of the selected eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

  RCONDV  (output) REAL
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace.
          Not referenced if SENSE = 'N' or 'E'.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).
          Also, if SENSE = 'E' or 'V' or 'B',
          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
          selected eigenvalues computed by this routine.  Note that
          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or

          'B' this may not be large enough.
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates upper bounds on the optimal sizes of the
          arrays WORK and IWORK, returns these values as the first
          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).

          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is

          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
          may not be large enough.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates upper bounds on the optimal sizes of

          the arrays WORK and IWORK, returns these values as the first

          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the transformation which
                   reduces A to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem

                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvs -
        sort -
        select -
        sense -
        n -
        a -
        lda -
        sdim -
        wr -
        wi -
        vs -
        ldvs -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        liwork -
        bwork -
        info -

      • sgeev

        public abstract void sgeev(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           float[] a,
                           int lda,
                           float[] wr,
                           float[] wi,
                           float[] vl,
                           int ldvl,
                           float[] vr,
                           int ldvr,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEEV computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues and, optionally, the left and/or right eigenvectors.

  The right eigenvector v(j) of A satisfies
                   A * v(j) = lambda(j) * v(j)
  where lambda(j) is its eigenvalue.
  The left eigenvector u(j) of A satisfies
                u(j)**H * A = lambda(j) * u(j)**H
  where u(j)**H denotes the conjugate transpose of u(j).

  The computed eigenvectors are normalized to have Euclidean norm
  equal to 1 and largest component real.

  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.

  JOBVR   (input) CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  WR      (output) REAL array, dimension (N)
  WI      (output) REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1; if
          JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N), and
          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good
          performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors have been computed;
                elements i+1:N of WR and WI contain eigenvalues which
                have converged.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • sgeevx

        public abstract void sgeevx(java.lang.String balanc,
                            java.lang.String jobvl,
                            java.lang.String jobvr,
                            java.lang.String sense,
                            int n,
                            float[] a,
                            int lda,
                            float[] wr,
                            float[] wi,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            float[] scale,
                            org.netlib.util.floatW abnrm,
                            float[] rconde,
                            float[] rcondv,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
  eigenvalues and, optionally, the left and/or right eigenvectors.

  Optionally also, it computes a balancing transformation to improve
  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
  SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
  (RCONDE), and reciprocal condition numbers for the right
  eigenvectors (RCONDV).

  The right eigenvector v(j) of A satisfies
                   A * v(j) = lambda(j) * v(j)
  where lambda(j) is its eigenvalue.
  The left eigenvector u(j) of A satisfies
                u(j)**H * A = lambda(j) * u(j)**H
  where u(j)**H denotes the conjugate transpose of u(j).

  The computed eigenvectors are normalized to have Euclidean norm
  equal to 1 and largest component real.

  Balancing a matrix means permuting the rows and columns to make it
  more nearly upper triangular, and applying a diagonal similarity
  transformation D * A * D**(-1), where D is a diagonal matrix, to
  make its rows and columns closer in norm and the condition numbers
  of its eigenvalues and eigenvectors smaller.  The computed
  reciprocal condition numbers correspond to the balanced matrix.
  Permuting rows and columns will not change the condition numbers
  (in exact arithmetic) but diagonal scaling will.  For further
  explanation of balancing, see section 4.10.2 of the LAPACK
  Users' Guide.

  Arguments
  =========

  BALANC  (input) CHARACTER*1
          Indicates how the input matrix should be diagonally scaled
          and/or permuted to improve the conditioning of its
          eigenvalues.
          = 'N': Do not diagonally scale or permute;
          = 'P': Perform permutations to make the matrix more nearly
                 upper triangular. Do not diagonally scale;
          = 'S': Diagonally scale the matrix, i.e. replace A by
                 D*A*D**(-1), where D is a diagonal matrix chosen
                 to make the rows and columns of A more equal in
                 norm. Do not permute;
          = 'B': Both diagonally scale and permute A.

          Computed reciprocal condition numbers will be for the matrix

          after balancing and/or permuting. Permuting does not change
          condition numbers (in exact arithmetic), but balancing does.


  JOBVL   (input) CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVL must = 'V'.

  JOBVR   (input) CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVR must = 'V'.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for eigenvalues only;
          = 'V': Computed for right eigenvectors only;
          = 'B': Computed for eigenvalues and right eigenvectors.

          If SENSE = 'E' or 'B', both left and right eigenvectors
          must also be computed (JOBVL = 'V' and JOBVR = 'V').

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.  If JOBVL = 'V' or
          JOBVR = 'V', A contains the real Schur form of the balanced
          version of the input matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  WR      (output) REAL array, dimension (N)
  WI      (output) REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues will appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          JOBVR = 'V', LDVR >= N.

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are integer values determined when A was
          balanced.  The balanced A(i,j) = 0 if I > J and 
          J = 1,...,ILO-1 or I = IHI+1,...,N.

  SCALE   (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          when balancing A.  If P(j) is the index of the row and column
          interchanged with row and column j, and D(j) is the scaling
          factor applied to row and column j, then
          SCALE(J) = P(J),    for J = 1,...,ILO-1
                   = D(J),    for J = ILO,...,IHI
                   = P(J)     for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  ABNRM   (output) REAL
          The one-norm of the balanced matrix (the maximum
          of the sum of absolute values of elements of any column).

  RCONDE  (output) REAL array, dimension (N)
          RCONDE(j) is the reciprocal condition number of the j-th
          eigenvalue.

  RCONDV  (output) REAL array, dimension (N)
          RCONDV(j) is the reciprocal condition number of the j-th
          right eigenvector.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.   If SENSE = 'N' or 'E',
          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (2*N-2)
          If SENSE = 'N' or 'E', not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors or condition numbers
                have been computed; elements 1:ILO-1 and i+1:N of WR
                and WI contain eigenvalues which have converged.

  =====================================================================

     .. Parameters ..
        Parameters:
        balanc -
        jobvl -
        jobvr -
        sense -
        n -
        a -
        lda -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        ilo -
        ihi -
        scale -
        abnrm -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        info -

      • sgegs

        public abstract void sgegs(java.lang.String jobvsl,
                           java.lang.String jobvsr,
                           int n,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float[] alphar,
                           float[] alphai,
                           float[] beta,
                           float[] vsl,
                           int ldvsl,
                           float[] vsr,
                           int ldvsr,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine SGGES.

  SGEGS computes the eigenvalues, real Schur form, and, optionally,
  left and or/right Schur vectors of a real matrix pair (A,B).
  Given two square matrices A and B, the generalized real Schur
  factorization has the form
  
    A = Q*S*Z**T,  B = Q*T*Z**T

  where Q and Z are orthogonal matrices, T is upper triangular, and S
  is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
  blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
  of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
  and the columns of Z are the right Schur vectors.
  
  If only the eigenvalues of (A,B) are needed, the driver routine
  SGEGV should be used instead.  See SGEGV for a description of the
  eigenvalues of the generalized nonsymmetric eigenvalue problem
  (GNEP).

  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors (returned in VSL).

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors (returned in VSR).

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the matrix A.
          On exit, the upper quasi-triangular matrix S from the
          generalized real Schur factorization.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the matrix B.
          On exit, the upper triangular matrix T from the generalized
          real Schur factorization.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP.

  ALPHAI  (output) REAL array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and (j+1)-st
          eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) REAL array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  VSL     (output) REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', the matrix of left Schur vectors Q.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', the matrix of right Schur vectors Z.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,4*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR
          The optimal LWORK is  2*N + N*(NB+1).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from SGGBAL
                =N+2: error return from SGEQRF
                =N+3: error return from SORMQR
                =N+4: error return from SORGQR
                =N+5: error return from SGGHRD
                =N+6: error return from SHGEQZ (other than failed
                                                iteration)
                =N+7: error return from SGGBAK (computing VSL)
                =N+8: error return from SGGBAK (computing VSR)
                =N+9: error return from SLASCL (various places)

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        work -
        lwork -
        info -

      • sgegv

        public abstract void sgegv(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float[] alphar,
                           float[] alphai,
                           float[] beta,
                           float[] vl,
                           int ldvl,
                           float[] vr,
                           int ldvr,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine SGGEV.

  SGEGV computes the eigenvalues and, optionally, the left and/or right
  eigenvectors of a real matrix pair (A,B).
  Given two square matrices A and B,
  the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
  eigenvalues lambda and corresponding (non-zero) eigenvectors x such
  that

     A*x = lambda*B*x.

  An alternate form is to find the eigenvalues mu and corresponding
  eigenvectors y such that

     mu*A*y = B*y.

  These two forms are equivalent with mu = 1/lambda and x = y if
  neither lambda nor mu is zero.  In order to deal with the case that
  lambda or mu is zero or small, two values alpha and beta are returned
  for each eigenvalue, such that lambda = alpha/beta and
  mu = beta/alpha.

  The vectors x and y in the above equations are right eigenvectors of

  the matrix pair (A,B).  Vectors u and v satisfying

     u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

  are left eigenvectors of (A,B).

  Note: this routine performs "full balancing" on A and B -- see
  "Further Details", below.

  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors (returned
                  in VL).

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors (returned

                  in VR).

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the matrix A.
          If JOBVL = 'V' or JOBVR = 'V', then on exit A
          contains the real Schur form of A from the generalized Schur

          factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only the diagonal
          blocks from the Schur form will be correct.  See SGGHRD and
          SHGEQZ for details.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the matrix B.
          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
          upper triangular matrix obtained from B in the generalized
          Schur factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only those elements of
          B corresponding to the diagonal blocks from the Schur form of
          A will be correct.  See SGGHRD and SHGEQZ for details.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue of
          GNEP.

  ALPHAI  (output) REAL array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) REAL array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored
          in the columns of VL, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then u(j) = VL(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
             u(j) = VL(:,j) + i*VL(:,j+1)
          and
            u(j+1) = VL(:,j) - i*VL(:,j+1).

          Each eigenvector is scaled so that its largest component has

          abs(real part) + abs(imag. part) = 1, except for eigenvectors
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors x(j) are stored
          in the columns of VR, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then x(j) = VR(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
            x(j) = VR(:,j) + i*VR(:,j+1)
          and
            x(j+1) = VR(:,j) - i*VR(:,j+1).

          Each eigenvector is scaled so that its largest component has

          abs(real part) + abs(imag. part) = 1, except for eigenvalues

          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;

          The optimal LWORK is:
              2*N + MAX( 6*N, N*(NB+1) ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from SGGBAL
                =N+2: error return from SGEQRF
                =N+3: error return from SORMQR
                =N+4: error return from SORGQR
                =N+5: error return from SGGHRD
                =N+6: error return from SHGEQZ (other than failed
                                                iteration)
                =N+7: error return from STGEVC
                =N+8: error return from SGGBAK (computing VL)
                =N+9: error return from SGGBAK (computing VR)
                =N+10: error return from SLASCL (various calls)

  Further Details
  ===============

  Balancing
  ---------

  This driver calls SGGBAL to both permute and scale rows and columns
  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
  and PL*B*R will be upper triangular except for the diagonal blocks
  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
  possible.  The diagonal scaling matrices DL and DR are chosen so
  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
  one (except for the elements that start out zero.)

  After the eigenvalues and eigenvectors of the balanced matrices
  have been computed, SGGBAK transforms the eigenvectors back to what
  they would have been (in perfect arithmetic) if they had not been
  balanced.

  Contents of A and B on Exit
  -------- -- - --- - -- ----

  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
  both), then on exit the arrays A and B will contain the real Schur
  form[*] of the "balanced" versions of A and B.  If no eigenvectors
  are computed, then only the diagonal blocks will be correct.

  [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
      by Golub & van Loan, pub. by Johns Hopkins U. Press.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • sgehd2

        public abstract void sgehd2(int n,
                            int ilo,
                            int ihi,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEHD2 reduces a real general matrix A to upper Hessenberg form H by

  an orthogonal similarity transformation:  Q' * A * Q = H .

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          It is assumed that A is already upper triangular in rows
          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
          set by a previous call to SGEBAL; otherwise they should be
          set to 1 and N respectively. See Further Details.
          1 <= ILO <= IHI <= max(1,N).

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the n by n general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          elements below the first subdiagonal, with the array TAU,
          represent the orthogonal matrix Q as a product of elementary

          reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of (ihi-ilo) elementary
  reflectors

     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
  exit in A(i+2:ihi,i), and tau in TAU(i).

  The contents of A are illustrated by the following example, with
  n = 7, ilo = 2 and ihi = 6:

  on entry,                        on exit,

  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
  (                         a )    (                          a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        info -

      • sgehrd

        public abstract void sgehrd(int n,
                            int ilo,
                            int ihi,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEHRD reduces a real general matrix A to upper Hessenberg form H by

  an orthogonal similarity transformation:  Q' * A * Q = H .

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          It is assumed that A is already upper triangular in rows
          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
          set by a previous call to SGEBAL; otherwise they should be
          set to 1 and N respectively. See Further Details.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          elements below the first subdiagonal, with the array TAU,
          represent the orthogonal matrix Q as a product of elementary

          reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
          zero.

  WORK    (workspace/output) REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of (ihi-ilo) elementary
  reflectors

     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
  exit in A(i+2:ihi,i), and tau in TAU(i).

  The contents of A are illustrated by the following example, with
  n = 7, ilo = 2 and ihi = 6:

  on entry,                        on exit,

  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
  (                         a )    (                          a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  This file is a slight modification of LAPACK-3.0's SGEHRD
  subroutine incorporating improvements proposed by Quintana-Orti and
  Van de Geijn (2005). 

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sgelq2

        public abstract void sgelq2(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELQ2 computes an LQ factorization of a real m by n matrix A:
  A = L * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, the elements on and below the diagonal of the array

          contain the m by min(m,n) lower trapezoidal matrix L (L is
          lower triangular if m <= n); the elements above the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) REAL array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

  and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • sgelqf

        public abstract void sgelqf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELQF computes an LQ factorization of a real M-by-N matrix A:
  A = L * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and below the diagonal of the array

          contain the m-by-min(m,n) lower trapezoidal matrix L (L is
          lower triangular if m <= n); the elements above the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

  and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sgels

        public abstract void sgels(java.lang.String trans,
                           int m,
                           int n,
                           int nrhs,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELS solves overdetermined or underdetermined real linear systems
  involving an M-by-N matrix A, or its transpose, using a QR or LQ
  factorization of A.  It is assumed that A has full rank.

  The following options are provided: 

  1. If TRANS = 'N' and m >= n:  find the least squares solution of
     an overdetermined system, i.e., solve the least squares problem
                  minimize || B - A*X ||.

  2. If TRANS = 'N' and m < n:  find the minimum norm solution of
     an underdetermined system A * X = B.

  3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
     an undetermined system A**T * X = B.

  4. If TRANS = 'T' and m < n:  find the least squares solution of
     an overdetermined system, i.e., solve the least squares problem
                  minimize || B - A**T * X ||.

  Several right hand side vectors b and solution vectors x can be 
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution 
  matrix X.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N': the linear system involves A;
          = 'T': the linear system involves A**T. 

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of the matrices B and X. NRHS >=0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
            if M >= N, A is overwritten by details of its QR
                       factorization as returned by SGEQRF;
            if M <  N, A is overwritten by details of its LQ
                       factorization as returned by SGELQF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the matrix B of right hand side vectors, stored
          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
          if TRANS = 'T'.  
          On exit, if INFO = 0, B is overwritten by the solution
          vectors, stored columnwise:
          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
          squares solution vectors; the residual sum of squares for the
          solution in each column is given by the sum of squares of
          elements N+1 to M in that column;
          if TRANS = 'N' and m < n, rows 1 to N of B contain the
          minimum norm solution vectors;
          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
          minimum norm solution vectors;
          if TRANS = 'T' and m < n, rows 1 to M of B contain the
          least squares solution vectors; the residual sum of squares
          for the solution in each column is given by the sum of
          squares of elements M+1 to N in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= MAX(1,M,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= max( 1, MN + max( MN, NRHS ) ).
          For optimal performance,
          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
          where MN = min(M,N) and NB is the optimum block size.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO =  i, the i-th diagonal element of the
                triangular factor of A is zero, so that A does not have
                full rank; the least squares solution could not be
                computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        work -
        lwork -
        info -

      • sgelsd

        public abstract void sgelsd(int m,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] s,
                            float rcond,
                            org.netlib.util.intW rank,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELSD computes the minimum-norm solution to a real linear least
  squares problem:
      minimize 2-norm(| b - A*x |)
  using the singular value decomposition (SVD) of A. A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The problem is solved in three steps:
  (1) Reduce the coefficient matrix A to bidiagonal form with
      Householder transformations, reducing the original problem
      into a "bidiagonal least squares problem" (BLS)
  (2) Solve the BLS using a divide and conquer approach.
  (3) Apply back all the Householder tranformations to solve
      the original least squares problem.

  The effective rank of A is determined by treating as zero those
  singular values which are less than RCOND times the largest singular

  value.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of A. M >= 0.

  N       (input) INTEGER
          The number of columns of A. N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X. NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, B is overwritten by the N-by-NRHS solution
          matrix X.  If m >= n and RANK = n, the residual
          sum-of-squares for the solution in the i-th column is given
          by the sum of squares of elements n+1:m in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,max(M,N)).

  S       (output) REAL array, dimension (min(M,N))
          The singular values of A in decreasing order.
          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

  RCOND   (input) REAL
          RCOND is used to determine the effective rank of A.
          Singular values S(i) <= RCOND*S(1) are treated as zero.
          If RCOND < 0, machine precision is used instead.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the number of singular values

          which are greater than RCOND*S(1).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK must be at least 1.
          The exact minimum amount of workspace needed depends on M,
          N and NRHS. As long as LWORK is at least
              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
          if M is greater than or equal to N or
              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
          if M is less than N, the code will execute correctly.
          SMLSIZ is returned by ILAENV and is equal to the maximum
          size of the subproblems at the bottom of the computation
          tree (usually about 25), and
             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the array WORK and the
          minimum size of the array IWORK, and returns these values as

          the first entries of the WORK and IWORK arrays, and no error

          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
          where MINMN = MIN( M,N ).
          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the algorithm for computing the SVD failed to converge;
                if INFO = i, i off-diagonal elements of an intermediate
                bidiagonal form did not converge to zero.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        s -
        rcond -
        rank -
        work -
        lwork -
        iwork -
        info -

      • sgelss

        public abstract void sgelss(int m,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] s,
                            float rcond,
                            org.netlib.util.intW rank,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELSS computes the minimum norm solution to a real linear least
  squares problem:

  Minimize 2-norm(| b - A*x |).

  using the singular value decomposition (SVD) of A. A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix

  X.

  The effective rank of A is determined by treating as zero those
  singular values which are less than RCOND times the largest singular

  value.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X. NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the first min(m,n) rows of A are overwritten with
          its right singular vectors, stored rowwise.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, B is overwritten by the N-by-NRHS solution
          matrix X.  If m >= n and RANK = n, the residual
          sum-of-squares for the solution in the i-th column is given
          by the sum of squares of elements n+1:m in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,max(M,N)).

  S       (output) REAL array, dimension (min(M,N))
          The singular values of A in decreasing order.
          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

  RCOND   (input) REAL
          RCOND is used to determine the effective rank of A.
          Singular values S(i) <= RCOND*S(1) are treated as zero.
          If RCOND < 0, machine precision is used instead.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the number of singular values

          which are greater than RCOND*S(1).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 1, and also:
          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the algorithm for computing the SVD failed to converge;
                if INFO = i, i off-diagonal elements of an intermediate
                bidiagonal form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        s -
        rcond -
        rank -
        work -
        lwork -
        info -

      • sgelsx

        public abstract void sgelsx(int m,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            int[] jpvt,
                            float rcond,
                            org.netlib.util.intW rank,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine SGELSY.

  SGELSX computes the minimum-norm solution to a real linear least
  squares problem:
      minimize || A * X - B ||
  using a complete orthogonal factorization of A.  A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be 
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The routine first computes a QR factorization with column pivoting:
      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]
  with R11 defined as the largest leading submatrix whose estimated
  condition number is less than 1/RCOND.  The order of R11, RANK,
  is the effective rank of A.

  Then, R22 is considered to be negligible, and R12 is annihilated
  by orthogonal transformations from the right, arriving at the
  complete orthogonal factorization:
     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]
  The minimum-norm solution is then
     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]
  where Q1 consists of the first RANK columns of Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of matrices B and X. NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been overwritten by details of its
          complete orthogonal factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, the N-by-NRHS solution matrix X.
          If m >= n and RANK = n, the residual sum-of-squares for
          the solution in the i-th column is given by the sum of
          squares of elements N+1:M in that column.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,M,N).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
          initial column, otherwise it is a free column.  Before
          the QR factorization of A, all initial columns are
          permuted to the leading positions; only the remaining
          free columns are moved as a result of column pivoting
          during the factorization.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  RCOND   (input) REAL
          RCOND is used to determine the effective rank of A, which
          is defined as the order of the largest leading triangular
          submatrix R11 in the QR factorization with pivoting of A,
          whose estimated condition number < 1/RCOND.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the order of the submatrix
          R11.  This is the same as the order of the submatrix T11
          in the complete orthogonal factorization of A.

  WORK    (workspace) REAL array, dimension
                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        jpvt -
        rcond -
        rank -
        work -
        info -

      • sgelsy

        public abstract void sgelsy(int m,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            int[] jpvt,
                            float rcond,
                            org.netlib.util.intW rank,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGELSY computes the minimum-norm solution to a real linear least
  squares problem:
      minimize || A * X - B ||
  using a complete orthogonal factorization of A.  A is an M-by-N
  matrix which may be rank-deficient.

  Several right hand side vectors b and solution vectors x can be
  handled in a single call; they are stored as the columns of the
  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
  matrix X.

  The routine first computes a QR factorization with column pivoting:
      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]
  with R11 defined as the largest leading submatrix whose estimated
  condition number is less than 1/RCOND.  The order of R11, RANK,
  is the effective rank of A.

  Then, R22 is considered to be negligible, and R12 is annihilated
  by orthogonal transformations from the right, arriving at the
  complete orthogonal factorization:
     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]
  The minimum-norm solution is then
     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]
  where Q1 consists of the first RANK columns of Q.

  This routine is basically identical to the original xGELSX except
  three differences:
    o The call to the subroutine xGEQPF has been substituted by the
      the call to the subroutine xGEQP3. This subroutine is a Blas-3
      version of the QR factorization with column pivoting.
    o Matrix B (the right hand side) is updated with Blas-3.
    o The permutation of matrix B (the right hand side) is faster and
      more simple.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of
          columns of matrices B and X. NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A has been overwritten by details of its
          complete orthogonal factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the M-by-NRHS right hand side matrix B.
          On exit, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,M,N).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of AP, otherwise column i is a free column.
          On exit, if JPVT(i) = k, then the i-th column of AP
          was the k-th column of A.

  RCOND   (input) REAL
          RCOND is used to determine the effective rank of A, which
          is defined as the order of the largest leading triangular
          submatrix R11 in the QR factorization with pivoting of A,
          whose estimated condition number < 1/RCOND.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the order of the submatrix
          R11.  This is the same as the order of the submatrix T11
          in the complete orthogonal factorization of A.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          The unblocked strategy requires that:
             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
          where MN = min( M, N ).
          The block algorithm requires that:
             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
          where NB is an upper bound on the blocksize returned
          by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,
          and SORMRZ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: If INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        jpvt -
        rcond -
        rank -
        work -
        lwork -
        info -

      • sgeql2

        public abstract void sgeql2(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEQL2 computes a QL factorization of a real m by n matrix A:
  A = Q * L.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, if m >= n, the lower triangle of the subarray
          A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
          if m <= n, the elements on and below the (n-m)-th
          superdiagonal contain the m by n lower trapezoidal matrix L;

          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of elementary reflectors
          (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

  A(1:m-k+i-1,n-k+i), and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • sgeqlf

        public abstract void sgeqlf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEQLF computes a QL factorization of a real M-by-N matrix A:
  A = Q * L.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if m >= n, the lower triangle of the subarray
          A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
          if m <= n, the elements on and below the (n-m)-th
          superdiagonal contain the M-by-N lower trapezoidal matrix L;

          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of elementary reflectors
          (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(k) . . . H(2) H(1), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

  A(1:m-k+i-1,n-k+i), and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sgeqp3

        public abstract void sgeqp3(int m,
                            int n,
                            float[] a,
                            int lda,
                            int[] jpvt,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEQP3 computes a QR factorization with column pivoting of a
  matrix A:  A*P = Q*R  using Level 3 BLAS.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of the array contains the
          min(M,N)-by-N upper trapezoidal matrix R; the elements below

          the diagonal, together with the array TAU, represent the
          orthogonal matrix Q as a product of min(M,N) elementary
          reflectors.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(J).ne.0, the J-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(J)=0,
          the J-th column of A is a free column.
          On exit, if JPVT(J)=K, then the J-th column of A*P was the
          the K-th column of A.

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO=0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 3*N+1.
          For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real/complex scalar, and v is a real/complex vector
  with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
  A(i+1:m,i), and tau in TAU(i).

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        jpvt -
        tau -
        work -
        lwork -
        info -

      • sgeqpf

        public abstract void sgeqpf(int m,
                            int n,
                            float[] a,
                            int lda,
                            int[] jpvt,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine SGEQP3.

  SGEQPF computes a QR factorization with column pivoting of a
  real M-by-N matrix A: A*P = Q*R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of the array contains the
          min(M,N)-by-N upper triangular matrix R; the elements
          below the diagonal, together with the array TAU,
          represent the orthogonal matrix Q as a product of
          min(m,n) elementary reflectors.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  WORK    (workspace) REAL array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(n)

  Each H(i) has the form

     H = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).


  The matrix P is represented in jpvt as follows: If
     jpvt(j) = i
  then the jth column of P is the ith canonical unit vector.

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        jpvt -
        tau -
        work -
        info -

      • sgeqr2

        public abstract void sgeqr2(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEQR2 computes a QR factorization of a real m by n matrix A:
  A = Q * R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(m,n) by n upper trapezoidal matrix R (R is
          upper triangular if m >= n); the elements below the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of elementary reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

  and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • sgeqrf

        public abstract void sgeqrf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGEQRF computes a QR factorization of a real M-by-N matrix A:
  A = Q * R.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
          upper triangular if m >= n); the elements below the diagonal,
          with the array TAU, represent the orthogonal matrix Q as a
          product of min(m,n) elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is 
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

  and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sgerfs

        public abstract void sgerfs(java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGERFS improves the computed solution to a system of linear
  equations and provides error bounds and backward error estimates for

  the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The original N-by-N matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) REAL array, dimension (LDAF,N)
          The factors L and U from the factorization A = P*L*U
          as computed by SGETRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from SGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SGETRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgerq2

        public abstract void sgerq2(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGERQ2 computes an RQ factorization of a real m by n matrix A:
  A = R * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the m by n upper trapezoidal matrix R; the remaining

          elements, with the array TAU, represent the orthogonal matrix
          Q as a product of elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace) REAL array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and tau in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        info -

      • sgerqf

        public abstract void sgerqf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGERQF computes an RQ factorization of a real M-by-N matrix A:
  A = R * Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the M-by-N upper trapezoidal matrix R;
          the remaining elements, with the array TAU, represent the
          orthogonal matrix Q as a product of min(m,n) elementary
          reflectors (see Further Details).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and tau in TAU(i).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sgesc2

        public abstract void sgesc2(int n,
                            float[] a,
                            int lda,
                            float[] rhs,
                            int[] ipiv,
                            int[] jpiv,
                            org.netlib.util.floatW scale)
        
     ..

  Purpose
  =======

  SGESC2 solves a system of linear equations

            A * X = scale* RHS

  with a general N-by-N matrix A using the LU factorization with
  complete pivoting computed by SGETC2.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  A       (input) REAL array, dimension (LDA,N)
          On entry, the  LU part of the factorization of the n-by-n
          matrix A computed by SGETC2:  A = P * L * U * Q

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1, N).

  RHS     (input/output) REAL array, dimension (N).
          On entry, the right hand side vector b.
          On exit, the solution vector X.

  IPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  SCALE    (output) REAL
           On exit, SCALE contains the scale factor. SCALE is chosen
           0 <= SCALE <= 1 to prevent owerflow in the solution.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        rhs -
        ipiv -
        jpiv -
        scale -

      • sgesdd

        public abstract void sgesdd(java.lang.String jobz,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] s,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGESDD computes the singular value decomposition (SVD) of a real
  M-by-N matrix A, optionally computing the left and right singular
  vectors.  If singular vectors are desired, it uses a
  divide-and-conquer algorithm.

  The SVD is written

       A = U * SIGMA * transpose(V)

  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
  are the singular values of A; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.

  Note that the routine returns VT = V**T, not V.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          Specifies options for computing all or part of the matrix U:

          = 'A':  all M columns of U and all N rows of V**T are
                  returned in the arrays U and VT;
          = 'S':  the first min(M,N) columns of U and the first
                  min(M,N) rows of V**T are returned in the arrays U
                  and VT;
          = 'O':  If M >= N, the first N columns of U are overwritten
                  on the array A and all rows of V**T are returned in
                  the array VT;
                  otherwise, all columns of U are returned in the
                  array U and the first M rows of V**T are overwritten

                  in the array A;
          = 'N':  no columns of U or rows of V**T are computed.

  M       (input) INTEGER
          The number of rows of the input matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the input matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBZ = 'O',  A is overwritten with the first N columns
                          of U (the left singular vectors, stored
                          columnwise) if M >= N;
                          A is overwritten with the first M rows
                          of V**T (the right singular vectors, stored
                          rowwise) otherwise.
          if JOBZ .ne. 'O', the contents of A are destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  S       (output) REAL array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

  U       (output) REAL array, dimension (LDU,UCOL)
          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
          UCOL = min(M,N) if JOBZ = 'S'.
          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M

          orthogonal matrix U;
          if JOBZ = 'S', U contains the first min(M,N) columns of U
          (the left singular vectors, stored columnwise);
          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

  VT      (output) REAL array, dimension (LDVT,N)
          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
          N-by-N orthogonal matrix V**T;
          if JOBZ = 'S', VT contains the first min(M,N) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
          if JOBZ = 'S', LDVT >= min(M,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= 1.
          If JOBZ = 'N',
            LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)).
          If JOBZ = 'O',
            LWORK >= 3*min(M,N)*min(M,N) + 
                     max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
          If JOBZ = 'S' or 'A'
            LWORK >= 3*min(M,N)*min(M,N) +
                     max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)).
          For good performance, LWORK should generally be larger.
          If LWORK = -1 but other input arguments are legal, WORK(1)
          returns the optimal LWORK.

  IWORK   (workspace) INTEGER array, dimension (8*min(M,N))

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  SBDSDC did not converge, updating process failed.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        m -
        n -
        a -
        lda -
        s -
        u -
        ldu -
        vt -
        ldvt -
        work -
        lwork -
        iwork -
        info -

      • sgesv

        public abstract void sgesv(int n,
                           int nrhs,
                           float[] a,
                           int lda,
                           int[] ipiv,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGESV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

  The LU decomposition with partial pivoting and row interchanges is
  used to factor A as
     A = P * L * U,
  where P is a permutation matrix, L is unit lower triangular, and U is
  upper triangular.  The factored form of A is then used to solve the
  system of equations A * X = B.

  Arguments
  =========

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N coefficient matrix A.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS matrix of right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                has been completed, but the factor U is exactly
                singular, so the solution could not be computed.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • sgesvd

        public abstract void sgesvd(java.lang.String jobu,
                            java.lang.String jobvt,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] s,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGESVD computes the singular value decomposition (SVD) of a real
  M-by-N matrix A, optionally computing the left and/or right singular

  vectors. The SVD is written

       A = U * SIGMA * transpose(V)

  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
  are the singular values of A; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.

  Note that the routine returns V**T, not V.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          Specifies options for computing all or part of the matrix U:

          = 'A':  all M columns of U are returned in array U:
          = 'S':  the first min(m,n) columns of U (the left singular
                  vectors) are returned in the array U;
          = 'O':  the first min(m,n) columns of U (the left singular
                  vectors) are overwritten on the array A;
          = 'N':  no columns of U (no left singular vectors) are
                  computed.

  JOBVT   (input) CHARACTER*1
          Specifies options for computing all or part of the matrix
          V**T:
          = 'A':  all N rows of V**T are returned in the array VT;
          = 'S':  the first min(m,n) rows of V**T (the right singular
                  vectors) are returned in the array VT;
          = 'O':  the first min(m,n) rows of V**T (the right singular
                  vectors) are overwritten on the array A;
          = 'N':  no rows of V**T (no right singular vectors) are
                  computed.

          JOBVT and JOBU cannot both be 'O'.

  M       (input) INTEGER
          The number of rows of the input matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the input matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBU = 'O',  A is overwritten with the first min(m,n)
                          columns of U (the left singular vectors,
                          stored columnwise);
          if JOBVT = 'O', A is overwritten with the first min(m,n)
                          rows of V**T (the right singular vectors,
                          stored rowwise);
          if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
                          are destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  S       (output) REAL array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

  U       (output) REAL array, dimension (LDU,UCOL)
          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
          If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
          if JOBU = 'S', U contains the first min(m,n) columns of U
          (the left singular vectors, stored columnwise);
          if JOBU = 'N' or 'O', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBU = 'S' or 'A', LDU >= M.

  VT      (output) REAL array, dimension (LDVT,N)
          If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
          V**T;
          if JOBVT = 'S', VT contains the first min(m,n) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBVT = 'N' or 'O', VT is not referenced.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
          if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
          superdiagonal elements of an upper bidiagonal matrix B
          whose diagonal is in S (not necessarily sorted). B
          satisfies A = U * B * VT, so it has the same singular values

          as A, and singular vectors related by U and VT.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)).
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if SBDSQR did not converge, INFO specifies how many
                superdiagonals of an intermediate bidiagonal form B
                did not converge to zero. See the description of WORK
                above for details.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobvt -
        m -
        n -
        a -
        lda -
        s -
        u -
        ldu -
        vt -
        ldvt -
        work -
        lwork -
        info -

      • sgesvx

        public abstract void sgesvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            int[] ipiv,
                            org.netlib.util.StringW equed,
                            float[] r,
                            float[] c,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGESVX uses the LU factorization to compute the solution to a real
  system of linear equations
     A * X = B,
  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
     or diag(C)*B (if TRANS = 'T' or 'C').

  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
     matrix A (after equilibration if FACT = 'E') as
        A = P * L * U,
     where P is a permutation matrix, L is a unit lower triangular
     matrix, and U is upper triangular.

  3. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
     that it solves the original system before equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AF and IPIV contain the factored form of A.
                  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  A, AF, and IPIV are not modified.
          = 'N':  The matrix A will be copied to AF and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AF and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
          not 'N', then A must have been equilibrated by the scaling
          factors in R and/or C.  A is not modified if FACT = 'F' or
          'N', or if FACT = 'E' and EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) REAL array, dimension (LDAF,N)
          If FACT = 'F', then AF is an input argument and on entry
          contains the factors L and U from the factorization
          A = P*L*U as computed by SGETRF.  If EQUED .ne. 'N', then
          AF is the factored form of the equilibrated matrix A.

          If FACT = 'N', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U

          of the original matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the factors L and U from the factorization A = P*L*U

          of the equilibrated matrix A (see the description of A for
          the form of the equilibrated matrix).

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = P*L*U
          as computed by SGETRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = P*L*U
          of the equilibrated matrix A.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  R       (input or output) REAL array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.

  C       (input or output) REAL array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is

          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit,
          if EQUED = 'N', B is not modified;
          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
          to the original system of equations.  Note that A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) REAL array, dimension (4*N)
          On exit, WORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If WORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization has
                       been completed, but the factor U is exactly
                       singular, so the solution and error bounds
                       could not be computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        equed -
        r -
        c -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgetc2

        public abstract void sgetc2(int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            int[] jpiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGETC2 computes an LU factorization with complete pivoting of the
  n-by-n matrix A. The factorization has the form A = P * L * U * Q,
  where P and Q are permutation matrices, L is lower triangular with
  unit diagonal elements and U is upper triangular.

  This is the Level 2 BLAS algorithm.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the n-by-n matrix A to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U*Q; the unit diagonal elements of L are not stored.

          If U(k, k) appears to be less than SMIN, U(k, k) is given the
          value of SMIN, i.e., giving a nonsingular perturbed system.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension(N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (output) INTEGER array, dimension(N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  INFO    (output) INTEGER
           = 0: successful exit
           > 0: if INFO = k, U(k, k) is likely to produce owerflow if
                we try to solve for x in Ax = b. So U is perturbed to
                avoid the overflow.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        ipiv -
        jpiv -
        info -

      • sgetf2

        public abstract void sgetf2(int m,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGETF2 computes an LU factorization of a general m-by-n matrix A
  using partial pivoting with row interchanges.

  The factorization has the form
     A = P * L * U
  where P is a permutation matrix, L is lower triangular with unit
  diagonal elements (lower trapezoidal if m > n), and U is upper
  triangular (upper trapezoidal if m < n).

  This is the right-looking Level 2 BLAS version of the algorithm.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n matrix to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, U(k,k) is exactly zero. The factorization
               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used

               to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        ipiv -
        info -

      • sgetrf

        public abstract void sgetrf(int m,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGETRF computes an LU factorization of a general M-by-N matrix A
  using partial pivoting with row interchanges.

  The factorization has the form
     A = P * L * U
  where P is a permutation matrix, L is lower triangular with unit
  diagonal elements (lower trapezoidal if m > n), and U is upper
  triangular (upper trapezoidal if m < n).

  This is the right-looking Level 3 BLAS version of the algorithm.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  IPIV    (output) INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization

                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        ipiv -
        info -

      • sgetri

        public abstract void sgetri(int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGETRI computes the inverse of a matrix using the LU factorization
  computed by SGETRF.

  This method inverts U and then computes inv(A) by solving the system

  inv(A)*L = inv(U) for inv(A).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the factors L and U from the factorization
          A = P*L*U as computed by SGETRF.
          On exit, if INFO = 0, the inverse of the original matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from SGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).
          For optimal performance LWORK >= N*NB, where NB is
          the optimal blocksize returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
                singular and its inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        ipiv -
        work -
        lwork -
        info -

      • sgetrs

        public abstract void sgetrs(java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGETRS solves a system of linear equations
     A * X = B  or  A' * X = B
  with a general N-by-N matrix A using the LU factorization computed
  by SGETRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The factors L and U from the factorization A = P*L*U
          as computed by SGETRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from SGETRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • sggbak

        public abstract void sggbak(java.lang.String job,
                            java.lang.String side,
                            int n,
                            int ilo,
                            int ihi,
                            float[] lscale,
                            float[] rscale,
                            int m,
                            float[] v,
                            int ldv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGBAK forms the right or left eigenvectors of a real generalized
  eigenvalue problem A*x = lambda*B*x, by backward transformation on
  the computed eigenvectors of the balanced pair of matrices output by

  SGGBAL.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the type of backward transformation required:
          = 'N':  do nothing, return immediately;
          = 'P':  do backward transformation for permutation only;
          = 'S':  do backward transformation for scaling only;
          = 'B':  do backward transformations for both permutation and

                  scaling.
          JOB must be the same as the argument JOB supplied to SGGBAL.


  SIDE    (input) CHARACTER*1
          = 'R':  V contains right eigenvectors;
          = 'L':  V contains left eigenvectors.

  N       (input) INTEGER
          The number of rows of the matrix V.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          The integers ILO and IHI determined by SGGBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  LSCALE  (input) REAL array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the left side of A and B, as returned by SGGBAL.

  RSCALE  (input) REAL array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the right side of A and B, as returned by SGGBAL.

  M       (input) INTEGER
          The number of columns of the matrix V.  M >= 0.

  V       (input/output) REAL array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by STGEVC.
          On exit, V is overwritten by the transformed eigenvectors.

  LDV     (input) INTEGER
          The leading dimension of the matrix V. LDV >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  See R.C. Ward, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        job -
        side -
        n -
        ilo -
        ihi -
        lscale -
        rscale -
        m -
        v -
        ldv -
        info -

      • sggbal

        public abstract void sggbal(java.lang.String job,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            float[] lscale,
                            float[] rscale,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGBAL balances a pair of general real matrices (A,B).  This
  involves, first, permuting A and B by similarity transformations to
  isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
  elements on the diagonal; and second, applying a diagonal similarity

  transformation to rows and columns ILO to IHI to make the rows
  and columns as close in norm as possible. Both steps are optional.

  Balancing may reduce the 1-norm of the matrices, and improve the
  accuracy of the computed eigenvalues and/or eigenvectors in the
  generalized eigenvalue problem A*x = lambda*B*x.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
                  and RSCALE(I) = 1.0 for i = 1,...,N.
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the input matrix A.
          On exit,  A is overwritten by the balanced matrix.
          If JOB = 'N', A is not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the input matrix B.
          On exit,  B is overwritten by the balanced matrix.
          If JOB = 'N', B is not referenced.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are set to integers such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If JOB = 'N' or 'S', ILO = 1 and IHI = N.

  LSCALE  (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If P(j) is the index of the
          row interchanged with row j, and D(j)
          is the scaling factor applied to row j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  RSCALE  (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If P(j) is the index of the
          column interchanged with column j, and D(j)
          is the scaling factor applied to column j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  WORK    (workspace) REAL array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and

          at least 1 when JOB = 'N' or 'P'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  See R.C. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        lda -
        b -
        ldb -
        ilo -
        ihi -
        lscale -
        rscale -
        work -
        info -

      • sgges

        public void sgges(java.lang.String jobvsl,
                  java.lang.String jobvsr,
                  java.lang.String sort,
                  java.lang.Object selctg,
                  int n,
                  float[] a,
                  int lda,
                  float[] b,
                  int ldb,
                  org.netlib.util.intW sdim,
                  float[] alphar,
                  float[] alphai,
                  float[] beta,
                  float[] vsl,
                  int ldvsl,
                  float[] vsr,
                  int ldvsr,
                  float[] work,
                  int lwork,
                  boolean[] bwork,
                  org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
  the generalized eigenvalues, the generalized real Schur form (S,T),
  optionally, the left and/or right matrices of Schur vectors (VSL and

  VSR). This gives the generalized Schur factorization

           (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

  Optionally, it also orders the eigenvalues so that a selected cluster
  of eigenvalues appears in the leading diagonal blocks of the upper
  quasi-triangular matrix S and the upper triangular matrix T.The
  leading columns of VSL and VSR then form an orthonormal basis for the
  corresponding left and right eigenspaces (deflating subspaces).

  (If only the generalized eigenvalues are needed, use the driver
  SGGEV instead, which is faster.)

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
  or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
  usually represented as the pair (alpha,beta), as there is a
  reasonable interpretation for beta=0 or both being zero.

  A pair of matrices (S,T) is in generalized real Schur form if T is
  upper triangular with non-negative diagonal and S is block upper
  triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
  to real generalized eigenvalues, while 2-by-2 blocks of S will be
  "standardized" by making the corresponding elements of T have the
  form:
          [  a  0  ]
          [  0  b  ]

  and the pair of corresponding 2-by-2 blocks in S and T will have a
  complex conjugate pair of generalized eigenvalues.


  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);

  SELCTG  (external procedure) LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),

          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

  ALPHAR  (output) REAL array, dimension (N)
  ALPHAI  (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

  VSL     (output) REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16).
          For good performance , LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        sort -
        selctg -
        n -
        a -
        lda -
        b -
        ldb -
        sdim -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        work -
        lwork -
        bwork -
        info -

      • sggesx

        public void sggesx(java.lang.String jobvsl,
                   java.lang.String jobvsr,
                   java.lang.String sort,
                   java.lang.Object selctg,
                   java.lang.String sense,
                   int n,
                   float[] a,
                   int lda,
                   float[] b,
                   int ldb,
                   org.netlib.util.intW sdim,
                   float[] alphar,
                   float[] alphai,
                   float[] beta,
                   float[] vsl,
                   int ldvsl,
                   float[] vsr,
                   int ldvsr,
                   float[] rconde,
                   float[] rcondv,
                   float[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   boolean[] bwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGESX computes for a pair of N-by-N real nonsymmetric matrices
  (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
  optionally, the left and/or right matrices of Schur vectors (VSL and

  VSR).  This gives the generalized Schur factorization

       (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

  Optionally, it also orders the eigenvalues so that a selected cluster
  of eigenvalues appears in the leading diagonal blocks of the upper
  quasi-triangular matrix S and the upper triangular matrix T; computes
  a reciprocal condition number for the average of the selected
  eigenvalues (RCONDE); and computes a reciprocal condition number for

  the right and left deflating subspaces corresponding to the selected

  eigenvalues (RCONDV). The leading columns of VSL and VSR then form
  an orthonormal basis for the corresponding left and right eigenspaces
  (deflating subspaces).

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
  or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
  usually represented as the pair (alpha,beta), as there is a
  reasonable interpretation for beta=0 or for both being zero.

  A pair of matrices (S,T) is in generalized real Schur form if T is
  upper triangular with non-negative diagonal and S is block upper
  triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
  to real generalized eigenvalues, while 2-by-2 blocks of S will be
  "standardized" by making the corresponding elements of T have the
  form:
          [  a  0  ]
          [  0  b  ]

  and the pair of corresponding 2-by-2 blocks in S and T will have a
  complex conjugate pair of generalized eigenvalues.


  Arguments
  =========

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

  SORT    (input) CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG).

  SELCTG  (external procedure) LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer satisfy
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,

          since ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned), in this
          case INFO is set to N+3.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N' : None are computed;
          = 'E' : Computed for average of selected eigenvalues only;
          = 'V' : Computed for selected deflating subspaces only;
          = 'B' : Computed for both.
          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  SDIM    (output) INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

  ALPHAR  (output) REAL array, dimension (N)
  ALPHAI  (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

  VSL     (output) REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

  VSR     (output) REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

  RCONDE  (output) REAL array, dimension ( 2 )
          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
          reciprocal condition numbers for the average of the selected

          eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

  RCONDV  (output) REAL array, dimension ( 2 )
          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
          reciprocal condition numbers for the selected deflating
          subspaces.
          Not referenced if SENSE = 'N' or 'E'.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
          LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
          LWORK >= max( 8*N, 6*N+16 ).
          Note that 2*SDIM*(N-SDIM) <= N*N/2.
          Note also that an error is only returned if
          LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
          this may not be large enough.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the bound on the optimal size of the WORK
          array and the minimum size of the IWORK array, returns these

          values as the first entries of the WORK and IWORK arrays, and
          no error message related to LWORK or LIWORK is issued by
          XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
          LIWORK >= N+6.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the bound on the optimal size of the

          WORK array and the minimum size of the IWORK array, returns
          these values as the first entries of the WORK and IWORK
          arrays, and no error message related to LWORK or LIWORK is
          issued by XERBLA.

  BWORK   (workspace) LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.

  Further details
  ===============

  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / RCONDE( 1 ).

  An approximate (asymptotic) bound on the maximum angular error in
  the computed deflating subspaces is

       EPS * norm((A, B)) / RCONDV( 2 ).

  See LAPACK User's Guide, section 4.11 for more information.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvsl -
        jobvsr -
        sort -
        selctg -
        sense -
        n -
        a -
        lda -
        b -
        ldb -
        sdim -
        alphar -
        alphai -
        beta -
        vsl -
        ldvsl -
        vsr -
        ldvsr -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        liwork -
        bwork -
        info -

      • sggev

        public abstract void sggev(java.lang.String jobvl,
                           java.lang.String jobvr,
                           int n,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float[] alphar,
                           float[] alphai,
                           float[] beta,
                           float[] vl,
                           int ldvl,
                           float[] vr,
                           int ldvr,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)

  the generalized eigenvalues, and optionally, the left and/or right
  generalized eigenvectors.

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
  singular. It is usually represented as the pair (alpha,beta), as
  there is a reasonable interpretation for beta=0, and even for both
  being zero.

  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)

  of (A,B) satisfies

                   A * v(j) = lambda(j) * B * v(j).

  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
  of (A,B) satisfies

                   u(j)**H * A  = lambda(j) * u(j)**H * B .

  where u(j)**H is the conjugate-transpose of u(j).


  Arguments
  =========

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
  ALPHAI  (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio
          alpha/beta.  However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).

          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).

          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: error return from STGEVC.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobvl -
        jobvr -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        work -
        lwork -
        info -

      • sggevx

        public abstract void sggevx(java.lang.String balanc,
                            java.lang.String jobvl,
                            java.lang.String jobvr,
                            java.lang.String sense,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] alphar,
                            float[] alphai,
                            float[] beta,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            org.netlib.util.intW ilo,
                            org.netlib.util.intW ihi,
                            float[] lscale,
                            float[] rscale,
                            org.netlib.util.floatW abnrm,
                            org.netlib.util.floatW bbnrm,
                            float[] rconde,
                            float[] rcondv,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            boolean[] bwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
  the generalized eigenvalues, and optionally, the left and/or right
  generalized eigenvectors.

  Optionally also, it computes a balancing transformation to improve
  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
  LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
  the eigenvalues (RCONDE), and reciprocal condition numbers for the
  right eigenvectors (RCONDV).

  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
  singular. It is usually represented as the pair (alpha,beta), as
  there is a reasonable interpretation for beta=0, and even for both
  being zero.

  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)

  of (A,B) satisfies

                   A * v(j) = lambda(j) * B * v(j) .

  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
  of (A,B) satisfies

                   u(j)**H * A  = lambda(j) * u(j)**H * B.

  where u(j)**H is the conjugate-transpose of u(j).


  Arguments
  =========

  BALANC  (input) CHARACTER*1
          Specifies the balance option to be performed.
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then A contains the first part of the real Schur
          form of the "balanced" versions of the input A and B.

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then B contains the second part of the real Schur
          form of the "balanced" versions of the input A and B.

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
  ALPHAI  (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.

          Thus, the user should avoid naively computing the ratio
          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).

          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).

          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

  ILO     (output) INTEGER
  IHI     (output) INTEGER
          ILO and IHI are integer values such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

  LSCALE  (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If PL(j) is the index of the
          row interchanged with row j, and DL(j) is the scaling
          factor applied to row j, then
            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                      = DL(j)  for j = ILO,...,IHI
                      = PL(j)  for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  RSCALE  (output) REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If PR(j) is the index of the
          column interchanged with column j, and DR(j) is the scaling
          factor applied to column j, then
            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                      = DR(j)  for j = ILO,...,IHI
                      = PR(j)  for j = IHI+1,...,N
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

  ABNRM   (output) REAL
          The one-norm of the balanced matrix A.

  BBNRM   (output) REAL
          The one-norm of the balanced matrix B.

  RCONDE  (output) REAL array, dimension (N)
          If SENSE = 'E' or 'B', the reciprocal condition numbers of
          the eigenvalues, stored in consecutive elements of the array.
          For a complex conjugate pair of eigenvalues two consecutive
          elements of RCONDE are set to the same value. Thus RCONDE(j),
          RCONDV(j), and the j-th columns of VL and VR all correspond
          to the j-th eigenpair.
          If SENSE = 'N' or 'V', RCONDE is not referenced.

  RCONDV  (output) REAL array, dimension (N)
          If SENSE = 'V' or 'B', the estimated reciprocal condition
          numbers of the eigenvectors, stored in consecutive elements
          of the array. For a complex eigenvector two consecutive
          elements of RCONDV are set to the same value. If the
          eigenvalues cannot be reordered to compute RCONDV(j),
          RCONDV(j) is set to 0; this can only occur when the true
          value would be very small anyway.
          If SENSE = 'N' or 'E', RCONDV is not referenced.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,2*N).
          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
          LWORK >= max(1,6*N).
          If SENSE = 'E', LWORK >= max(1,10*N).
          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N+6)
          If SENSE = 'E', IWORK is not referenced.

  BWORK   (workspace) LOGICAL array, dimension (N)
          If SENSE = 'N', BWORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: error return from STGEVC.

  Further Details
  ===============

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity

  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

  =====================================================================

     .. Parameters ..
        Parameters:
        balanc -
        jobvl -
        jobvr -
        sense -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        vl -
        ldvl -
        vr -
        ldvr -
        ilo -
        ihi -
        lscale -
        rscale -
        abnrm -
        bbnrm -
        rconde -
        rcondv -
        work -
        lwork -
        iwork -
        bwork -
        info -

      • sggglm

        public abstract void sggglm(int n,
                            int m,
                            int p,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] d,
                            float[] x,
                            float[] y,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGGLM solves a general Gauss-Markov linear model (GLM) problem:

          minimize || y ||_2   subject to   d = A*x + B*y
              x

  where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
  given N-vector. It is assumed that M <= N <= M+P, and

             rank(A) = M    and    rank( A B ) = N.

  Under these assumptions, the constrained equation is always
  consistent, and there is a unique solution x and a minimal 2-norm
  solution y, which is obtained using a generalized QR factorization
  of the matrices (A, B) given by

     A = Q*(R),   B = Q*T*Z.
           (0)

  In particular, if matrix B is square nonsingular, then the problem
  GLM is equivalent to the following weighted linear least squares
  problem

               minimize || inv(B)*(d-A*x) ||_2
                   x

  where inv(B) denotes the inverse of B.

  Arguments
  =========

  N       (input) INTEGER
          The number of rows of the matrices A and B.  N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix A.  0 <= M <= N.

  P       (input) INTEGER
          The number of columns of the matrix B.  P >= N-M.

  A       (input/output) REAL array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the upper triangular part of the array A contains
          the M-by-M upper triangular matrix R.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)th subdiagonal
          contain the N-by-P upper trapezoidal matrix T.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  D       (input/output) REAL array, dimension (N)
          On entry, D is the left hand side of the GLM equation.
          On exit, D is destroyed.

  X       (output) REAL array, dimension (M)
  Y       (output) REAL array, dimension (P)
          On exit, X and Y are the solutions of the GLM problem.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N+M+P).
          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
          where NB is an upper bound for the optimal blocksizes for
          SGEQRF, SGERQF, SORMQR and SORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the upper triangular factor R associated with A in the

                generalized QR factorization of the pair (A, B) is
                singular, so that rank(A) < M; the least squares
                solution could not be computed.
          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                factor T associated with B in the generalized QR
                factorization of the pair (A, B) is singular, so that
                rank( A B ) < N; the least squares solution could not
                be computed.

  ===================================================================

     .. Parameters ..
        Parameters:
        n -
        m -
        p -
        a -
        lda -
        b -
        ldb -
        d -
        x -
        y -
        work -
        lwork -
        info -

      • sgghrd

        public abstract void sgghrd(java.lang.String compq,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] q,
                            int ldq,
                            float[] z,
                            int ldz,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGHRD reduces a pair of real matrices (A,B) to generalized upper
  Hessenberg form using orthogonal transformations, where A is a
  general matrix and B is upper triangular.  The form of the
  generalized eigenvalue problem is
     A*x = lambda*B*x,
  and B is typically made upper triangular by computing its QR
  factorization and moving the orthogonal matrix Q to the left side
  of the equation.

  This subroutine simultaneously reduces A to a Hessenberg matrix H:
     Q**T*A*Z = H
  and transforms B to another upper triangular matrix T:
     Q**T*B*Z = T
  in order to reduce the problem to its standard form
     H*y = lambda*T*y
  where y = Z**T*x.

  The orthogonal matrices Q and Z are determined as products of Givens

  rotations.  They may either be formed explicitly, or they may be
  postmultiplied into input matrices Q1 and Z1, so that

       Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

       Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

  If Q1 is the orthogonal matrix from the QR factorization of B in the

  original equation A*x = lambda*B*x, then SGGHRD reduces the original

  problem to generalized Hessenberg form.

  Arguments
  =========

  COMPQ   (input) CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 orthogonal matrix Q is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry,
                 and the product Q1*Q is returned.

  COMPZ   (input) CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 orthogonal matrix Z is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry,
                 and the product Z1*Z is returned.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to SGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**T B Z.  The
          elements below the diagonal are set to zero.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  Q       (input/output) REAL array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
          typically from the QR factorization of B.
          On exit, if COMPQ='I', the orthogonal matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

  Z       (input/output) REAL array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
          On exit, if COMPZ='I', the orthogonal matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and Van Loan (Johns Hopkins Press.)

  =====================================================================

     .. Parameters ..
        Parameters:
        compq -
        compz -
        n -
        ilo -
        ihi -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        info -

      • sgglse

        public abstract void sgglse(int m,
                            int n,
                            int p,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] c,
                            float[] d,
                            float[] x,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGLSE solves the linear equality-constrained least squares (LSE)
  problem:

          minimize || c - A*x ||_2   subject to   B*x = d

  where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
  M-vector, and d is a given P-vector. It is assumed that
  P <= N <= M+P, and

           rank(B) = P and  rank( (A) ) = N.
                                ( (B) )

  These conditions ensure that the LSE problem has a unique solution,
  which is obtained using a generalized RQ factorization of the
  matrices (B, A) given by

     B = (0 R)*Q,   A = Z*T*Q.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B. N >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B. 0 <= P <= N <= M+P.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(M,N)-by-N upper trapezoidal matrix T.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
          contains the P-by-P upper triangular matrix R.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  C       (input/output) REAL array, dimension (M)
          On entry, C contains the right hand side vector for the
          least squares part of the LSE problem.
          On exit, the residual sum of squares for the solution
          is given by the sum of squares of elements N-P+1 to M of
          vector C.

  D       (input/output) REAL array, dimension (P)
          On entry, D contains the right hand side vector for the
          constrained equation.
          On exit, D is destroyed.

  X       (output) REAL array, dimension (N)
          On exit, X is the solution of the LSE problem.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M+N+P).
          For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
          where NB is an upper bound for the optimal blocksizes for
          SGEQRF, SGERQF, SORMQR and SORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the upper triangular factor R associated with B in the

                generalized RQ factorization of the pair (B, A) is
                singular, so that rank(B) < P; the least squares
                solution could not be computed.
          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
                T associated with A in the generalized RQ factorization
                of the pair (B, A) is singular, so that
                rank( (A) ) < N; the least squares solution could not
                    ( (B) )
                be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        p -
        a -
        lda -
        b -
        ldb -
        c -
        d -
        x -
        work -
        lwork -
        info -

      • sggqrf

        public abstract void sggqrf(int n,
                            int m,
                            int p,
                            float[] a,
                            int lda,
                            float[] taua,
                            float[] b,
                            int ldb,
                            float[] taub,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGQRF computes a generalized QR factorization of an N-by-M matrix A

  and an N-by-P matrix B:

              A = Q*R,        B = Q*T*Z,

  where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
  matrix, and R and T assume one of the forms:

  if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                  (  0  ) N-M                         N   M-N
                     M

  where R11 is upper triangular, and

  if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                   P-N  N                           ( T21 ) P
                                                       P

  where T12 or T21 is upper triangular.

  In particular, if B is square and nonsingular, the GQR factorization

  of A and B implicitly gives the QR factorization of inv(B)*A:

               inv(B)*A = Z'*(inv(T)*R)

  where inv(B) denotes the inverse of the matrix B, and Z' denotes the

  transpose of the matrix Z.

  Arguments
  =========

  N       (input) INTEGER
          The number of rows of the matrices A and B. N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of columns of the matrix B.  P >= 0.

  A       (input/output) REAL array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the elements on and above the diagonal of the array

          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
          upper triangular if N >= M); the elements below the diagonal,
          with the array TAUA, represent the orthogonal matrix Q as a
          product of min(N,M) elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAUA    (output) REAL array, dimension (min(N,M))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

  B       (input/output) REAL array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)-th subdiagonal

          contain the N-by-P upper trapezoidal matrix T; the remaining

          elements, with the array TAUB, represent the orthogonal
          matrix Z as a product of elementary reflectors (see Further
          Details).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  TAUB    (output) REAL array, dimension (min(N,P))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the QR factorization
          of an N-by-M matrix, NB2 is the optimal blocksize for the
          RQ factorization of an N-by-P matrix, and NB3 is the optimal

          blocksize for a call of SORMQR.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(n,m).

  Each H(i) has the form

     H(i) = I - taua * v * v'

  where taua is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),

  and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine SORGQR.
  To use Q to update another matrix, use LAPACK subroutine SORMQR.

  The matrix Z is represented as a product of elementary reflectors

     Z = H(1) H(2) . . . H(k), where k = min(n,p).

  Each H(i) has the form

     H(i) = I - taub * v * v'

  where taub is a real scalar, and v is a real vector with
  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in

  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine SORGRQ.
  To use Z to update another matrix, use LAPACK subroutine SORMRQ.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        m -
        p -
        a -
        lda -
        taua -
        b -
        ldb -
        taub -
        work -
        lwork -
        info -

      • sggrqf

        public abstract void sggrqf(int m,
                            int p,
                            int n,
                            float[] a,
                            int lda,
                            float[] taua,
                            float[] b,
                            int ldb,
                            float[] taub,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGRQF computes a generalized RQ factorization of an M-by-N matrix A

  and a P-by-N matrix B:

              A = R*Q,        B = Z*T*Q,

  where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
  matrix, and R and T assume one of the forms:

  if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                   N-M  M                           ( R21 ) N
                                                       N

  where R12 or R21 is upper triangular, and

  if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                  (  0  ) P-N                         P   N-P
                     N

  where T11 is upper triangular.

  In particular, if B is square and nonsingular, the GRQ factorization

  of A and B implicitly gives the RQ factorization of A*inv(B):

               A*inv(B) = (R*inv(T))*Z'

  where inv(B) denotes the inverse of the matrix B, and Z' denotes the

  transpose of the matrix Z.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, if M <= N, the upper triangle of the subarray
          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
          if M > N, the elements on and above the (M-N)-th subdiagonal

          contain the M-by-N upper trapezoidal matrix R; the remaining

          elements, with the array TAUA, represent the orthogonal
          matrix Q as a product of elementary reflectors (see Further
          Details).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  TAUA    (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the elements on and above the diagonal of the array

          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
          upper triangular if P >= N); the elements below the diagonal,
          with the array TAUB, represent the orthogonal matrix Z as a
          product of elementary reflectors (see Further Details).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TAUB    (output) REAL array, dimension (min(P,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the RQ factorization
          of an M-by-N matrix, NB2 is the optimal blocksize for the
          QR factorization of a P-by-N matrix, and NB3 is the optimal
          blocksize for a call of SORMRQ.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INF0= -i, the i-th argument had an illegal value.

  Further Details
  ===============

  The matrix Q is represented as a product of elementary reflectors

     Q = H(1) H(2) . . . H(k), where k = min(m,n).

  Each H(i) has the form

     H(i) = I - taua * v * v'

  where taua is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine SORGRQ.
  To use Q to update another matrix, use LAPACK subroutine SORMRQ.

  The matrix Z is represented as a product of elementary reflectors

     Z = H(1) H(2) . . . H(k), where k = min(p,n).

  Each H(i) has the form

     H(i) = I - taub * v * v'

  where taub is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),

  and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine SORGQR.
  To use Z to update another matrix, use LAPACK subroutine SORMQR.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        m -
        p -
        n -
        a -
        lda -
        taua -
        b -
        ldb -
        taub -
        work -
        lwork -
        info -

      • sggsvd

        public abstract void sggsvd(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int n,
                            int p,
                            org.netlib.util.intW k,
                            org.netlib.util.intW l,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] alpha,
                            float[] beta,
                            float[] u,
                            int ldu,
                            float[] v,
                            int ldv,
                            float[] q,
                            int ldq,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGSVD computes the generalized singular value decomposition (GSVD)
  of an M-by-N real matrix A and P-by-N real matrix B:

      U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

  where U, V and Q are orthogonal matrices, and Z' is the transpose
  of Z.  Let K+L = the effective numerical rank of the matrix (A',B')',
  then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
  D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
  following structures, respectively:

  If M-K-L >= 0,

                      K  L
         D1 =     K ( I  0 )
                  L ( 0  C )
              M-K-L ( 0  0 )

                    K  L
         D2 =   L ( 0  S )
              P-L ( 0  0 )

                  N-K-L  K    L
    ( 0 R ) = K (  0   R11  R12 )
              L (  0    0   R22 )

  where

    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
    S = diag( BETA(K+1),  ... , BETA(K+L) ),
    C**2 + S**2 = I.

    R is stored in A(1:K+L,N-K-L+1:N) on exit.

  If M-K-L < 0,

                    K M-K K+L-M
         D1 =   K ( I  0    0   )
              M-K ( 0  C    0   )

                      K M-K K+L-M
         D2 =   M-K ( 0  S    0  )
              K+L-M ( 0  0    I  )
                P-L ( 0  0    0  )

                     N-K-L  K   M-K  K+L-M
    ( 0 R ) =     K ( 0    R11  R12  R13  )
                M-K ( 0     0   R22  R23  )
              K+L-M ( 0     0    0   R33  )

  where

    C = diag( ALPHA(K+1), ... , ALPHA(M) ),
    S = diag( BETA(K+1),  ... , BETA(M) ),
    C**2 + S**2 = I.

    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
    ( 0  R22 R23 )
    in B(M-K+1:L,N+M-K-L+1:N) on exit.

  The routine computes C, S, R, and optionally the orthogonal
  transformation matrices U, V and Q.

  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
  A and B implicitly gives the SVD of A*inv(B):
                       A*inv(B) = U*(D1*inv(D2))*V'.
  If ( A',B')' has orthonormal columns, then the GSVD of A and B is
  also equal to the CS decomposition of A and B. Furthermore, the GSVD

  can be used to derive the solution of the eigenvalue problem:
                       A'*A x = lambda* B'*B x.
  In some literature, the GSVD of A and B is presented in the form
                   U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
  where U and V are orthogonal and X is nonsingular, D1 and D2 are
  ``diagonal''.  The former GSVD form can be converted to the latter
  form by taking the nonsingular matrix X as

                       X = Q*( I   0    )
                             ( 0 inv(R) ).

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  K       (output) INTEGER
  L       (output) INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in the Purpose section.
          K + L = effective numerical rank of (A',B')'.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular matrix R, or part of R.
          See Purpose for details.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix R if M-K-L < 0.
          See Purpose for details.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  ALPHA   (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = C,
            BETA(K+1:K+L)  = S,
          or if M-K-L < 0,
            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
            BETA(K+1:M) =S, BETA(M+1:K+L) =1
          and
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0

  U       (output) REAL array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (output) REAL array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (output) REAL array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  WORK    (workspace) REAL array,
                      dimension (max(3*N,M,P)+N)

  IWORK   (workspace/output) INTEGER array, dimension (N)
          On exit, IWORK stores the sorting information. More
          precisely, the following loop will sort ALPHA
             for I = K+1, min(M,K+L)
                 swap ALPHA(I) and ALPHA(IWORK(I))
             endfor
          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, the Jacobi-type procedure failed to
                converge.  For further details, see subroutine STGSJA.


  Internal Parameters
  ===================

  TOLA    REAL
  TOLB    REAL
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A',B')'. Generally, they are set to
                   TOLA = MAX(M,N)*norm(A)*MACHEPS,
                   TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

  Further Details
  ===============

  2-96 Based on modifications by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        n -
        p -
        k -
        l -
        a -
        lda -
        b -
        ldb -
        alpha -
        beta -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        work -
        iwork -
        info -

      • sggsvp

        public abstract void sggsvp(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int p,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float tola,
                            float tolb,
                            org.netlib.util.intW k,
                            org.netlib.util.intW l,
                            float[] u,
                            int ldu,
                            float[] v,
                            int ldv,
                            float[] q,
                            int ldq,
                            int[] iwork,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGGSVP computes orthogonal matrices U, V and Q such that

                   N-K-L  K    L
   U'*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                L ( 0     0   A23 )
            M-K-L ( 0     0    0  )

                   N-K-L  K    L
          =     K ( 0    A12  A13 )  if M-K-L < 0;
              M-K ( 0     0   A23 )

                 N-K-L  K    L
   V'*B*Q =   L ( 0     0   B13 )
            P-L ( 0     0    0  )

  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
  otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
  numerical rank of the (M+P)-by-N matrix (A',B')'.  Z' denotes the
  transpose of Z.

  This decomposition is the preprocessing step for computing the
  Generalized Singular Value Decomposition (GSVD), see subroutine
  SGGSVD.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular (or trapezoidal) matrix
          described in the Purpose section.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix described in
          the Purpose section.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TOLA    (input) REAL
  TOLB    (input) REAL
          TOLA and TOLB are the thresholds to determine the effective
          numerical rank of matrix B and a subblock of A. Generally,
          they are set to
             TOLA = MAX(M,N)*norm(A)*MACHEPS,
             TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

  K       (output) INTEGER
  L       (output) INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in Purpose.
          K + L = effective numerical rank of (A',B')'.

  U       (output) REAL array, dimension (LDU,M)
          If JOBU = 'U', U contains the orthogonal matrix U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (output) REAL array, dimension (LDV,M)
          If JOBV = 'V', V contains the orthogonal matrix V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (output) REAL array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  IWORK   (workspace) INTEGER array, dimension (N)

  TAU     (workspace) REAL array, dimension (N)

  WORK    (workspace) REAL array, dimension (max(3*N,M,P))

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.


  Further Details
  ===============

  The subroutine uses LAPACK subroutine SGEQPF for the QR factorization
  with column pivoting to detect the effective numerical rank of the
  a matrix. It may be replaced by a better rank determination strategy.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        p -
        n -
        a -
        lda -
        b -
        ldb -
        tola -
        tolb -
        k -
        l -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        iwork -
        tau -
        work -
        info -

      • sgtcon

        public abstract void sgtcon(java.lang.String norm,
                            int n,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] du2,
                            int[] ipiv,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTCON estimates the reciprocal of the condition number of a real
  tridiagonal matrix A using the LU factorization as computed by
  SGTTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by SGTTRF.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

  DU2     (input) REAL array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  ANORM   (input) REAL
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (2*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        dl -
        d -
        du -
        du2 -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • sgtrfs

        public abstract void sgtrfs(java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] dlf,
                            float[] df,
                            float[] duf,
                            float[] du2,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is tridiagonal, and provides
  error bounds and backward error estimates for the solution.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

  D       (input) REAL array, dimension (N)
          The diagonal elements of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

  DLF     (input) REAL array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by SGTTRF.

  DF      (input) REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DUF     (input) REAL array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

  DU2     (input) REAL array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SGTTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        dlf -
        df -
        duf -
        du2 -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgtsv

        public abstract void sgtsv(int n,
                           int nrhs,
                           float[] dl,
                           float[] d,
                           float[] du,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTSV  solves the equation

     A*X = B,

  where A is an n by n tridiagonal matrix, by Gaussian elimination with
  partial pivoting.

  Note that the equation  A'*X = B  may be solved by interchanging the

  order of the arguments DU and DL.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input/output) REAL array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of

          A.

          On exit, DL is overwritten by the (n-2) elements of the
          second super-diagonal of the upper triangular matrix U from
          the LU factorization of A, in DL(1), ..., DL(n-2).

  D       (input/output) REAL array, dimension (N)
          On entry, D must contain the diagonal elements of A.

          On exit, D is overwritten by the n diagonal elements of U.

  DU      (input/output) REAL array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.

          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix of right hand side matrix B.
          On exit, if INFO = 0, the N by NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, U(i,i) is exactly zero, and the solution
               has not been computed.  The factorization has not been
               completed unless i = N.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        nrhs -
        dl -
        d -
        du -
        b -
        ldb -
        info -

      • sgtsvx

        public abstract void sgtsvx(java.lang.String fact,
                            java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] dlf,
                            float[] df,
                            float[] duf,
                            float[] du2,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTSVX uses the LU factorization to compute the solution to a real
  system of linear equations A * X = B or A**T * X = B,
  where A is a tridiagonal matrix of order N and X and B are N-by-NRHS

  matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the LU decomposition is used to factor the matrix A
     as A = L * U, where L is a product of permutation and unit lower
     bidiagonal matrices and U is upper triangular with nonzeros in
     only the main diagonal and first two superdiagonals.

  2. If some U(i,i)=0, so that U is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
                  will not be modified.
          = 'N':  The matrix will be copied to DLF, DF, and DUF
                  and factored.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

  DLF     (input or output) REAL array, dimension (N-1)
          If FACT = 'F', then DLF is an input argument and on entry
          contains the (n-1) multipliers that define the matrix L from

          the LU factorization of A as computed by SGTTRF.

          If FACT = 'N', then DLF is an output argument and on exit
          contains the (n-1) multipliers that define the matrix L from

          the LU factorization of A.

  DF      (input or output) REAL array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.

          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.

  DUF     (input or output) REAL array, dimension (N-1)
          If FACT = 'F', then DUF is an input argument and on entry
          contains the (n-1) elements of the first superdiagonal of U.


          If FACT = 'N', then DUF is an output argument and on exit
          contains the (n-1) elements of the first superdiagonal of U.


  DU2     (input or output) REAL array, dimension (N-2)
          If FACT = 'F', then DU2 is an input argument and on entry
          contains the (n-2) elements of the second superdiagonal of
          U.

          If FACT = 'N', then DU2 is an output argument and on exit
          contains the (n-2) elements of the second superdiagonal of
          U.

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the LU factorization of A as

          computed by SGTTRF.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the LU factorization of A;
          row i of the matrix was interchanged with row IPIV(i).
          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
          a row interchange was not required.

  B       (input) REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has not been completed unless i = N, but the
                       factor U is exactly singular, so the solution
                       and error bounds could not be computed.
                       RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        dlf -
        df -
        duf -
        du2 -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sgttrf

        public abstract void sgttrf(int n,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] du2,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTTRF computes an LU factorization of a real tridiagonal matrix A
  using elimination with partial pivoting and row interchanges.

  The factorization has the form
     A = L * U
  where L is a product of permutation and unit lower bidiagonal
  matrices and U is upper triangular with nonzeros in only the main
  diagonal and first two superdiagonals.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  DL      (input/output) REAL array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of

          A.

          On exit, DL is overwritten by the (n-1) multipliers that
          define the matrix L from the LU factorization of A.

  D       (input/output) REAL array, dimension (N)
          On entry, D must contain the diagonal elements of A.

          On exit, D is overwritten by the n diagonal elements of the
          upper triangular matrix U from the LU factorization of A.

  DU      (input/output) REAL array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.

          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

  DU2     (output) REAL array, dimension (N-2)
          On exit, DU2 is overwritten by the (n-2) elements of the
          second super-diagonal of U.

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
          > 0:  if INFO = k, U(k,k) is exactly zero. The factorization

                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        dl -
        d -
        du -
        du2 -
        ipiv -
        info -

      • sgttrs

        public abstract void sgttrs(java.lang.String trans,
                            int n,
                            int nrhs,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] du2,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SGTTRS solves one of the systems of equations
     A*X = B  or  A'*X = B,
  with a tridiagonal matrix A using the LU factorization computed
  by SGTTRF.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B  (No transpose)
          = 'T':  A'* X = B  (Transpose)
          = 'C':  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

  DU2     (input) REAL array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        trans -
        n -
        nrhs -
        dl -
        d -
        du -
        du2 -
        ipiv -
        b -
        ldb -
        info -

      • sgtts2

        public abstract void sgtts2(int itrans,
                            int n,
                            int nrhs,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] du2,
                            int[] ipiv,
                            float[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  SGTTS2 solves one of the systems of equations
     A*X = B  or  A'*X = B,
  with a tridiagonal matrix A using the LU factorization computed
  by SGTTRF.

  Arguments
  =========

  ITRANS  (input) INTEGER
          Specifies the form of the system of equations.
          = 0:  A * X = B  (No transpose)
          = 1:  A'* X = B  (Transpose)
          = 2:  A'* X = B  (Conjugate transpose = Transpose)

  N       (input) INTEGER
          The order of the matrix A.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

  DU2     (input) REAL array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, row i of the matrix was
          interchanged with row IPIV(i).  IPIV(i) will always be either
          i or i+1; IPIV(i) = i indicates a row interchange was not
          required.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        itrans -
        n -
        nrhs -
        dl -
        d -
        du -
        du2 -
        ipiv -
        b -
        ldb -

      • shgeqz

        public abstract void shgeqz(java.lang.String job,
                            java.lang.String compq,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            float[] h,
                            int ldh,
                            float[] t,
                            int ldt,
                            float[] alphar,
                            float[] alphai,
                            float[] beta,
                            float[] q,
                            int ldq,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
  where H is an upper Hessenberg matrix and T is upper triangular,
  using the double-shift QZ method.
  Matrix pairs of this type are produced by the reduction to
  generalized upper Hessenberg form of a real matrix pair (A,B):

     A = Q1*H*Z1**T,  B = Q1*T*Z1**T,

  as computed by SGGHRD.

  If JOB='S', then the Hessenberg-triangular pair (H,T) is
  also reduced to generalized Schur form,
  
     H = Q*S*Z**T,  T = Q*P*Z**T,
  
  where Q and Z are orthogonal matrices, P is an upper triangular
  matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
  diagonal blocks.

  The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
  (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of

  eigenvalues.

  Additionally, the 2-by-2 upper triangular diagonal blocks of P
  corresponding to 2-by-2 blocks of S are reduced to positive diagonal

  form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
  P(j,j) > 0, and P(j+1,j+1) > 0.

  Optionally, the orthogonal matrix Q from the generalized Schur
  factorization may be postmultiplied into an input matrix Q1, and the

  orthogonal matrix Z may be postmultiplied into an input matrix Z1.
  If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
  the matrix pair (A,B) to generalized upper Hessenberg form, then the

  output matrices Q1*Q and Z1*Z are the orthogonal factors from the
  generalized Schur factorization of (A,B):

     A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
  
  To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently
  of (A,B)) are computed as a pair of values (alpha,beta), where alpha 
  complex and beta real.
  If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
  generalized nonsymmetric eigenvalue problem (GNEP)
     A*x = lambda*B*x
  and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
  alternate form of the GNEP
     mu*A*y = B*y.
  Real eigenvalues can be read directly from the generalized Schur
  form: 
    alpha = S(i,i), beta = P(i,i).

  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix

       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
       pp. 241--256.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form. 

  COMPQ   (input) CHARACTER*1
          = 'N': Left Schur vectors (Q) are not computed;
          = 'I': Q is initialized to the unit matrix and the matrix Q
                 of left Schur vectors of (H,T) is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry and
                 the product Q1*Q is returned.

  COMPZ   (input) CHARACTER*1
          = 'N': Right Schur vectors (Z) are not computed;
          = 'I': Z is initialized to the unit matrix and the matrix Z
                 of right Schur vectors of (H,T) is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry and
                 the product Z1*Z is returned.

  N       (input) INTEGER
          The order of the matrices H, T, Q, and Z.  N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI mark the rows and columns of H which are in
          Hessenberg form.  It is assumed that A is already upper
          triangular in rows and columns 1:ILO-1 and IHI+1:N.
          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

  H       (input/output) REAL array, dimension (LDH, N)
          On entry, the N-by-N upper Hessenberg matrix H.
          On exit, if JOB = 'S', H contains the upper quasi-triangular

          matrix S from the generalized Schur factorization;
          2-by-2 diagonal blocks (corresponding to complex conjugate
          pairs of eigenvalues) are returned in standard form, with
          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
          If JOB = 'E', the diagonal blocks of H match those of S, but

          the rest of H is unspecified.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max( 1, N ).

  T       (input/output) REAL array, dimension (LDT, N)
          On entry, the N-by-N upper triangular matrix T.
          On exit, if JOB = 'S', T contains the upper triangular
          matrix P from the generalized Schur factorization;
          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of
          are reduced to positive diagonal form, i.e., if H(j+1,j) is
          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
          T(j+1,j+1) > 0.
          If JOB = 'E', the diagonal blocks of T match those of P, but

          the rest of T is unspecified.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= max( 1, N ).

  ALPHAR  (output) REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP.

  ALPHAI  (output) REAL array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

  BETA    (output) REAL array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

  Q       (input/output) REAL array, dimension (LDQ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
          the reduction of (A,B) to generalized Hessenberg form.
          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur

          vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
          of left Schur vectors of (A,B).
          Not referenced if COMPZ = 'N'.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= 1.
          If COMPQ='V' or 'I', then LDQ >= N.

  Z       (input/output) REAL array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
          the reduction of (A,B) to generalized Hessenberg form.
          On exit, if COMPZ = 'I', the orthogonal matrix of
          right Schur vectors of (H,T), and if COMPZ = 'V', the
          orthogonal matrix of right Schur vectors of (A,B).
          Not referenced if COMPZ = 'N'.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If COMPZ='V' or 'I', then LDZ >= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO+1,...,N should be correct.
          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO-N+1,...,N should be correct.

  Further Details
  ===============

  Iteration counters:

  JITER  -- counts iterations.
  IITER  -- counts iterations run since ILAST was last
            changed.  This is therefore reset only when a 1-by-1 or
            2-by-2 block deflates off the bottom.

  =====================================================================

     .. Parameters ..
    $                     SAFETY = 1.0E+0 )
        Parameters:
        job -
        compq -
        compz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        t -
        ldt -
        alphar -
        alphai -
        beta -
        q -
        ldq -
        z -
        ldz -
        work -
        lwork -
        info -

      • shsein

        public abstract void shsein(java.lang.String side,
                            java.lang.String eigsrc,
                            java.lang.String initv,
                            boolean[] select,
                            int n,
                            float[] h,
                            int ldh,
                            float[] wr,
                            float[] wi,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            float[] work,
                            int[] ifaill,
                            int[] ifailr,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SHSEIN uses inverse iteration to find specified right and/or left
  eigenvectors of a real upper Hessenberg matrix H.

  The right eigenvector x and the left eigenvector y of the matrix H
  corresponding to an eigenvalue w are defined by:

               H * x = w * x,     y**h * H = w * y**h

  where y**h denotes the conjugate transpose of the vector y.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

  EIGSRC  (input) CHARACTER*1
          Specifies the source of eigenvalues supplied in (WR,WI):
          = 'Q': the eigenvalues were found using SHSEQR; thus, if
                 H has zero subdiagonal elements, and so is
                 block-triangular, then the j-th eigenvalue can be
                 assumed to be an eigenvalue of the block containing
                 the j-th row/column.  This property allows SHSEIN to
                 perform inverse iteration on just one diagonal block.

          = 'N': no assumptions are made on the correspondence
                 between eigenvalues and diagonal blocks.  In this
                 case, SHSEIN must always perform inverse iteration
                 using the whole matrix H.

  INITV   (input) CHARACTER*1
          = 'N': no initial vectors are supplied;
          = 'U': user-supplied initial vectors are stored in the arrays
                 VL and/or VR.

  SELECT  (input/output) LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the
          real eigenvector corresponding to a real eigenvalue WR(j),
          SELECT(j) must be set to .TRUE.. To select the complex
          eigenvector corresponding to a complex eigenvalue
          (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
          .FALSE..

  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  H       (input) REAL array, dimension (LDH,N)
          The upper Hessenberg matrix H.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

  WR      (input/output) REAL array, dimension (N)
  WI      (input) REAL array, dimension (N)
          On entry, the real and imaginary parts of the eigenvalues of

          H; a complex conjugate pair of eigenvalues must be stored in

          consecutive elements of WR and WI.
          On exit, WR may have been altered since close eigenvalues
          are perturbed slightly in searching for independent
          eigenvectors.

  VL      (input/output) REAL array, dimension (LDVL,MM)
          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
          contain starting vectors for the inverse iteration for the
          left eigenvectors; the starting vector for each eigenvector
          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'L' or 'B', the left eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VL, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is

          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'R', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.
          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

  VR      (input/output) REAL array, dimension (LDVR,MM)
          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
          contain starting vectors for the inverse iteration for the
          right eigenvectors; the starting vector for each eigenvector

          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'R' or 'B', the right eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VR, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is

          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'L', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.
          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR required to

          store the eigenvectors; each selected real eigenvector
          occupies one column and each selected complex eigenvector
          occupies two columns.

  WORK    (workspace) REAL array, dimension ((N+2)*N)

  IFAILL  (output) INTEGER array, dimension (MM)
          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
          eigenvector in the i-th column of VL (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VL hold a complex eigenvector, then IFAILL(i) and

          IFAILL(i+1) are set to the same value.
          If SIDE = 'R', IFAILL is not referenced.

  IFAILR  (output) INTEGER array, dimension (MM)
          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
          eigenvector in the i-th column of VR (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VR hold a complex eigenvector, then IFAILR(i) and

          IFAILR(i+1) are set to the same value.
          If SIDE = 'L', IFAILR is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, i is the number of eigenvectors which
                failed to converge; see IFAILL and IFAILR for further
                details.

  Further Details
  ===============

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x|+|y|.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        eigsrc -
        initv -
        select -
        n -
        h -
        ldh -
        wr -
        wi -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        ifaill -
        ifailr -
        info -

      • shseqr

        public abstract void shseqr(java.lang.String job,
                            java.lang.String compz,
                            int n,
                            int ilo,
                            int ihi,
                            float[] h,
                            int ldh,
                            float[] wr,
                            float[] wi,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..
     Purpose
     =======

     SHSEQR computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     JOB   (input) CHARACTER*1
           = 'E':  compute eigenvalues only;
           = 'S':  compute eigenvalues and the Schur form T.

     COMPZ (input) CHARACTER*1
           = 'N':  no Schur vectors are computed;
           = 'I':  Z is initialized to the unit matrix and the matrix Z
                   of Schur vectors of H is returned;
           = 'V':  Z must contain an orthogonal matrix Q on entry, and

                   the product Q*Z is returned.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
           set by a previous call to SGEBAL, and then passed to SGEHRD

           when the matrix output by SGEBAL is reduced to Hessenberg
           form. Otherwise ILO and IHI should be set to 1 and N
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) REAL array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and JOB = 'S', then H contains the
           upper quasi-triangular matrix T from the Schur decomposition
           (the Schur form); 2-by-2 diagonal blocks (corresponding to
           complex conjugate pairs of eigenvalues) are returned in
           standard form, with H(i,i) = H(i+1,i+1) and
           H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
           contents of H are unspecified on exit.  (The output value of
           H when INFO.GT.0 is given under the description of INFO
           below.)

           Unlike earlier versions of SHSEQR, this subroutine may
           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
           or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) REAL array, dimension (N)
     WI    (output) REAL array, dimension (N)
           The real and imaginary parts, respectively, of the computed

           eigenvalues. If two eigenvalues are computed as a complex
           conjugate pair, they are stored in consecutive elements of
           WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
           WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in

           the same order as on the diagonal of the Schur form returned
           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     Z     (input/output) REAL array, dimension (LDZ,N)
           If COMPZ = 'N', Z is not referenced.
           If COMPZ = 'I', on entry Z need not be set and on exit,
           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
           vectors of H.  If COMPZ = 'V', on entry Z must contain an
           N-by-N matrix Q, which is assumed to be equal to the unit
           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
           if INFO = 0, Z contains Q*Z.
           Normally Q is the orthogonal matrix generated by SORGHR
           after the call to SGEHRD which formed the Hessenberg matrix

           H. (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if COMPZ = 'I' or
           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) REAL array, dimension (LWORK)
           On exit, if INFO = 0, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then SHSEQR does a workspace query.
           In this case, SHSEQR checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .LT. 0:  if INFO = -i, the i-th argument had an illegal
                    value
           .GT. 0:  if INFO = i, SHSEQR failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and JOB = 'E', then on exit, the
                remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and JOB   = 'S', then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and COMPZ = 'V', then on exit

                  (final value of Z)  =  (initial value of Z)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)

                If INFO .GT. 0 and COMPZ = 'I', then on exit
                      (final value of Z)  = U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)

                If INFO .GT. 0 and COMPZ = 'N', then Z is not
                accessed.

     ================================================================
             Default values supplied by
             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
             It is suggested that these defaults be adjusted in order
             to attain best performance in each particular
             computational environment.

            ISPEC=1:  The SLAHQR vs SLAQR0 crossover point.
                      Default: 75. (Must be at least 11.)

            ISPEC=2:  Recommended deflation window size.
                      This depends on ILO, IHI and NS.  NS is the
                      number of simultaneous shifts returned
                      by ILAENV(ISPEC=4).  (See ISPEC=4 below.)
                      The default for (IHI-ILO+1).LE.500 is NS.
                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.

            ISPEC=3:  Nibble crossover point. (See ILAENV for
                      details.)  Default: 14% of deflation window
                      size.

            ISPEC=4:  Number of simultaneous shifts, NS, in
                      a multi-shift QR iteration.

                      If IHI-ILO+1 is ...

                      greater than      ...but less    ... the
                      or equal to ...      than        default is

                           1               30          NS -   2(+)
                          30               60          NS -   4(+)
                          60              150          NS =  10(+)
                         150              590          NS =  **
                         590             3000          NS =  64
                        3000             6000          NS = 128
                        6000             infinity      NS = 256

                  (+)  By default some or all matrices of this order 
                       are passed to the implicit double shift routine

                       SLAHQR and NS is ignored.  See ISPEC=1 above 
                       and comments in IPARM for details.

                       The asterisks (**) indicate an ad-hoc
                       function of N increasing from 10 to 64.

            ISPEC=5:  Select structured matrix multiply.
                      (See ILAENV for details.) Default: 3.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    SLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== NL allocates some local workspace to help small matrices
     .    through a rare SLAHQR failure.  NL .GT. NTINY = 11 is
     .    required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom-
     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
     .    allows up to six simultaneous shifts and a 16-by-16
     .    deflation window.  ====
        Parameters:
        job -
        compz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        z -
        ldz -
        work -
        lwork -
        info -

      • sisnan

        public boolean sisnan(float sin)
        
     ..

  Purpose
  =======

  SISNAN returns .TRUE. if its argument is NaN, and .FALSE.
  otherwise.  To be replaced by the Fortran 2003 intrinsic in the
  future.

  Arguments
  =========

  SIN      (input) REAL
          Input to test for NaN.

  =====================================================================

  .. External Functions ..
        Parameters:
        sin -

      • slabad

        public abstract void slabad(org.netlib.util.floatW small,
                            org.netlib.util.floatW large)
        
     ..

  Purpose
  =======

  SLABAD takes as input the values computed by SLAMCH for underflow and
  overflow, and returns the square root of each of these values if the

  log of LARGE is sufficiently large.  This subroutine is intended to
  identify machines with a large exponent range, such as the Crays, and
  redefine the underflow and overflow limits to be the square roots of

  the values computed by SLAMCH.  This subroutine is needed because
  SLAMCH does not compensate for poor arithmetic in the upper half of
  the exponent range, as is found on a Cray.

  Arguments
  =========

  SMALL   (input/output) REAL
          On entry, the underflow threshold as computed by SLAMCH.
          On exit, if LOG10(LARGE) is sufficiently large, the square
          root of SMALL, otherwise unchanged.

  LARGE   (input/output) REAL
          On entry, the overflow threshold as computed by SLAMCH.
          On exit, if LOG10(LARGE) is sufficiently large, the square
          root of LARGE, otherwise unchanged.

  =====================================================================

     .. Intrinsic Functions ..
        Parameters:
        small -
        large -

      • slabrd

        public abstract void slabrd(int m,
                            int n,
                            int nb,
                            float[] a,
                            int lda,
                            float[] d,
                            float[] e,
                            float[] tauq,
                            float[] taup,
                            float[] x,
                            int ldx,
                            float[] y,
                            int ldy)
        
     ..

  Purpose
  =======

  SLABRD reduces the first NB rows and columns of a real general
  m by n matrix A to upper or lower bidiagonal form by an orthogonal
  transformation Q' * A * P, and returns the matrices X and Y which
  are needed to apply the transformation to the unreduced part of A.

  If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower

  bidiagonal form.

  This is an auxiliary routine called by SGEBRD

  Arguments
  =========

  M       (input) INTEGER
          The number of rows in the matrix A.

  N       (input) INTEGER
          The number of columns in the matrix A.

  NB      (input) INTEGER
          The number of leading rows and columns of A to be reduced.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit, the first NB rows and columns of the matrix are
          overwritten; the rest of the array is unchanged.
          If m >= n, elements on and below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the

            array TAUP, represent the orthogonal matrix P as a product

            of elementary reflectors.
          If m < n, elements below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors, and
            elements on and above the diagonal in the first NB rows,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  D       (output) REAL array, dimension (NB)
          The diagonal elements of the first NB rows and columns of
          the reduced matrix.  D(i) = A(i,i).

  E       (output) REAL array, dimension (NB)
          The off-diagonal elements of the first NB rows and columns of
          the reduced matrix.

  TAUQ    (output) REAL array dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

  TAUP    (output) REAL array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

  X       (output) REAL array, dimension (LDX,NB)
          The m-by-nb matrix X required to update the unreduced part
          of A.

  LDX     (input) INTEGER
          The leading dimension of the array X. LDX >= M.

  Y       (output) REAL array, dimension (LDY,NB)
          The n-by-nb matrix Y required to update the unreduced part
          of A.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrices Q and P are represented as products of elementary
  reflectors:

     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

  where tauq and taup are real scalars, and v and u are real vectors.

  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  The elements of the vectors v and u together form the m-by-nb matrix

  V and the nb-by-n matrix U' which are needed, with X and Y, to apply

  the transformation to the unreduced part of the matrix, using a block
  update of the form:  A := A - V*Y' - X*U'.

  The contents of A on exit are illustrated by the following examples
  with nb = 2:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )

  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element

  of the vector defining G(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        nb -
        a -
        lda -
        d -
        e -
        tauq -
        taup -
        x -
        ldx -
        y -
        ldy -

      • slacn2

        public void slacn2(int n,
                   float[] v,
                   float[] x,
                   int[] isgn,
                   org.netlib.util.floatW est,
                   org.netlib.util.intW kase,
                   int[] isave)
        
     ..

  Purpose
  =======

  SLACN2 estimates the 1-norm of a square, real matrix A.
  Reverse communication is used for evaluating matrix-vector products.


  Arguments
  =========

  N      (input) INTEGER
         The order of the matrix.  N >= 1.

  V      (workspace) REAL array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

  X      (input/output) REAL array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A' * X,  if KASE=2,
         and SLACN2 must be re-called with all the other parameters
         unchanged.

  ISGN   (workspace) INTEGER array, dimension (N)

  EST    (input/output) REAL
         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
         unchanged from the previous call to SLACN2.
         On exit, EST is an estimate (a lower bound) for norm(A). 

  KASE   (input/output) INTEGER
         On the initial call to SLACN2, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A' * X.
         On the final return from SLACN2, KASE will again be 0.

  ISAVE  (input/output) INTEGER array, dimension (3)
         ISAVE is used to save variables between calls to SLACN2

  Further Details
  ======= =======

  Contributed by Nick Higham, University of Manchester.
  Originally named SONEST, dated March 16, 1988.

  Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of
  a real or complex matrix, with applications to condition estimation",
  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

  This is a thread safe version of SLACON, which uses the array ISAVE
  in place of a SAVE statement, as follows:

     SLACON     SLACN2
      JUMP     ISAVE(1)
      J        ISAVE(2)
      ITER     ISAVE(3)

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        v -
        x -
        isgn -
        est -
        kase -
        isave -

      • slacon

        public abstract void slacon(int n,
                            float[] v,
                            float[] x,
                            int[] isgn,
                            org.netlib.util.floatW est,
                            org.netlib.util.intW kase)
        
     ..

  Purpose
  =======

  SLACON estimates the 1-norm of a square, real matrix A.
  Reverse communication is used for evaluating matrix-vector products.


  Arguments
  =========

  N      (input) INTEGER
         The order of the matrix.  N >= 1.

  V      (workspace) REAL array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

  X      (input/output) REAL array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A' * X,  if KASE=2,
         and SLACON must be re-called with all the other parameters
         unchanged.

  ISGN   (workspace) INTEGER array, dimension (N)

  EST    (input/output) REAL
         On entry with KASE = 1 or 2 and JUMP = 3, EST should be
         unchanged from the previous call to SLACON.
         On exit, EST is an estimate (a lower bound) for norm(A). 

  KASE   (input/output) INTEGER
         On the initial call to SLACON, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A' * X.
         On the final return from SLACON, KASE will again be 0.

  Further Details
  ======= =======

  Contributed by Nick Higham, University of Manchester.
  Originally named SONEST, dated March 16, 1988.

  Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of
  a real or complex matrix, with applications to condition estimation",
  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        v -
        x -
        isgn -
        est -
        kase -

      • slacpy

        public abstract void slacpy(java.lang.String uplo,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  SLACPY copies all or part of a two-dimensional matrix A to another
  matrix B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies the part of the matrix A to be copied to B.
          = 'U':      Upper triangular part
          = 'L':      Lower triangular part
          Otherwise:  All of the matrix A

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The m by n matrix A.  If UPLO = 'U', only the upper triangle

          or trapezoid is accessed; if UPLO = 'L', only the lower
          triangle or trapezoid is accessed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (output) REAL array, dimension (LDB,N)
          On exit, B = A in the locations specified by UPLO.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        m -
        n -
        a -
        lda -
        b -
        ldb -

      • sladiv

        public abstract void sladiv(float a,
                            float b,
                            float c,
                            float d,
                            org.netlib.util.floatW p,
                            org.netlib.util.floatW q)
        
     ..

  Purpose
  =======

  SLADIV performs complex division in  real arithmetic

                        a + i*b
             p + i*q = ---------
                        c + i*d

  The algorithm is due to Robert L. Smith and can be found
  in D. Knuth, The art of Computer Programming, Vol.2, p.195

  Arguments
  =========

  A       (input) REAL
  B       (input) REAL
  C       (input) REAL
  D       (input) REAL
          The scalars a, b, c, and d in the above expression.

  P       (output) REAL
  Q       (output) REAL
          The scalars p and q in the above expression.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        a -
        b -
        c -
        d -
        p -
        q -

      • slae2

        public abstract void slae2(float a,
                           float b,
                           float c,
                           org.netlib.util.floatW rt1,
                           org.netlib.util.floatW rt2)
        
     ..

  Purpose
  =======

  SLAE2  computes the eigenvalues of a 2-by-2 symmetric matrix
     [  A   B  ]
     [  B   C  ].
  On return, RT1 is the eigenvalue of larger absolute value, and RT2
  is the eigenvalue of smaller absolute value.

  Arguments
  =========

  A       (input) REAL
          The (1,1) element of the 2-by-2 matrix.

  B       (input) REAL
          The (1,2) and (2,1) elements of the 2-by-2 matrix.

  C       (input) REAL
          The (2,2) element of the 2-by-2 matrix.

  RT1     (output) REAL
          The eigenvalue of larger absolute value.

  RT2     (output) REAL
          The eigenvalue of smaller absolute value.

  Further Details
  ===============

  RT1 is accurate to a few ulps barring over/underflow.

  RT2 may be inaccurate if there is massive cancellation in the
  determinant A*C-B*B; higher precision or correctly rounded or
  correctly truncated arithmetic would be needed to compute RT2
  accurately in all cases.

  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  Underflow is harmless if the input data is 0 or exceeds
     underflow_threshold / macheps.

 =====================================================================


     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        rt1 -
        rt2 -

      • slaebz

        public abstract void slaebz(int ijob,
                            int nitmax,
                            int n,
                            int mmax,
                            int minp,
                            int nbmin,
                            float abstol,
                            float reltol,
                            float pivmin,
                            float[] d,
                            float[] e,
                            float[] e2,
                            int[] nval,
                            float[] ab,
                            float[] c,
                            org.netlib.util.intW mout,
                            int[] nab,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAEBZ contains the iteration loops which compute and use the
  function N(w), which is the count of eigenvalues of a symmetric
  tridiagonal matrix T less than or equal to its argument  w.  It
  performs a choice of two types of loops:

  IJOB=1, followed by
  IJOB=2: It takes as input a list of intervals and returns a list of
          sufficiently small intervals whose union contains the same
          eigenvalues as the union of the original intervals.
          The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
          The output interval (AB(j,1),AB(j,2)] will contain
          eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.

  IJOB=3: It performs a binary search in each input interval
          (AB(j,1),AB(j,2)] for a point  w(j)  such that
          N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
          the search.  If such a w(j) is found, then on output
          AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output

          (AB(j,1),AB(j,2)] will be a small interval containing the
          point where N(w) jumps through NVAL(j), unless that point
          lies outside the initial interval.

  Note that the intervals are in all cases half-open intervals,
  i.e., of the form  (a,b] , which includes  b  but not  a .

  To avoid underflow, the matrix should be scaled so that its largest
  element is no greater than  overflow**(1/2) * underflow**(1/4)
  in absolute value.  To assure the most accurate computation
  of small eigenvalues, the matrix should be scaled to be
  not much smaller than that, either.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966

  Note: the arguments are, in general, *not* checked for unreasonable
  values.

  Arguments
  =========

  IJOB    (input) INTEGER
          Specifies what is to be done:
          = 1:  Compute NAB for the initial intervals.
          = 2:  Perform bisection iteration to find eigenvalues of T.
          = 3:  Perform bisection iteration to invert N(w), i.e.,
                to find a point which has a specified number of
                eigenvalues of T to its left.
          Other values will cause SLAEBZ to return with INFO=-1.

  NITMAX  (input) INTEGER
          The maximum number of "levels" of bisection to be
          performed, i.e., an interval of width W will not be made
          smaller than 2^(-NITMAX) * W.  If not all intervals
          have converged after NITMAX iterations, then INFO is set
          to the number of non-converged intervals.

  N       (input) INTEGER
          The dimension n of the tridiagonal matrix T.  It must be at
          least 1.

  MMAX    (input) INTEGER
          The maximum number of intervals.  If more than MMAX intervals
          are generated, then SLAEBZ will quit with INFO=MMAX+1.

  MINP    (input) INTEGER
          The initial number of intervals.  It may not be greater than

          MMAX.

  NBMIN   (input) INTEGER
          The smallest number of intervals that should be processed
          using a vector loop.  If zero, then only the scalar loop
          will be used.

  ABSTOL  (input) REAL
          The minimum (absolute) width of an interval.  When an
          interval is narrower than ABSTOL, or than RELTOL times the
          larger (in magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  This must be at least
          zero.

  RELTOL  (input) REAL
          The minimum relative width of an interval.  When an interval

          is narrower than ABSTOL, or than RELTOL times the larger (in

          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  PIVMIN  (input) REAL
          The minimum absolute value of a "pivot" in the Sturm
          sequence loop.  This *must* be at least  max |e(j)**2| *
          safe_min  and at least safe_min, where safe_min is at least
          the smallest number that can divide one without overflow.

  D       (input) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (input) REAL array, dimension (N)
          The offdiagonal elements of the tridiagonal matrix T in
          positions 1 through N-1.  E(N) is arbitrary.

  E2      (input) REAL array, dimension (N)
          The squares of the offdiagonal elements of the tridiagonal
          matrix T.  E2(N) is ignored.

  NVAL    (input/output) INTEGER array, dimension (MINP)
          If IJOB=1 or 2, not referenced.
          If IJOB=3, the desired values of N(w).  The elements of NVAL

          will be reordered to correspond with the intervals in AB.
          Thus, NVAL(j) on output will not, in general be the same as
          NVAL(j) on input, but it will correspond with the interval
          (AB(j,1),AB(j,2)] on output.

  AB      (input/output) REAL array, dimension (MMAX,2)
          The endpoints of the intervals.  AB(j,1) is  a(j), the left
          endpoint of the j-th interval, and AB(j,2) is b(j), the
          right endpoint of the j-th interval.  The input intervals
          will, in general, be modified, split, and reordered by the
          calculation.

  C       (input/output) REAL array, dimension (MMAX)
          If IJOB=1, ignored.
          If IJOB=2, workspace.
          If IJOB=3, then on input C(j) should be initialized to the
          first search point in the binary search.

  MOUT    (output) INTEGER
          If IJOB=1, the number of eigenvalues in the intervals.
          If IJOB=2 or 3, the number of intervals output.
          If IJOB=3, MOUT will equal MINP.

  NAB     (input/output) INTEGER array, dimension (MMAX,2)
          If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
          If IJOB=2, then on input, NAB(i,j) should be set.  It must
             satisfy the condition:
             N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
             which means that in interval i only eigenvalues
             NAB(i,1)+1,...,NAB(i,2) will be considered.  Usually,
             NAB(i,j)=N(AB(i,j)), from a previous call to SLAEBZ with
             IJOB=1.
             On output, NAB(i,j) will contain
             max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of

             the input interval that the output interval
             (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
             the input values of NAB(k,1) and NAB(k,2).
          If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
             unless N(w) > NVAL(i) for all search points  w , in which

             case NAB(i,1) will not be modified, i.e., the output
             value will be the same as the input value (modulo
             reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)

             for all search points  w , in which case NAB(i,2) will
             not be modified.  Normally, NAB should be set to some
             distinctive value(s) before SLAEBZ is called.

  WORK    (workspace) REAL array, dimension (MMAX)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (MMAX)
          Workspace.

  INFO    (output) INTEGER
          = 0:       All intervals converged.
          = 1--MMAX: The last INFO intervals did not converge.
          = MMAX+1:  More than MMAX intervals were generated.

  Further Details
  ===============

      This routine is intended to be called only by other LAPACK
  routines, thus the interface is less user-friendly.  It is intended
  for two purposes:

  (a) finding eigenvalues.  In this case, SLAEBZ should have one or
      more initial intervals set up in AB, and SLAEBZ should be called

      with IJOB=1.  This sets up NAB, and also counts the eigenvalues.

      Intervals with no eigenvalues would usually be thrown out at
      this point.  Also, if not all the eigenvalues in an interval i
      are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
      For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
      eigenvalue.  SLAEBZ is then called with IJOB=2 and MMAX
      no smaller than the value of MOUT returned by the call with
      IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1
      through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
      tolerance specified by ABSTOL and RELTOL.

  (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
      In this case, start with a Gershgorin interval  (a,b).  Set up
      AB to contain 2 search intervals, both initially (a,b).  One
      NVAL element should contain  f-1  and the other should contain  l
      , while C should contain a and b, resp.  NAB(i,1) should be -1
      and NAB(i,2) should be N+1, to flag an error if the desired
      interval does not lie in (a,b).  SLAEBZ is then called with
      IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals --

      j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
      if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
      >= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and
      N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and
      w(l-r)=...=w(l+k) are handled similarly.

  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        nitmax -
        n -
        mmax -
        minp -
        nbmin -
        abstol -
        reltol -
        pivmin -
        d -
        e -
        e2 -
        nval -
        ab -
        c -
        mout -
        nab -
        work -
        iwork -
        info -

      • slaed0

        public abstract void slaed0(int icompq,
                            int qsiz,
                            int n,
                            float[] d,
                            float[] e,
                            float[] q,
                            int ldq,
                            float[] qstore,
                            int ldqs,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED0 computes all eigenvalues and corresponding eigenvectors of a
  symmetric tridiagonal matrix using the divide and conquer method.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.
          = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                matrix.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D      (input/output) REAL array, dimension (N)
         On entry, the main diagonal of the tridiagonal matrix.
         On exit, its eigenvalues.

  E      (input) REAL array, dimension (N-1)
         The off-diagonal elements of the tridiagonal matrix.
         On exit, E has been destroyed.

  Q      (input/output) REAL array, dimension (LDQ, N)
         On entry, Q must contain an N-by-N orthogonal matrix.
         If ICOMPQ = 0    Q is not referenced.
         If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                          orthogonal matrix used to reduce the full
                          matrix to tridiagonal form corresponding to
                          the subset of the full matrix which is being

                          decomposed at this time.
         If ICOMPQ = 2    On entry, Q will be the identity matrix.
                          On exit, Q contains the eigenvectors of the
                          tridiagonal matrix.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  If eigenvectors are
         desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

  QSTORE (workspace) REAL array, dimension (LDQS, N)
         Referenced only when ICOMPQ = 1.  Used to store parts of
         the eigenvector matrix when the updating matrix multiplies
         take place.

  LDQS   (input) INTEGER
         The leading dimension of the array QSTORE.  If ICOMPQ = 1,
         then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

  WORK   (workspace) REAL array,
         If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                     1 + 3*N + 2*N*lg N + 2*N**2
                     ( lg( N ) = smallest integer k
                                 such that 2^k >= N )
         If ICOMPQ = 2, the dimension of WORK must be at least
                     4*N + N**2.

  IWORK  (workspace) INTEGER array,
         If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                        6 + 6*N + 5*N*lg N.
                        ( lg( N ) = smallest integer k
                                    such that 2^k >= N )
         If ICOMPQ = 2, the dimension of IWORK must be at least
                        3 + 5*N.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        qsiz -
        n -
        d -
        e -
        q -
        ldq -
        qstore -
        ldqs -
        work -
        iwork -
        info -

      • slaed1

        public abstract void slaed1(int n,
                            float[] d,
                            float[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.floatW rho,
                            int cutpnt,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED1 computes the updated eigensystem of a diagonal
  matrix after modification by a rank-one symmetric matrix.  This
  routine is used only for the eigenproblem which requires all
  eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
  the case in which eigenvalues only or eigenvalues and eigenvectors
  of a full symmetric matrix (which was reduced to tridiagonal form)
  are desired.

    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

     where Z = Q'u, u is a vector of length N with ones in the
     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

     The eigenvectors of the original matrix are stored in Q, and the
     eigenvalues are in D.  The algorithm consists of three stages:

        The first stage consists of deflating the size of the problem
        when there are multiple eigenvalues or if there is a zero in
        the Z vector.  For each such occurence the dimension of the
        secular equation problem is reduced by one.  This stage is
        performed by the routine SLAED2.

        The second stage consists of calculating the updated
        eigenvalues. This is done by finding the roots of the secular
        equation via the routine SLAED4 (as called by SLAED3).
        This routine also calculates the eigenvectors of the current
        problem.

        The final stage consists of computing the updated eigenvectors

        directly using the updated eigenvalues.  The eigenvectors for
        the current problem are multiplied with the eigenvectors from
        the overall problem.

  Arguments
  =========

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D      (input/output) REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

  Q      (input/output) REAL array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.


  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input/output) INTEGER array, dimension (N)
         On entry, the permutation which separately sorts the two
         subproblems in D into ascending order.
         On exit, the permutation which will reintegrate the
         subproblems back into sorted order,
         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

  RHO    (input) REAL
         The subdiagonal entry used to create the rank-1 modification.


  CUTPNT (input) INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= CUTPNT <= N/2.

  WORK   (workspace) REAL array, dimension (4*N + N**2)

  IWORK  (workspace) INTEGER array, dimension (4*N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        work -
        iwork -
        info -

      • slaed2

        public abstract void slaed2(org.netlib.util.intW k,
                            int n,
                            int n1,
                            float[] d,
                            float[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.floatW rho,
                            float[] z,
                            float[] dlamda,
                            float[] w,
                            float[] q2,
                            int[] indx,
                            int[] indxc,
                            int[] indxp,
                            int[] coltyp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED2 merges the two sets of eigenvalues together into a single
  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  eigenvalues are close together or if there is a tiny entry in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  Arguments
  =========

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation. 0 <= K <=N.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  N1     (input) INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= N1 <= N/2.

  D      (input/output) REAL array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.
         On exit, D contains the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

  Q      (input/output) REAL array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in
         the two square blocks with corners at (1,1), (N1,N1)
         and (N1+1, N1+1), (N,N).
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input/output) INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have N1 added to their
         values. Destroyed on exit.

  RHO    (input/output) REAL
         On entry, the off-diagonal element associated with the rank-1

         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         SLAED3.

  Z      (input) REAL array, dimension (N)
         On entry, Z contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).
         On exit, the contents of Z have been destroyed by the updating
         process.

  DLAMDA (output) REAL array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         SLAED3 to form the secular equation.

  W      (output) REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector
         which will be passed to SLAED3.

  Q2     (output) REAL array, dimension (N1**2+(N-N1)**2)
         A copy of the first K eigenvectors which will be used by
         SLAED3 in a matrix multiply (SGEMM) to solve for the new
         eigenvectors.

  INDX   (workspace) INTEGER array, dimension (N)
         The permutation used to sort the contents of DLAMDA into
         ascending order.

  INDXC  (output) INTEGER array, dimension (N)
         The permutation used to arrange the columns of the deflated
         Q matrix into three groups:  the first group contains non-zero
         elements only at and above N1, the second contains
         non-zero elements only below N1, and the third is dense.

  INDXP  (workspace) INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end

         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

  COLTYP (workspace/output) INTEGER array, dimension (N)
         During execution, a label which will indicate which of the
         following types a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : dense;
         3 : non-zero in the lower half only;
         4 : deflated.
         On exit, COLTYP(i) is the number of columns of type i,
         for i=1 to 4 only.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        k -
        n -
        n1 -
        d -
        q -
        ldq -
        indxq -
        rho -
        z -
        dlamda -
        w -
        q2 -
        indx -
        indxc -
        indxp -
        coltyp -
        info -

      • slaed3

        public abstract void slaed3(int k,
                            int n,
                            int n1,
                            float[] d,
                            float[] q,
                            int ldq,
                            float rho,
                            float[] dlamda,
                            float[] q2,
                            int[] indx,
                            int[] ctot,
                            float[] w,
                            float[] s,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED3 finds the roots of the secular equation, as defined by the
  values in D, W, and RHO, between 1 and K.  It makes the
  appropriate calls to SLAED4 and then updates the eigenvectors by
  multiplying the matrix of eigenvectors of the pair of eigensystems
  being combined by the matrix of eigenvectors of the K-by-K system
  which is solved here.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  Arguments
  =========

  K       (input) INTEGER
          The number of terms in the rational function to be solved by

          SLAED4.  K >= 0.

  N       (input) INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (deflation may result in N>K).

  N1      (input) INTEGER
          The location of the last eigenvalue in the leading submatrix.
          min(1,N) <= N1 <= N/2.

  D       (output) REAL array, dimension (N)
          D(I) contains the updated eigenvalues for
          1 <= I <= K.

  Q       (output) REAL array, dimension (LDQ,N)
          Initially the first K columns are used as workspace.
          On output the columns 1 to K contain
          the updated eigenvectors.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max(1,N).

  RHO     (input) REAL
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

  DLAMDA  (input/output) REAL array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation. May be changed on output by
          having lowest order bit set to zero on Cray X-MP, Cray Y-MP,

          Cray-2, or Cray C-90, as described above.

  Q2      (input) REAL array, dimension (LDQ2, N)
          The first K columns of this matrix contain the non-deflated
          eigenvectors for the split problem.

  INDX    (input) INTEGER array, dimension (N)
          The permutation used to arrange the columns of the deflated
          Q matrix into three groups (see SLAED2).
          The rows of the eigenvectors found by SLAED4 must be likewise
          permuted before the matrix multiply can take place.

  CTOT    (input) INTEGER array, dimension (4)
          A count of the total number of the various types of columns
          in Q, as described in INDX.  The fourth column type is any
          column which has been deflated.

  W       (input/output) REAL array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector. Destroyed on
          output.

  S       (workspace) REAL array, dimension (N1 + 1)*K
          Will contain the eigenvectors of the repaired matrix which
          will be multiplied by the previously accumulated eigenvectors
          to update the system.

  LDS     (input) INTEGER
          The leading dimension of S.  LDS >= max(1,K).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        k -
        n -
        n1 -
        d -
        q -
        ldq -
        rho -
        dlamda -
        q2 -
        indx -
        ctot -
        w -
        s -
        info -

      • slaed4

        public abstract void slaed4(int n,
                            int i,
                            float[] d,
                            float[] z,
                            float[] delta,
                            float rho,
                            org.netlib.util.floatW dlam,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This subroutine computes the I-th updated eigenvalue of a symmetric
  rank-one modification to a diagonal matrix whose elements are
  given in the array d, and that

             D(i) < D(j)  for  i < j

  and that RHO > 0.  This is arranged by the calling routine, and is
  no loss in generality.  The rank-one modified system is thus

             diag( D )  +  RHO *  Z * Z_transpose.

  where we assume the Euclidean norm of Z is 1.

  The method consists of approximating the rational functions in the
  secular equation by simpler interpolating rational functions.

  Arguments
  =========

  N      (input) INTEGER
         The length of all arrays.

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

  D      (input) REAL array, dimension (N)
         The original eigenvalues.  It is assumed that they are in
         order, D(I) < D(J)  for I < J.

  Z      (input) REAL array, dimension (N)
         The components of the updating vector.

  DELTA  (output) REAL array, dimension (N)
         If N .GT. 2, DELTA contains (D(j) - lambda_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5

         for detail. The vector DELTA contains the information necessar
         to construct the eigenvectors by SLAED3 and SLAED9.

  RHO    (input) REAL
         The scalar in the symmetric updating formula.

  DLAM   (output) REAL
         The computed lambda_I, the I-th updated eigenvalue.

  INFO   (output) INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Internal Parameters
  ===================

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.

            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1

   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
   if we are working with THREE poles!

   MAXIT is the maximum number of iterations allowed for each
   eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        i -
        d -
        z -
        delta -
        rho -
        dlam -
        info -

      • slaed5

        public abstract void slaed5(int i,
                            float[] d,
                            float[] z,
                            float[] delta,
                            float rho,
                            org.netlib.util.floatW dlam)
        
     ..

  Purpose
  =======

  This subroutine computes the I-th eigenvalue of a symmetric rank-one

  modification of a 2-by-2 diagonal matrix

             diag( D )  +  RHO *  Z * transpose(Z) .

  The diagonal elements in the array D are assumed to satisfy

             D(i) < D(j)  for  i < j .

  We also assume RHO > 0 and that the Euclidean norm of the vector
  Z is one.

  Arguments
  =========

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

  D      (input) REAL array, dimension (2)
         The original eigenvalues.  We assume D(1) < D(2).

  Z      (input) REAL array, dimension (2)
         The components of the updating vector.

  DELTA  (output) REAL array, dimension (2)
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

  RHO    (input) REAL
         The scalar in the symmetric updating formula.

  DLAM   (output) REAL
         The computed lambda_I, the I-th updated eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        i -
        d -
        z -
        delta -
        rho -
        dlam -

      • slaed6

        public abstract void slaed6(int kniter,
                            boolean orgati,
                            float rho,
                            float[] d,
                            float[] z,
                            float finit,
                            org.netlib.util.floatW tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED6 computes the positive or negative root (closest to the origin)
  of
                   z(1)        z(2)        z(3)
  f(x) =   rho + --------- + ---------- + ---------
                  d(1)-x      d(2)-x      d(3)-x

  It is assumed that

        if ORGATI = .true. the root is between d(2) and d(3);
        otherwise it is between d(1) and d(2)

  This routine will be called by SLAED4 when necessary. In most cases,

  the root sought is the smallest in magnitude, though it might not be

  in some extremely rare situations.

  Arguments
  =========

  KNITER       (input) INTEGER
               Refer to SLAED4 for its significance.

  ORGATI       (input) LOGICAL
               If ORGATI is true, the needed root is between d(2) and
               d(3); otherwise it is between d(1) and d(2).  See
               SLAED4 for further details.

  RHO          (input) REAL            
               Refer to the equation f(x) above.

  D            (input) REAL array, dimension (3)
               D satisfies d(1) < d(2) < d(3).

  Z            (input) REAL array, dimension (3)
               Each of the elements in z must be positive.

  FINIT        (input) REAL            
               The value of f at 0. It is more accurate than the one
               evaluated inside this routine (if someone wants to do
               so).

  TAU          (output) REAL            
               The root of the equation f(x).

  INFO         (output) INTEGER
               = 0: successful exit
               > 0: if INFO = 1, failure to converge

  Further Details
  ===============

  30/06/99: Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  10/02/03: This version has a few statements commented out for thread 
     (machine parameters are computed on each entry). SJH.

  05/10/06: Modified from a new version of Ren-Cang Li, use
     Gragg-Thornton-Warner cubic convergent scheme for better stability

  =====================================================================

     .. Parameters ..
        Parameters:
        kniter -
        orgati -
        rho -
        d -
        z -
        finit -
        tau -
        info -

      • slaed7

        public abstract void slaed7(int icompq,
                            int n,
                            int qsiz,
                            int tlvls,
                            int curlvl,
                            int curpbm,
                            float[] d,
                            float[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.floatW rho,
                            int cutpnt,
                            float[] qstore,
                            int[] qptr,
                            int[] prmptr,
                            int[] perm,
                            int[] givptr,
                            int[] givcol,
                            float[] givnum,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED7 computes the updated eigensystem of a diagonal
  matrix after modification by a rank-one symmetric matrix. This
  routine is used only for the eigenproblem which requires all
  eigenvalues and optionally eigenvectors of a dense symmetric matrix
  that has been reduced to tridiagonal form.  SLAED1 handles
  the case in which all eigenvalues and eigenvectors of a symmetric
  tridiagonal matrix are desired.

    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

     where Z = Q'u, u is a vector of length N with ones in the
     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

     The eigenvectors of the original matrix are stored in Q, and the
     eigenvalues are in D.  The algorithm consists of three stages:

        The first stage consists of deflating the size of the problem
        when there are multiple eigenvalues or if there is a zero in
        the Z vector.  For each such occurence the dimension of the
        secular equation problem is reduced by one.  This stage is
        performed by the routine SLAED8.

        The second stage consists of calculating the updated
        eigenvalues. This is done by finding the roots of the secular
        equation via the routine SLAED4 (as called by SLAED9).
        This routine also calculates the eigenvectors of the current
        problem.

        The final stage consists of computing the updated eigenvectors

        directly using the updated eigenvalues.  The eigenvectors for
        the current problem are multiplied with the eigenvectors from
        the overall problem.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  TLVLS  (input) INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

  CURLVL (input) INTEGER
         The current level in the overall merge routine,
         0 <= CURLVL <= TLVLS.

  CURPBM (input) INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

  D      (input/output) REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

  Q      (input/output) REAL array, dimension (LDQ, N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.


  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (output) INTEGER array, dimension (N)
         The permutation which will reintegrate the subproblem just
         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
         will be in ascending order.

  RHO    (input) REAL
         The subdiagonal element used to create the rank-1
         modification.

  CUTPNT (input) INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

  QSTORE (input/output) REAL array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.

  QPTR   (input/output) INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.

  PRMPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.

  PERM   (input) INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

  GIVPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)

         indicates the number of Givens rotations.

  GIVCOL (input) INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (input) REAL array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  WORK   (workspace) REAL array, dimension (3*N+QSIZ*N)

  IWORK  (workspace) INTEGER array, dimension (4*N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        n -
        qsiz -
        tlvls -
        curlvl -
        curpbm -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        qstore -
        qptr -
        prmptr -
        perm -
        givptr -
        givcol -
        givnum -
        work -
        iwork -
        info -

      • slaed8

        public abstract void slaed8(int icompq,
                            org.netlib.util.intW k,
                            int n,
                            int qsiz,
                            float[] d,
                            float[] q,
                            int ldq,
                            int[] indxq,
                            org.netlib.util.floatW rho,
                            int cutpnt,
                            float[] z,
                            float[] dlamda,
                            float[] q2,
                            int ldq2,
                            float[] w,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            float[] givnum,
                            int[] indxp,
                            int[] indx,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED8 merges the two sets of eigenvalues together into a single
  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  eigenvalues are close together or if there is a tiny element in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used

                to reduce the original matrix to tridiagonal form.

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  D      (input/output) REAL array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be
         combined.  On exit, the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

  Q      (input/output) REAL array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise,
         on entry, Q contains the eigenvectors of the partially solved

         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

  INDXQ  (input) INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have CUTPNT added to
         their values in order to be accurate.

  RHO    (input/output) REAL
         On entry, the off-diagonal element associated with the rank-1

         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         SLAED3.

  CUTPNT (input) INTEGER
         The location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

  Z      (input) REAL array, dimension (N)
         On entry, Z contains the updating vector (the last row of
         the first sub-eigenvector matrix and the first row of the
         second sub-eigenvector matrix).
         On exit, the contents of Z are destroyed by the updating
         process.

  DLAMDA (output) REAL array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         SLAED3 to form the secular equation.

  Q2     (output) REAL array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         a copy of the first K eigenvectors which will be used by
         SLAED7 in a matrix multiply (SGEMM) to update the new
         eigenvectors.

  LDQ2   (input) INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max(1,N).

  W      (output) REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector and
         will be passed to SLAED3.

  PERM   (output) INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied
         to each eigenblock.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem.

  GIVCOL (output) INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (output) REAL array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  INDXP  (workspace) INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end

         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

  INDX   (workspace) INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending

         order.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        k -
        n -
        qsiz -
        d -
        q -
        ldq -
        indxq -
        rho -
        cutpnt -
        z -
        dlamda -
        q2 -
        ldq2 -
        w -
        perm -
        givptr -
        givcol -
        givnum -
        indxp -
        indx -
        info -

      • slaed9

        public abstract void slaed9(int k,
                            int kstart,
                            int kstop,
                            int n,
                            float[] d,
                            float[] q,
                            int ldq,
                            float rho,
                            float[] dlamda,
                            float[] w,
                            float[] s,
                            int lds,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAED9 finds the roots of the secular equation, as defined by the
  values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
  appropriate calls to SLAED4 and then stores the new matrix of
  eigenvectors for use in calculating the next level of Z vectors.

  Arguments
  =========

  K       (input) INTEGER
          The number of terms in the rational function to be solved by

          SLAED4.  K >= 0.

  KSTART  (input) INTEGER
  KSTOP   (input) INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed.  1 <= KSTART <= KSTOP <= K.

  N       (input) INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (delation may result in N > K).

  D       (output) REAL array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP.

  Q       (workspace) REAL array, dimension (LDQ,N)

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max( 1, N ).

  RHO     (input) REAL
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

  DLAMDA  (input) REAL array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

  W       (input) REAL array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector.

  S       (output) REAL array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system.

  LDS     (input) INTEGER
          The leading dimension of S.  LDS >= max( 1, K ).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        k -
        kstart -
        kstop -
        n -
        d -
        q -
        ldq -
        rho -
        dlamda -
        w -
        s -
        lds -
        info -

      • slaeda

        public abstract void slaeda(int n,
                            int tlvls,
                            int curlvl,
                            int curpbm,
                            int[] prmptr,
                            int[] perm,
                            int[] givptr,
                            int[] givcol,
                            float[] givnum,
                            float[] q,
                            int[] qptr,
                            float[] z,
                            float[] ztemp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAEDA computes the Z vector corresponding to the merge step in the
  CURLVLth step of the merge process with TLVLS steps for the CURPBMth

  problem.

  Arguments
  =========

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  TLVLS  (input) INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

  CURLVL (input) INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.

  CURPBM (input) INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

  PRMPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and incidentally the
         size of the full, non-deflated problem.

  PERM   (input) INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

  GIVPTR (input) INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)

         indicates the number of Givens rotations.

  GIVCOL (input) INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

  GIVNUM (input) REAL array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

  Q      (input) REAL array, dimension (N**2)
         Contains the square eigenblocks from previous levels, the
         starting positions for blocks are given by QPTR.

  QPTR   (input) INTEGER array, dimension (N+2)
         Contains a list of pointers which indicate where in Q an
         eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
         the size of the block.

  Z      (output) REAL array, dimension (N)
         On output this vector contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).

  ZTEMP  (workspace) REAL array, dimension (N)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        tlvls -
        curlvl -
        curpbm -
        prmptr -
        perm -
        givptr -
        givcol -
        givnum -
        q -
        qptr -
        z -
        ztemp -
        info -

      • slaein

        public abstract void slaein(boolean rightv,
                            boolean noinit,
                            int n,
                            float[] h,
                            int ldh,
                            float wr,
                            float wi,
                            float[] vr,
                            float[] vi,
                            float[] b,
                            int ldb,
                            float[] work,
                            float eps3,
                            float smlnum,
                            float bignum,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAEIN uses inverse iteration to find a right or left eigenvector
  corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
  matrix H.

  Arguments
  =========

  RIGHTV   (input) LOGICAL
          = .TRUE. : compute right eigenvector;
          = .FALSE.: compute left eigenvector.

  NOINIT   (input) LOGICAL
          = .TRUE. : no initial vector supplied in (VR,VI).
          = .FALSE.: initial vector supplied in (VR,VI).

  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  H       (input) REAL array, dimension (LDH,N)
          The upper Hessenberg matrix H.

  LDH     (input) INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

  WR      (input) REAL
  WI      (input) REAL
          The real and imaginary parts of the eigenvalue of H whose
          corresponding right or left eigenvector is to be computed.

  VR      (input/output) REAL array, dimension (N)
  VI      (input/output) REAL array, dimension (N)
          On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
          a real starting vector for inverse iteration using the real
          eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
          must contain the real and imaginary parts of a complex
          starting vector for inverse iteration using the complex
          eigenvalue (WR,WI); otherwise VR and VI need not be set.
          On exit, if WI = 0.0 (real eigenvalue), VR contains the
          computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
          VR and VI contain the real and imaginary parts of the
          computed complex eigenvector. The eigenvector is normalized
          so that the component of largest magnitude has magnitude 1;
          here the magnitude of a complex number (x,y) is taken to be
          |x| + |y|.
          VI is not referenced if WI = 0.0.

  B       (workspace) REAL array, dimension (LDB,N)

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= N+1.

  WORK   (workspace) REAL array, dimension (N)

  EPS3    (input) REAL
          A small machine-dependent value which is used to perturb
          close eigenvalues, and to replace zero pivots.

  SMLNUM  (input) REAL
          A machine-dependent value close to the underflow threshold.

  BIGNUM  (input) REAL
          A machine-dependent value close to the overflow threshold.

  INFO    (output) INTEGER
          = 0:  successful exit
          = 1:  inverse iteration did not converge; VR is set to the
                last iterate, and so is VI if WI.ne.0.0.

  =====================================================================

     .. Parameters ..
        Parameters:
        rightv -
        noinit -
        n -
        h -
        ldh -
        wr -
        wi -
        vr -
        vi -
        b -
        ldb -
        work -
        eps3 -
        smlnum -
        bignum -
        info -

      • slaev2

        public abstract void slaev2(float a,
                            float b,
                            float c,
                            org.netlib.util.floatW rt1,
                            org.netlib.util.floatW rt2,
                            org.netlib.util.floatW cs1,
                            org.netlib.util.floatW sn1)
        
     ..

  Purpose
  =======

  SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
     [  A   B  ]
     [  B   C  ].
  On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
  eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
  eigenvector for RT1, giving the decomposition

     [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
     [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].

  Arguments
  =========

  A       (input) REAL
          The (1,1) element of the 2-by-2 matrix.

  B       (input) REAL
          The (1,2) element and the conjugate of the (2,1) element of
          the 2-by-2 matrix.

  C       (input) REAL
          The (2,2) element of the 2-by-2 matrix.

  RT1     (output) REAL
          The eigenvalue of larger absolute value.

  RT2     (output) REAL
          The eigenvalue of smaller absolute value.

  CS1     (output) REAL
  SN1     (output) REAL
          The vector (CS1, SN1) is a unit right eigenvector for RT1.

  Further Details
  ===============

  RT1 is accurate to a few ulps barring over/underflow.

  RT2 may be inaccurate if there is massive cancellation in the
  determinant A*C-B*B; higher precision or correctly rounded or
  correctly truncated arithmetic would be needed to compute RT2
  accurately in all cases.

  CS1 and SN1 are accurate to a few ulps barring over/underflow.

  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  Underflow is harmless if the input data is 0 or exceeds
     underflow_threshold / macheps.

 =====================================================================


     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        rt1 -
        rt2 -
        cs1 -
        sn1 -

      • slaexc

        public abstract void slaexc(boolean wantq,
                            int n,
                            float[] t,
                            int ldt,
                            float[] q,
                            int ldq,
                            int j1,
                            int n1,
                            int n2,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in

  an upper quasi-triangular matrix T by an orthogonal similarity
  transformation.

  T must be in Schur canonical form, that is, block upper triangular
  with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
  has its diagonal elemnts equal and its off-diagonal elements of
  opposite sign.

  Arguments
  =========

  WANTQ   (input) LOGICAL
          = .TRUE. : accumulate the transformation in the matrix Q;
          = .FALSE.: do not accumulate the transformation.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) REAL array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, the updated matrix T, again in Schur canonical form.

  LDT     (input)  INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
          On exit, if WANTQ is .TRUE., the updated matrix Q.
          If WANTQ is .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

  J1      (input) INTEGER
          The index of the first row of the first block T11.

  N1      (input) INTEGER
          The order of the first block T11. N1 = 0, 1 or 2.

  N2      (input) INTEGER
          The order of the second block T22. N2 = 0, 1 or 2.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          = 1: the transformed matrix T would be too far from Schur
               form; the blocks are not swapped and T and Q are
               unchanged.

  =====================================================================

     .. Parameters ..
        Parameters:
        wantq -
        n -
        t -
        ldt -
        q -
        ldq -
        j1 -
        n1 -
        n2 -
        work -
        info -

      • slag2

        public abstract void slag2(float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float safmin,
                           org.netlib.util.floatW scale1,
                           org.netlib.util.floatW scale2,
                           org.netlib.util.floatW wr1,
                           org.netlib.util.floatW wr2,
                           org.netlib.util.floatW wi)
        
     ..

  Purpose
  =======

  SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
  problem  A - w B, with scaling as necessary to avoid over-/underflow.

  The scaling factor "s" results in a modified eigenvalue equation

      s A - w B

  where  s  is a non-negative scaling factor chosen so that  w,  w B,
  and  s A  do not overflow and, if possible, do not underflow, either.

  Arguments
  =========

  A       (input) REAL array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm

          is less than 1/SAFMIN.  Entries less than
          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= 2.

  B       (input) REAL array, dimension (LDB, 2)
          On entry, the 2 x 2 upper triangular matrix B.  It is
          assumed that the one-norm of B is less than 1/SAFMIN.  The
          diagonals should be at least sqrt(SAFMIN) times the largest
          element of B (in absolute value); if a diagonal is smaller
          than that, then  +/- sqrt(SAFMIN) will be used instead of
          that diagonal.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= 2.

  SAFMIN  (input) REAL
          The smallest positive number s.t. 1/SAFMIN does not
          overflow.  (This should always be SLAMCH('S') -- it is an
          argument in order to avoid having to call SLAMCH frequently.)

  SCALE1  (output) REAL
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the first eigenvalue.  If
          the eigenvalues are complex, then the eigenvalues are
          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
          exponent range of the machine), SCALE1=SCALE2, and SCALE1
          will always be positive.  If the eigenvalues are real, then
          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
          overflow or underflow, and in fact, SCALE1 may be zero or
          less than the underflow threshhold if the exact eigenvalue
          is sufficiently large.

  SCALE2  (output) REAL
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the second eigenvalue.  If

          the eigenvalues are complex, then SCALE2=SCALE1.  If the
          eigenvalues are real, then the second (real) eigenvalue is
          WR2 / SCALE2 , but this may overflow or underflow, and in
          fact, SCALE2 may be zero or less than the underflow
          threshhold if the exact eigenvalue is sufficiently large.

  WR1     (output) REAL
          If the eigenvalue is real, then WR1 is SCALE1 times the
          eigenvalue closest to the (2,2) element of A B**(-1).  If the
          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real

          part of the eigenvalues.

  WR2     (output) REAL
          If the eigenvalue is real, then WR2 is SCALE2 times the
          other eigenvalue.  If the eigenvalue is complex, then
          WR1=WR2 is SCALE1 times the real part of the eigenvalues.

  WI      (output) REAL
          If the eigenvalue is real, then WI is zero.  If the
          eigenvalue is complex, then WI is SCALE1 times the imaginary

          part of the eigenvalues.  WI will always be non-negative.

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        lda -
        b -
        ldb -
        safmin -
        scale1 -
        scale2 -
        wr1 -
        wr2 -
        wi -

      • slag2d

        public void slag2d(int m,
                   int n,
                   float[] sa,
                   int ldsa,
                   double[] a,
                   int lda,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAG2D converts a SINGLE PRECISION matrix, SA, to a DOUBLE
  PRECISION matrix, A.

  Note that while it is possible to overflow while converting 
  from double to single, it is not possible to overflow when
  converting from single to double. 

  This is a helper routine so there is no argument checking.

  Arguments
  =========

  M       (input) INTEGER
          The number of lines of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  SA      (output) REAL array, dimension (LDSA,N)
          On exit, the M-by-N coefficient matrix SA.

  LDSA    (input) INTEGER
          The leading dimension of the array SA.  LDSA >= max(1,M).

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N coefficient matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  INFO    (output) INTEGER
          = 0:  successful exit
  =========

     .. Local Scalars ..
        Parameters:
        m -
        n -
        sa -
        ldsa -
        a -
        lda -
        info -

      • slags2

        public abstract void slags2(boolean upper,
                            float a1,
                            float a2,
                            float a3,
                            float b1,
                            float b2,
                            float b3,
                            org.netlib.util.floatW csu,
                            org.netlib.util.floatW snu,
                            org.netlib.util.floatW csv,
                            org.netlib.util.floatW snv,
                            org.netlib.util.floatW csq,
                            org.netlib.util.floatW snq)
        
     ..

  Purpose
  =======

  SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
  that if ( UPPER ) then

            U'*A*Q = U'*( A1 A2 )*Q = ( x  0  )
                        ( 0  A3 )     ( x  x  )
  and
            V'*B*Q = V'*( B1 B2 )*Q = ( x  0  )
                        ( 0  B3 )     ( x  x  )

  or if ( .NOT.UPPER ) then

            U'*A*Q = U'*( A1 0  )*Q = ( x  x  )
                        ( A2 A3 )     ( 0  x  )
  and
            V'*B*Q = V'*( B1 0  )*Q = ( x  x  )
                        ( B2 B3 )     ( 0  x  )

  The rows of the transformed A and B are parallel, where

    U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
        ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )

  Z' denotes the transpose of Z.


  Arguments
  =========

  UPPER   (input) LOGICAL
          = .TRUE.: the input matrices A and B are upper triangular.
          = .FALSE.: the input matrices A and B are lower triangular.

  A1      (input) REAL
  A2      (input) REAL
  A3      (input) REAL
          On entry, A1, A2 and A3 are elements of the input 2-by-2
          upper (lower) triangular matrix A.

  B1      (input) REAL
  B2      (input) REAL
  B3      (input) REAL
          On entry, B1, B2 and B3 are elements of the input 2-by-2
          upper (lower) triangular matrix B.

  CSU     (output) REAL
  SNU     (output) REAL
          The desired orthogonal matrix U.

  CSV     (output) REAL
  SNV     (output) REAL
          The desired orthogonal matrix V.

  CSQ     (output) REAL
  SNQ     (output) REAL
          The desired orthogonal matrix Q.

  =====================================================================

     .. Parameters ..
        Parameters:
        upper -
        a1 -
        a2 -
        a3 -
        b1 -
        b2 -
        b3 -
        csu -
        snu -
        csv -
        snv -
        csq -
        snq -

      • slagtf

        public abstract void slagtf(int n,
                            float[] a,
                            float lambda,
                            float[] b,
                            float[] c,
                            float tol,
                            float[] d,
                            int[] in,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
  tridiagonal matrix and lambda is a scalar, as

     T - lambda*I = PLU,

  where P is a permutation matrix, L is a unit lower tridiagonal matrix
  with at most one non-zero sub-diagonal elements per column and U is
  an upper triangular matrix with at most two non-zero super-diagonal
  elements per column.

  The factorization is obtained by Gaussian elimination with partial
  pivoting and implicit row scaling.

  The parameter LAMBDA is included in the routine so that SLAGTF may
  be used, in conjunction with SLAGTS, to obtain eigenvectors of T by
  inverse iteration.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix T.

  A       (input/output) REAL array, dimension (N)
          On entry, A must contain the diagonal elements of T.

          On exit, A is overwritten by the n diagonal elements of the
          upper triangular matrix U of the factorization of T.

  LAMBDA  (input) REAL
          On entry, the scalar lambda.

  B       (input/output) REAL array, dimension (N-1)
          On entry, B must contain the (n-1) super-diagonal elements of
          T.

          On exit, B is overwritten by the (n-1) super-diagonal
          elements of the matrix U of the factorization of T.

  C       (input/output) REAL array, dimension (N-1)
          On entry, C must contain the (n-1) sub-diagonal elements of
          T.

          On exit, C is overwritten by the (n-1) sub-diagonal elements

          of the matrix L of the factorization of T.

  TOL     (input) REAL
          On entry, a relative tolerance used to indicate whether or
          not the matrix (T - lambda*I) is nearly singular. TOL should

          normally be chose as approximately the largest relative error
          in the elements of T. For example, if the elements of T are
          correct to about 4 significant figures, then TOL should be
          set to about 5*10**(-4). If TOL is supplied as less than eps,
          where eps is the relative machine precision, then the value
          eps is used in place of TOL.

  D       (output) REAL array, dimension (N-2)
          On exit, D is overwritten by the (n-2) second super-diagonal

          elements of the matrix U of the factorization of T.

  IN      (output) INTEGER array, dimension (N)
          On exit, IN contains details of the permutation matrix P. If

          an interchange occurred at the kth step of the elimination,
          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
          returns the smallest positive integer j such that

             abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,

          where norm( A(j) ) denotes the sum of the absolute values of

          the jth row of the matrix A. If no such j exists then IN(n)
          is returned as zero. If IN(n) is returned as positive, then a
          diagonal element of U is small, indicating that
          (T - lambda*I) is singular or nearly singular,

  INFO    (output) INTEGER
          = 0   : successful exit
          .lt. 0: if INFO = -k, the kth argument had an illegal value

 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        a -
        lambda -
        b -
        c -
        tol -
        d -
        in -
        info -

      • slagtm

        public abstract void slagtm(java.lang.String trans,
                            int n,
                            int nrhs,
                            float alpha,
                            float[] dl,
                            float[] d,
                            float[] du,
                            float[] x,
                            int ldx,
                            float beta,
                            float[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  SLAGTM performs a matrix-vector product of the form

     B := alpha * A * X + beta * B

  where A is a tridiagonal matrix of order N, B and X are N by NRHS
  matrices, and alpha and beta are real scalars, each of which may be
  0., 1., or -1.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  No transpose, B := alpha * A * X + beta * B
          = 'T':  Transpose,    B := alpha * A'* X + beta * B
          = 'C':  Conjugate transpose = Transpose

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices X and B.

  ALPHA   (input) REAL
          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
          it is assumed to be 0.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) sub-diagonal elements of T.

  D       (input) REAL array, dimension (N)
          The diagonal elements of T.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) super-diagonal elements of T.

  X       (input) REAL array, dimension (LDX,NRHS)
          The N by NRHS matrix X.
  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(N,1).

  BETA    (input) REAL
          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
          it is assumed to be 1.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix B.
          On exit, B is overwritten by the matrix expression
          B := alpha * A * X + beta * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(N,1).

  =====================================================================

     .. Parameters ..
        Parameters:
        trans -
        n -
        nrhs -
        alpha -
        dl -
        d -
        du -
        x -
        ldx -
        beta -
        b -
        ldb -

      • slagts

        public abstract void slagts(int job,
                            int n,
                            float[] a,
                            float[] b,
                            float[] c,
                            float[] d,
                            int[] in,
                            float[] y,
                            org.netlib.util.floatW tol,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAGTS may be used to solve one of the systems of equations

     (T - lambda*I)*x = y   or   (T - lambda*I)'*x = y,

  where T is an n by n tridiagonal matrix, for x, following the
  factorization of (T - lambda*I) as

     (T - lambda*I) = P*L*U ,

  by routine SLAGTF. The choice of equation to be solved is
  controlled by the argument JOB, and in each case there is an option
  to perturb zero or very small diagonal elements of U, this option
  being intended for use in applications such as inverse iteration.

  Arguments
  =========

  JOB     (input) INTEGER
          Specifies the job to be performed by SLAGTS as follows:
          =  1: The equations  (T - lambda*I)x = y  are to be solved,
                but diagonal elements of U are not to be perturbed.
          = -1: The equations  (T - lambda*I)x = y  are to be solved
                and, if overflow would otherwise occur, the diagonal
                elements of U are to be perturbed. See argument TOL
                below.
          =  2: The equations  (T - lambda*I)'x = y  are to be solved,

                but diagonal elements of U are not to be perturbed.
          = -2: The equations  (T - lambda*I)'x = y  are to be solved
                and, if overflow would otherwise occur, the diagonal
                elements of U are to be perturbed. See argument TOL
                below.

  N       (input) INTEGER
          The order of the matrix T.

  A       (input) REAL array, dimension (N)
          On entry, A must contain the diagonal elements of U as
          returned from SLAGTF.

  B       (input) REAL array, dimension (N-1)
          On entry, B must contain the first super-diagonal elements of
          U as returned from SLAGTF.

  C       (input) REAL array, dimension (N-1)
          On entry, C must contain the sub-diagonal elements of L as
          returned from SLAGTF.

  D       (input) REAL array, dimension (N-2)
          On entry, D must contain the second super-diagonal elements
          of U as returned from SLAGTF.

  IN      (input) INTEGER array, dimension (N)
          On entry, IN must contain details of the matrix P as returned
          from SLAGTF.

  Y       (input/output) REAL array, dimension (N)
          On entry, the right hand side vector y.
          On exit, Y is overwritten by the solution vector x.

  TOL     (input/output) REAL
          On entry, with  JOB .lt. 0, TOL should be the minimum
          perturbation to be made to very small diagonal elements of U.
          TOL should normally be chosen as about eps*norm(U), where eps
          is the relative machine precision, but if TOL is supplied as

          non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
          If  JOB .gt. 0  then TOL is not referenced.

          On exit, TOL is changed as described above, only if TOL is
          non-positive on entry. Otherwise TOL is unchanged.

  INFO    (output) INTEGER
          = 0   : successful exit
          .lt. 0: if INFO = -i, the i-th argument had an illegal value

          .gt. 0: overflow would occur when computing the INFO(th)
                  element of the solution vector x. This can only occur
                  when JOB is supplied as positive and either means
                  that a diagonal element of U is very small, or that
                  the elements of the right-hand side vector y are very
                  large.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        n -
        a -
        b -
        c -
        d -
        in -
        y -
        tol -
        info -

      • slagv2

        public abstract void slagv2(float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] alphar,
                            float[] alphai,
                            float[] beta,
                            org.netlib.util.floatW csl,
                            org.netlib.util.floatW snl,
                            org.netlib.util.floatW csr,
                            org.netlib.util.floatW snr)
        
     ..

  Purpose
  =======

  SLAGV2 computes the Generalized Schur factorization of a real 2-by-2

  matrix pencil (A,B) where B is upper triangular. This routine
  computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
  SNR such that

  1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
     types), then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],

  2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
     then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]

     where b11 >= b22 > 0.


  Arguments
  =========

  A       (input/output) REAL array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.
          On exit, A is overwritten by the ``A-part'' of the
          generalized Schur form.

  LDA     (input) INTEGER
          THe leading dimension of the array A.  LDA >= 2.

  B       (input/output) REAL array, dimension (LDB, 2)
          On entry, the upper triangular 2 x 2 matrix B.
          On exit, B is overwritten by the ``B-part'' of the
          generalized Schur form.

  LDB     (input) INTEGER
          THe leading dimension of the array B.  LDB >= 2.

  ALPHAR  (output) REAL array, dimension (2)
  ALPHAI  (output) REAL array, dimension (2)
  BETA    (output) REAL array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
          be zero.

  CSL     (output) REAL
          The cosine of the left rotation matrix.

  SNL     (output) REAL
          The sine of the left rotation matrix.

  CSR     (output) REAL
          The cosine of the right rotation matrix.

  SNR     (output) REAL
          The sine of the right rotation matrix.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        csl -
        snl -
        csr -
        snr -

      • slahqr

        public abstract void slahqr(boolean wantt,
                            boolean wantz,
                            int n,
                            int ilo,
                            int ihi,
                            float[] h,
                            int ldh,
                            float[] wr,
                            float[] wi,
                            int iloz,
                            int ihiz,
                            float[] z,
                            int ldz,
                            org.netlib.util.intW info)
        
     ..

     Purpose
     =======

     SLAHQR is an auxiliary routine called by SHSEQR to update the
     eigenvalues and Schur decomposition already computed by SHSEQR, by
     dealing with the Hessenberg submatrix in rows and columns ILO to
     IHI.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N       (input) INTEGER
          The order of the matrix H.  N >= 0.

     ILO     (input) INTEGER
     IHI     (input) INTEGER
          It is assumed that H is already upper quasi-triangular in
          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
          ILO = 1). SLAHQR works primarily with the Hessenberg
          submatrix in rows and columns ILO to IHI, but applies
          transformations to all of H if WANTT is .TRUE..
          1 <= ILO <= max(1,IHI); IHI <= N.

     H       (input/output) REAL array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.
          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
          quasi-triangular in rows and columns ILO:IHI, with any
          2-by-2 diagonal blocks in standard form. If INFO is zero
          and WANTT is .FALSE., the contents of H are unspecified on
          exit.  The output state of H if INFO is nonzero is given
          below under the description of INFO.

     LDH     (input) INTEGER
          The leading dimension of the array H. LDH >= max(1,N).

     WR      (output) REAL array, dimension (N)
     WI      (output) REAL array, dimension (N)
          The real and imaginary parts, respectively, of the computed
          eigenvalues ILO to IHI are stored in the corresponding
          elements of WR and WI. If two eigenvalues are computed as a
          complex conjugate pair, they are stored in consecutive
          elements of WR and WI, say the i-th and (i+1)th, with
          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
          eigenvalues are stored in the same order as on the diagonal
          of the Schur form returned in H, with WR(i) = H(i,i), and, if
          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE..
          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

     Z       (input/output) REAL array, dimension (LDZ,N)
          If WANTZ is .TRUE., on entry Z must contain the current
          matrix Z of transformations accumulated by SHSEQR, and on
          exit Z has been updated; transformations are applied only to

          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
          If WANTZ is .FALSE., Z is not referenced.

     LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= max(1,N).

     INFO    (output) INTEGER
           =   0: successful exit
          .GT. 0: If INFO = i, SLAHQR failed to compute all the
                  eigenvalues ILO to IHI in a total of 30 iterations
                  per eigenvalue; elements i+1:ihi of WR and WI
                  contain those eigenvalues which have been
                  successfully computed.

                  If INFO .GT. 0 and WANTT is .FALSE., then on exit,
                  the remaining unconverged eigenvalues are the
                  eigenvalues of the upper Hessenberg matrix rows
                  and columns ILO thorugh INFO of the final, output
                  value of H.

                  If INFO .GT. 0 and WANTT is .TRUE., then on exit
          (*)       (initial value of H)*U  = U*(final value of H)
                  where U is an orthognal matrix.    The final
                  value of H is upper Hessenberg and triangular in
                  rows and columns INFO+1 through IHI.

                  If INFO .GT. 0 and WANTZ is .TRUE., then on exit
                      (final value of Z)  = (initial value of Z)*U
                  where U is the orthogonal matrix in (*)
                  (regardless of the value of WANTT.)

     Further Details
     ===============

     02-96 Based on modifications by
     David Day, Sandia National Laboratory, USA

     12-04 Further modifications by
     Ralph Byers, University of Kansas, USA

       This is a modified version of SLAHQR from LAPACK version 3.0.
       It is (1) more robust against overflow and underflow and
       (2) adopts the more conservative Ahues & Tisseur stopping
       criterion (LAWN 122, 1997).

     =========================================================

     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        info -

      • slahr2

        public void slahr2(int n,
                   int k,
                   int nb,
                   float[] a,
                   int lda,
                   float[] tau,
                   float[] t,
                   int ldt,
                   float[] y,
                   int ldy)
        
     ..

  Purpose
  =======

  SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
  matrix A so that elements below the k-th subdiagonal are zero. The
  reduction is performed by an orthogonal similarity transformation
  Q' * A * Q. The routine returns the matrices V and T which determine

  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.

  This is an auxiliary routine called by SGEHRD.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  K       (input) INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.

  NB      (input) INTEGER
          The number of columns to be reduced.

  A       (input/output) REAL array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a

          product of elementary reflectors. The other columns of A are

          unchanged. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) REAL array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further

          Details.

  T       (output) REAL array, dimension (LDT,NB)
          The upper triangular matrix T.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= NB.

  Y       (output) REAL array, dimension (LDY,NB)
          The n-by-nb matrix Y.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrix Q is represented as a product of nb elementary reflectors


     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix

  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V') * (A - Y*V').

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  This file is a slight modification of LAPACK-3.0's SLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK routine. This function is
  not backward compatible with LAPACK3.0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        k -
        nb -
        a -
        lda -
        tau -
        t -
        ldt -
        y -
        ldy -

      • slahrd

        public abstract void slahrd(int n,
                            int k,
                            int nb,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] t,
                            int ldt,
                            float[] y,
                            int ldy)
        
     ..

  Purpose
  =======

  SLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
  matrix A so that elements below the k-th subdiagonal are zero. The
  reduction is performed by an orthogonal similarity transformation
  Q' * A * Q. The routine returns the matrices V and T which determine

  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.

  This is an OBSOLETE auxiliary routine. 
  This routine will be 'deprecated' in a  future release.
  Please use the new routine SLAHR2 instead.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.

  K       (input) INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.

  NB      (input) INTEGER
          The number of columns to be reduced.

  A       (input/output) REAL array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a

          product of elementary reflectors. The other columns of A are

          unchanged. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  TAU     (output) REAL array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further

          Details.

  T       (output) REAL array, dimension (LDT,NB)
          The upper triangular matrix T.

  LDT     (input) INTEGER
          The leading dimension of the array T.  LDT >= NB.

  Y       (output) REAL array, dimension (LDY,NB)
          The n-by-nb matrix Y.

  LDY     (input) INTEGER
          The leading dimension of the array Y. LDY >= N.

  Further Details
  ===============

  The matrix Q is represented as a product of nb elementary reflectors


     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix

  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V') * (A - Y*V').

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an

  element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        k -
        nb -
        a -
        lda -
        tau -
        t -
        ldt -
        y -
        ldy -

      • slaic1

        public abstract void slaic1(int job,
                            int j,
                            float[] x,
                            float sest,
                            float[] w,
                            float gamma,
                            org.netlib.util.floatW sestpr,
                            org.netlib.util.floatW s,
                            org.netlib.util.floatW c)
        
     ..

  Purpose
  =======

  SLAIC1 applies one step of incremental condition estimation in
  its simplest version:

  Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
  lower triangular matrix L, such that
           twonorm(L*x) = sest
  Then SLAIC1 computes sestpr, s, c such that
  the vector
                  [ s*x ]
           xhat = [  c  ]
  is an approximate singular vector of
                  [ L     0  ]
           Lhat = [ w' gamma ]
  in the sense that
           twonorm(Lhat*xhat) = sestpr.

  Depending on JOB, an estimate for the largest or smallest singular
  value is computed.

  Note that [s c]' and sestpr**2 is an eigenpair of the system

      diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                            [ gamma ]

  where  alpha =  x'*w.

  Arguments
  =========

  JOB     (input) INTEGER
          = 1: an estimate for the largest singular value is computed.

          = 2: an estimate for the smallest singular value is computed.

  J       (input) INTEGER
          Length of X and W

  X       (input) REAL array, dimension (J)
          The j-vector x.

  SEST    (input) REAL
          Estimated singular value of j by j matrix L

  W       (input) REAL array, dimension (J)
          The j-vector w.

  GAMMA   (input) REAL
          The diagonal element gamma.

  SESTPR  (output) REAL
          Estimated singular value of (j+1) by (j+1) matrix Lhat.

  S       (output) REAL
          Sine needed in forming xhat.

  C       (output) REAL
          Cosine needed in forming xhat.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        j -
        x -
        sest -
        w -
        gamma -
        sestpr -
        s -
        c -

      • slaisnan

        public boolean slaisnan(float sin1,
                        float sin2)
        
     ..

  Purpose
  =======

  This routine is not for general use.  It exists solely to avoid
  over-optimization in SISNAN.

  SLAISNAN checks for NaNs by comparing its two arguments for
  inequality.  NaN is the only floating-point value where NaN != NaN
  returns .TRUE.  To check for NaNs, pass the same variable as both
  arguments.

  Strictly speaking, Fortran does not allow aliasing of function
  arguments. So a compiler must assume that the two arguments are
  not the same variable, and the test will not be optimized away.
  Interprocedural or whole-program optimization may delete this
  test.  The ISNAN functions will be replaced by the correct
  Fortran 03 intrinsic once the intrinsic is widely available.

  Arguments
  =========

  SIN1     (input) REAL
  SIN2     (input) REAL
          Two numbers to compare for inequality.

  =====================================================================

  .. Executable Statements ..
        Parameters:
        sin1 -
        sin2 -

      • slaln2

        public abstract void slaln2(boolean ltrans,
                            int na,
                            int nw,
                            float smin,
                            float ca,
                            float[] a,
                            int lda,
                            float d1,
                            float d2,
                            float[] b,
                            int ldb,
                            float wr,
                            float wi,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW scale,
                            org.netlib.util.floatW xnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLALN2 solves a system of the form  (ca A - w D ) X = s B
  or (ca A' - w D) X = s B   with possible scaling ("s") and
  perturbation of A.  (A' means A-transpose.)

  A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
  real diagonal matrix, w is a real or complex value, and X and B are
  NA x 1 matrices -- real if w is real, complex if w is complex.  NA
  may be 1 or 2.

  If w is complex, X and B are represented as NA x 2 matrices,
  the first column of each being the real part and the second
  being the imaginary part.

  "s" is a scaling factor (.LE. 1), computed by SLALN2, which is
  so chosen that X can be computed without overflow.  X is further
  scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
  than overflow.

  If both singular values of (ca A - w D) are less than SMIN,
  SMIN*identity will be used instead of (ca A - w D).  If only one
  singular value is less than SMIN, one element of (ca A - w D) will be
  perturbed enough to make the smallest singular value roughly SMIN.
  If both singular values are at least SMIN, (ca A - w D) will not be
  perturbed.  In any case, the perturbation will be at most some small

  multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
  are computed by infinity-norm approximations, and thus will only be
  correct to a factor of 2 or so.

  Note: all input quantities are assumed to be smaller than overflow
  by a reasonable factor.  (See BIGNUM.)

  Arguments
  ==========

  LTRANS  (input) LOGICAL
          =.TRUE.:  A-transpose will be used.
          =.FALSE.: A will be used (not transposed.)

  NA      (input) INTEGER
          The size of the matrix A.  It may (only) be 1 or 2.

  NW      (input) INTEGER
          1 if "w" is real, 2 if "w" is complex.  It may only be 1
          or 2.

  SMIN    (input) REAL
          The desired lower bound on the singular values of A.  This
          should be a safe distance away from underflow or overflow,
          say, between (underflow/machine precision) and  (machine
          precision * overflow ).  (See BIGNUM and ULP.)

  CA      (input) REAL
          The coefficient c, which A is multiplied by.

  A       (input) REAL array, dimension (LDA,NA)
          The NA x NA matrix A.

  LDA     (input) INTEGER
          The leading dimension of A.  It must be at least NA.

  D1      (input) REAL
          The 1,1 element in the diagonal matrix D.

  D2      (input) REAL
          The 2,2 element in the diagonal matrix D.  Not used if NW=1.


  B       (input) REAL array, dimension (LDB,NW)
          The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
          complex), column 1 contains the real part of B and column 2
          contains the imaginary part.

  LDB     (input) INTEGER
          The leading dimension of B.  It must be at least NA.

  WR      (input) REAL
          The real part of the scalar "w".

  WI      (input) REAL
          The imaginary part of the scalar "w".  Not used if NW=1.

  X       (output) REAL array, dimension (LDX,NW)
          The NA x NW matrix X (unknowns), as computed by SLALN2.
          If NW=2 ("w" is complex), on exit, column 1 will contain
          the real part of X and column 2 will contain the imaginary
          part.

  LDX     (input) INTEGER
          The leading dimension of X.  It must be at least NA.

  SCALE   (output) REAL
          The scale factor that B must be multiplied by to insure
          that overflow does not occur when computing X.  Thus,
          (ca A - w D) X  will be SCALE*B, not B (ignoring
          perturbations of A.)  It will be at most 1.

  XNORM   (output) REAL
          The infinity-norm of X, when X is regarded as an NA x NW
          real matrix.

  INFO    (output) INTEGER
          An error flag.  It will be set to zero if no error occurs,
          a negative number if an argument is in error, or a positive
          number if  ca A - w D  had to be perturbed.
          The possible values are:
          = 0: No error occurred, and (ca A - w D) did not have to be
                 perturbed.
          = 1: (ca A - w D) had to be perturbed to make its smallest
               (or only) singular value greater than SMIN.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltrans -
        na -
        nw -
        smin -
        ca -
        a -
        lda -
        d1 -
        d2 -
        b -
        ldb -
        wr -
        wi -
        x -
        ldx -
        scale -
        xnorm -
        info -

      • slals0

        public abstract void slals0(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            int nrhs,
                            float[] b,
                            int ldb,
                            float[] bx,
                            int ldbx,
                            int[] perm,
                            int givptr,
                            int[] givcol,
                            int ldgcol,
                            float[] givnum,
                            int ldgnum,
                            float[] poles,
                            float[] difl,
                            float[] difr,
                            float[] z,
                            int k,
                            float c,
                            float s,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLALS0 applies back the multiplying factors of either the left or the
  right singular vector matrix of a diagonal matrix appended by a row
  to the right hand side matrix B in solving the least squares problem

  using the divide-and-conquer SVD approach.

  For the left singular vector matrix, three types of orthogonal
  matrices are involved:

  (1L) Givens rotations: the number of such rotations is GIVPTR; the
       pairs of columns/rows they were applied to are stored in GIVCOL;
       and the C- and S-values of these rotations are stored in GIVNUM.

  (2L) Permutation. The (NL+1)-st row of B is to be moved to the first

       row, and for J=2:N, PERM(J)-th row of B is to be moved to the
       J-th row.

  (3L) The left singular vector matrix of the remaining matrix.

  For the right singular vector matrix, four types of orthogonal
  matrices are involved:

  (1R) The right singular vector matrix of the remaining matrix.

  (2R) If SQRE = 1, one extra Givens rotation to generate the right
       null space.

  (3R) The inverse transformation of (2L).

  (4R) The inverse transformation of (1L).

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Left singular vector matrix.
         = 1: Right singular vector matrix.

  NL     (input) INTEGER
         The row dimension of the upper block. NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block. NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  NRHS   (input) INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

  B      (input/output) REAL array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M. On output, B contains
         the solution X in rows 1 through N.

  LDB    (input) INTEGER
         The leading dimension of B. LDB must be at least
         max(1,MAX( M, N ) ).

  BX     (workspace) REAL array, dimension ( LDBX, NRHS )

  LDBX   (input) INTEGER
         The leading dimension of BX.

  PERM   (input) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) applied
         to the two blocks.

  GIVPTR (input) INTEGER
         The number of Givens rotations which took place in this
         subproblem.

  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of rows/columns
         involved in a Givens rotation.

  LDGCOL (input) INTEGER
         The leading dimension of GIVCOL, must be at least N.

  GIVNUM (input) REAL array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value used in the
         corresponding Givens rotation.

  LDGNUM (input) INTEGER
         The leading dimension of arrays DIFR, POLES and
         GIVNUM, must be at least K.

  POLES  (input) REAL array, dimension ( LDGNUM, 2 )
         On entry, POLES(1:K, 1) contains the new singular
         values obtained from solving the secular equation, and
         POLES(1:K, 2) is an array containing the poles in the secular

         equation.

  DIFL   (input) REAL array, dimension ( K ).
         On entry, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

  DIFR   (input) REAL array, dimension ( LDGNUM, 2 ).
         On entry, DIFR(I, 1) contains the distances between I-th
         updated (undeflated) singular value and the I+1-th
         (undeflated) old singular value. And DIFR(I, 2) is the
         normalizing factor for the I-th right singular vector.

  Z      (input) REAL array, dimension ( K )
         Contain the components of the deflation-adjusted updating row

         vector.

  K      (input) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  C      (input) REAL
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (input) REAL
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  WORK   (workspace) REAL array, dimension ( K )

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        nrhs -
        b -
        ldb -
        bx -
        ldbx -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        poles -
        difl -
        difr -
        z -
        k -
        c -
        s -
        work -
        info -

      • slalsa

        public abstract void slalsa(int icompq,
                            int smlsiz,
                            int n,
                            int nrhs,
                            float[] b,
                            int ldb,
                            float[] bx,
                            int ldbx,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int[] k,
                            float[] difl,
                            float[] difr,
                            float[] z,
                            float[] poles,
                            int[] givptr,
                            int[] givcol,
                            int ldgcol,
                            int[] perm,
                            float[] givnum,
                            float[] c,
                            float[] s,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLALSA is an itermediate step in solving the least squares problem
  by computing the SVD of the coefficient matrix in compact form (The
  singular vectors are computed as products of simple orthorgonal
  matrices.).

  If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
  matrix of an upper bidiagonal matrix to the right hand side; and if
  ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
  right hand side. The singular vector matrices were generated in
  compact form by SLALSA.

  Arguments
  =========


  ICOMPQ (input) INTEGER
         Specifies whether the left or the right singular vector
         matrix is involved.
         = 0: Left singular vector matrix
         = 1: Right singular vector matrix

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The row and column dimensions of the upper bidiagonal matrix.


  NRHS   (input) INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

  B      (input/output) REAL array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M.
         On output, B contains the solution X in rows 1 through N.

  LDB    (input) INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,MAX( M, N ) ).

  BX     (output) REAL array, dimension ( LDBX, NRHS )
         On exit, the result of applying the left or right singular
         vector matrix to B.

  LDBX   (input) INTEGER
         The leading dimension of BX.

  U      (input) REAL array, dimension ( LDU, SMLSIZ ).
         On entry, U contains the left singular vector matrices of all

         subproblems at the bottom level.

  LDU    (input) INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR,
         POLES, GIVNUM, and Z.

  VT     (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
         On entry, VT' contains the right singular vector matrices of
         all subproblems at the bottom level.

  K      (input) INTEGER array, dimension ( N ).

  DIFL   (input) REAL array, dimension ( LDU, NLVL ).
         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

  DIFR   (input) REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
         distances between singular values on the I-th level and
         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
         record the normalizing factors of the right singular vectors
         matrices of subproblems on I-th level.

  Z      (input) REAL array, dimension ( LDU, NLVL ).
         On entry, Z(1, I) contains the components of the deflation-
         adjusted updating row vector for subproblems on the I-th
         level.

  POLES  (input) REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
         singular values involved in the secular equations on the I-th

         level.

  GIVPTR (input) INTEGER array, dimension ( N ).
         On entry, GIVPTR( I ) records the number of Givens
         rotations performed on the I-th problem on the computation
         tree.

  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the

         locations of Givens rotations performed on the I-th level on
         the computation tree.

  LDGCOL (input) INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

  PERM   (input) INTEGER array, dimension ( LDGCOL, NLVL ).
         On entry, PERM(*, I) records permutations done on the I-th
         level of the computation tree.

  GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
         values of Givens rotations performed on the I-th level on the

         computation tree.

  C      (input) REAL array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  S      (input) REAL array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         S( I ) contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  WORK   (workspace) REAL array.
         The dimension must be at least N.

  IWORK  (workspace) INTEGER array.
         The dimension must be at least 3 * N

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        smlsiz -
        n -
        nrhs -
        b -
        ldb -
        bx -
        ldbx -
        u -
        ldu -
        vt -
        k -
        difl -
        difr -
        z -
        poles -
        givptr -
        givcol -
        ldgcol -
        perm -
        givnum -
        c -
        s -
        work -
        iwork -
        info -

      • slalsd

        public abstract void slalsd(java.lang.String uplo,
                            int smlsiz,
                            int n,
                            int nrhs,
                            float[] d,
                            float[] e,
                            float[] b,
                            int ldb,
                            float rcond,
                            org.netlib.util.intW rank,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLALSD uses the singular value decomposition of A to solve the least

  squares problem of finding X to minimize the Euclidean norm of each
  column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
  are N-by-NRHS. The solution X overwrites B.

  The singular values of A smaller than RCOND times the largest
  singular value are treated as zero in solving the least squares
  problem; in this case a minimum norm solution is returned.
  The actual singular values are returned in D in ascending order.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  Arguments
  =========

  UPLO   (input) CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

  NRHS   (input) INTEGER
         The number of columns of B. NRHS must be at least 1.

  D      (input/output) REAL array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.


  E      (input/output) REAL array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.

         On exit, E has been destroyed.

  B      (input/output) REAL array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

  LDB    (input) INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

  RCOND  (input) REAL
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

  RANK   (output) INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

  WORK   (workspace) REAL array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

  IWORK  (workspace) INTEGER array, dimension at least
         (3*N*NLVL + 11*N)

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute an singular value while

               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Ren-Cang Li, Computer Science Division, University of

       California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        smlsiz -
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        rcond -
        rank -
        work -
        iwork -
        info -

      • slamrg

        public abstract void slamrg(int n1,
                            int n2,
                            float[] a,
                            int strd1,
                            int strd2,
                            int[] index)
        
     ..

  Purpose
  =======

  SLAMRG will create a permutation list which will merge the elements
  of A (which is composed of two independently sorted sets) into a
  single set which is sorted in ascending order.

  Arguments
  =========

  N1     (input) INTEGER
  N2     (input) INTEGER
         These arguements contain the respective lengths of the two
         sorted lists to be merged.

  A      (input) REAL array, dimension (N1+N2)
         The first N1 elements of A contain a list of numbers which
         are sorted in either ascending or descending order.  Likewise

         for the final N2 elements.

  STRD1  (input) INTEGER
  STRD2  (input) INTEGER
         These are the strides to be taken through the array A.
         Allowable strides are 1 and -1.  They indicate whether a
         subset of A is sorted in ascending (STRDx = 1) or descending
         (STRDx = -1) order.

  INDEX  (output) INTEGER array, dimension (N1+N2)
         On exit this array will contain a permutation such that
         if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
         sorted in ascending order.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n1 -
        n2 -
        a -
        strd1 -
        strd2 -
        index -

      • slaneg

        public int slaneg(int n,
                  float[] d,
                  float[] lld,
                  float sigma,
                  float pivmin,
                  int r)
        
     ..

  Purpose
  =======

  SLANEG computes the Sturm count, the number of negative pivots
  encountered while factoring tridiagonal T - sigma I = L D L^T.
  This implementation works directly on the factors without forming
  the tridiagonal matrix T.  The Sturm count is also the number of
  eigenvalues of T less than sigma.

  This routine is called from SLARRB.

  The current routine does not use the PIVMIN parameter but rather
  requires IEEE-754 propagation of Infinities and NaNs.  This
  routine also has no input range restrictions but does require
  default exception handling such that x/0 produces Inf when x is
  non-zero, and Inf/Inf produces NaN.  For more information, see:

    Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
    Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on
    Scientific Computing, v28, n5, 2006.  DOI 10.1137/050641624
    (Tech report version in LAWN 172 with the same title.)

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) REAL             array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  LLD     (input) REAL             array, dimension (N-1)
          The (N-1) elements L(i)*L(i)*D(i).

  SIGMA   (input) REAL            
          Shift amount in T - sigma I = L D L^T.

  PIVMIN  (input) REAL            
          The minimum pivot in the Sturm sequence.  May be used
          when zero pivots are encountered on non-IEEE-754
          architectures.

  R       (input) INTEGER
          The twist index for the twisted factorization that is used
          for the negcount.

  Further Details
  ===============

  Based on contributions by
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Jason Riedy, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        lld -
        sigma -
        pivmin -
        r -

      • slangb

        public float slangb(java.lang.String norm,
                    int n,
                    int kl,
                    int ku,
                    float[] ab,
                    int ldab,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANGB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n band matrix  A,  with kl sub-diagonals and ku super-diagonals.

  Description
  ===========

  SLANGB returns the value

     SLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANGB as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANGB is
          set to zero.

  KL      (input) INTEGER
          The number of sub-diagonals of the matrix A.  KL >= 0.

  KU      (input) INTEGER
          The number of super-diagonals of the matrix A.  KU >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================



     .. Parameters ..
        Parameters:
        norm -
        n -
        kl -
        ku -
        ab -
        ldab -
        work -

      • slange

        public float slange(java.lang.String norm,
                    int m,
                    int n,
                    float[] a,
                    int lda,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANGE  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real matrix A.

  Description
  ===========

  SLANGE returns the value

     SLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANGE as described
          above.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.  When M = 0,
          SLANGE is set to zero.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.  When N = 0,

          SLANGE is set to zero.

  A       (input) REAL array, dimension (LDA,N)
          The m by n matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        m -
        n -
        a -
        lda -
        work -

      • slangt

        public float slangt(java.lang.String norm,
                    int n,
                    float[] dl,
                    float[] d,
                    float[] du)
        
     ..

  Purpose
  =======

  SLANGT  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real tridiagonal matrix A.

  Description
  ===========

  SLANGT returns the value

     SLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANGT as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANGT is
          set to zero.

  DL      (input) REAL array, dimension (N-1)
          The (n-1) sub-diagonal elements of A.

  D       (input) REAL array, dimension (N)
          The diagonal elements of A.

  DU      (input) REAL array, dimension (N-1)
          The (n-1) super-diagonal elements of A.

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        dl -
        d -
        du -

      • slanhs

        public float slanhs(java.lang.String norm,
                    int n,
                    float[] a,
                    int lda,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANHS  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  Hessenberg matrix A.

  Description
  ===========

  SLANHS returns the value

     SLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANHS as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANHS is
          set to zero.

  A       (input) REAL array, dimension (LDA,N)
          The n by n upper Hessenberg matrix A; the part of A below the
          first sub-diagonal is not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        n -
        a -
        lda -
        work -

      • slansb

        public float slansb(java.lang.String norm,
                    java.lang.String uplo,
                    int n,
                    int k,
                    float[] ab,
                    int ldab,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANSB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n symmetric band matrix A,  with k super-diagonals.

  Description
  ===========

  SLANSB returns the value

     SLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANSB as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          band matrix A is supplied.
          = 'U':  Upper triangular part is supplied
          = 'L':  Lower triangular part is supplied

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANSB is
          set to zero.

  K       (input) INTEGER
          The number of super-diagonals or sub-diagonals of the
          band matrix A.  K >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first K+1 rows of AB.  The j-th column of A is

          stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        k -
        ab -
        ldab -
        work -

      • slansp

        public float slansp(java.lang.String norm,
                    java.lang.String uplo,
                    int n,
                    float[] ap,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANSP  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric matrix A,  supplied in packed form.

  Description
  ===========

  SLANSP returns the value

     SLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANSP as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is supplied.
          = 'U':  Upper triangular part of A is supplied
          = 'L':  Lower triangular part of A is supplied

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANSP is
          set to zero.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        ap -
        work -

      • slanst

        public float slanst(java.lang.String norm,
                    int n,
                    float[] d,
                    float[] e)
        
     ..

  Purpose
  =======

  SLANST  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric tridiagonal matrix A.

  Description
  ===========

  SLANST returns the value

     SLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANST as described
          above.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANST is
          set to zero.

  D       (input) REAL array, dimension (N)
          The diagonal elements of A.

  E       (input) REAL array, dimension (N-1)
          The (n-1) sub-diagonal or super-diagonal elements of A.

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        n -
        d -
        e -

      • slansy

        public float slansy(java.lang.String norm,
                    java.lang.String uplo,
                    int n,
                    float[] a,
                    int lda,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANSY  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  real symmetric matrix A.

  Description
  ===========

  SLANSY returns the value

     SLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANSY as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is to be referenced.
          = 'U':  Upper triangular part of A is referenced
          = 'L':  Lower triangular part of A is referenced

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANSY is
          set to zero.

  A       (input) REAL array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading n by n
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading n by n lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        n -
        a -
        lda -
        work -

      • slantb

        public float slantb(java.lang.String norm,
                    java.lang.String uplo,
                    java.lang.String diag,
                    int n,
                    int k,
                    float[] ab,
                    int ldab,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANTB  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the element of  largest absolute value  of an
  n by n triangular band matrix A,  with ( k + 1 ) diagonals.

  Description
  ===========

  SLANTB returns the value

     SLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANTB as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANTB is
          set to zero.

  K       (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals of the matrix A if UPLO = 'L'.
          K >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first k+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
          Note that when DIAG = 'U', the elements of the array AB
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        k -
        ab -
        ldab -
        work -

      • slantp

        public float slantp(java.lang.String norm,
                    java.lang.String uplo,
                    java.lang.String diag,
                    int n,
                    float[] ap,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANTP  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  triangular matrix A, supplied in packed form.

  Description
  ===========

  SLANTP returns the value

     SLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANTP as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, SLANTP is
          set to zero.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          Note that when DIAG = 'U', the elements of the array AP
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        ap -
        work -

      • slantr

        public float slantr(java.lang.String norm,
                    java.lang.String uplo,
                    java.lang.String diag,
                    int m,
                    int n,
                    float[] a,
                    int lda,
                    float[] work)
        
     ..

  Purpose
  =======

  SLANTR  returns the value of the one norm,  or the Frobenius norm, or
  the  infinity norm,  or the  element of  largest absolute value  of a
  trapezoidal or triangular matrix A.

  Description
  ===========

  SLANTR returns the value

     SLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
              (
              ( norm1(A),         NORM = '1', 'O' or 'o'
              (
              ( normI(A),         NORM = 'I' or 'i'
              (
              ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

  where  norm1  denotes the  one norm of a matrix (maximum column sum),
  normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
  normF  denotes the  Frobenius norm of a matrix (square root of sum of
  squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix no

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies the value to be returned in SLANTR as described
          above.

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower trapezoidal.
          = 'U':  Upper trapezoidal
          = 'L':  Lower trapezoidal
          Note that A is triangular instead of trapezoidal if M = N.

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A has unit diagonal.
          = 'N':  Non-unit diagonal
          = 'U':  Unit diagonal

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0, and if
          UPLO = 'U', M <= N.  When M = 0, SLANTR is set to zero.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0, and if
          UPLO = 'L', N <= M.  When N = 0, SLANTR is set to zero.

  A       (input) REAL array, dimension (LDA,N)
          The trapezoidal matrix A (A is triangular if M = N).
          If UPLO = 'U', the leading m by n upper trapezoidal part of
          the array A contains the upper trapezoidal matrix, and the
          strictly lower triangular part of A is not referenced.
          If UPLO = 'L', the leading m by n lower trapezoidal part of
          the array A contains the lower trapezoidal matrix, and the
          strictly upper triangular part of A is not referenced.  Note

          that when DIAG = 'U', the diagonal elements of A are not
          referenced and are assumed to be one.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.

 =====================================================================


     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        m -
        n -
        a -
        lda -
        work -

      • slanv2

        public abstract void slanv2(org.netlib.util.floatW a,
                            org.netlib.util.floatW b,
                            org.netlib.util.floatW c,
                            org.netlib.util.floatW d,
                            org.netlib.util.floatW rt1r,
                            org.netlib.util.floatW rt1i,
                            org.netlib.util.floatW rt2r,
                            org.netlib.util.floatW rt2i,
                            org.netlib.util.floatW cs,
                            org.netlib.util.floatW sn)
        
     ..

  Purpose
  =======

  SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
  matrix in standard form:

       [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
       [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]

  where either
  1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
  2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
  conjugate eigenvalues.

  Arguments
  =========

  A       (input/output) REAL            
  B       (input/output) REAL            
  C       (input/output) REAL            
  D       (input/output) REAL            
          On entry, the elements of the input matrix.
          On exit, they are overwritten by the elements of the
          standardised Schur form.

  RT1R    (output) REAL 
  RT1I    (output) REAL            
  RT2R    (output) REAL            
  RT2I    (output) REAL            
          The real and imaginary parts of the eigenvalues. If the
          eigenvalues are a complex conjugate pair, RT1I > 0.

  CS      (output) REAL            
  SN      (output) REAL            
          Parameters of the rotation matrix.

  Further Details
  ===============

  Modified by V. Sima, Research Institute for Informatics, Bucharest,
  Romania, to reduce the risk of cancellation errors,
  when computing real eigenvalues, and to ensure, if possible, that
  abs(RT1R) >= abs(RT2R).

  =====================================================================

     .. Parameters ..
        Parameters:
        a -
        b -
        c -
        d -
        rt1r -
        rt1i -
        rt2r -
        rt2i -
        cs -
        sn -

      • slapll

        public abstract void slapll(int n,
                            float[] x,
                            int incx,
                            float[] y,
                            int incy,
                            org.netlib.util.floatW ssmin)
        
     ..

  Purpose
  =======

  Given two column vectors X and Y, let

                       A = ( X Y ).

  The subroutine first computes the QR factorization of A = Q*R,
  and then computes the SVD of the 2-by-2 upper triangular matrix R.
  The smaller singular value of R is returned in SSMIN, which is used
  as the measurement of the linear dependency of the vectors X and Y.

  Arguments
  =========

  N       (input) INTEGER
          The length of the vectors X and Y.

  X       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          On entry, X contains the N-vector X.
          On exit, X is overwritten.

  INCX    (input) INTEGER
          The increment between successive elements of X. INCX > 0.

  Y       (input/output) REAL array,
                         dimension (1+(N-1)*INCY)
          On entry, Y contains the N-vector Y.
          On exit, Y is overwritten.

  INCY    (input) INTEGER
          The increment between successive elements of Y. INCY > 0.

  SSMIN   (output) REAL
          The smallest singular value of the N-by-2 matrix A = ( X Y ).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        ssmin -

      • slapmt

        public abstract void slapmt(boolean forwrd,
                            int m,
                            int n,
                            float[] x,
                            int ldx,
                            int[] k)
        
     ..

  Purpose
  =======

  SLAPMT rearranges the columns of the M by N matrix X as specified
  by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
  If FORWRD = .TRUE.,  forward permutation:

       X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.

  If FORWRD = .FALSE., backward permutation:

       X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

  Arguments
  =========

  FORWRD  (input) LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

  M       (input) INTEGER
          The number of rows of the matrix X. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix X. N >= 0.

  X       (input/output) REAL array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X, LDX >= MAX(1,M).

  K       (input/output) INTEGER array, dimension (N)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        forwrd -
        m -
        n -
        x -
        ldx -
        k -

      • slapy2

        public float slapy2(float x,
                    float y)
        
     ..

  Purpose
  =======

  SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary

  overflow.

  Arguments
  =========

  X       (input) REAL
  Y       (input) REAL
          X and Y specify the values x and y.

  =====================================================================

     .. Parameters ..
        Parameters:
        x -
        y -

      • slapy3

        public float slapy3(float x,
                    float y,
                    float z)
        
     ..

  Purpose
  =======

  SLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause
  unnecessary overflow.

  Arguments
  =========

  X       (input) REAL
  Y       (input) REAL
  Z       (input) REAL
          X, Y and Z specify the values x, y and z.

  =====================================================================

     .. Parameters ..
        Parameters:
        x -
        y -
        z -

      • slaqgb

        public abstract void slaqgb(int m,
                            int n,
                            int kl,
                            int ku,
                            float[] ab,
                            int ldab,
                            float[] r,
                            float[] c,
                            float rowcnd,
                            float colcnd,
                            float amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  SLAQGB equilibrates a general M by N band matrix A with KL
  subdiagonals and KU superdiagonals using the row and scaling factors

  in the vectors R and C.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.


  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, the equilibrated matrix, in the same storage format

          as A.  See EQUED for the form of the equilibrated matrix.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDA >= KL+KU+1.

  R       (input) REAL array, dimension (M)
          The row scale factors for A.

  C       (input) REAL array, dimension (N)
          The column scale factors for A.

  ROWCND  (input) REAL
          Ratio of the smallest R(i) to the largest R(i).

  COLCND  (input) REAL
          Ratio of the smallest C(i) to the largest C(i).

  AMAX    (input) REAL
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if row or column scaling
  should be done based on the ratio of the row or column scaling
  factors.  If ROWCND < THRESH, row scaling is done, and if
  COLCND < THRESH, column scaling is done.

  LARGE and SMALL are threshold values used to decide if row scaling
  should be done based on the absolute size of the largest matrix
  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        kl -
        ku -
        ab -
        ldab -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        equed -

      • slaqge

        public abstract void slaqge(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] r,
                            float[] c,
                            float rowcnd,
                            float colcnd,
                            float amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  SLAQGE equilibrates a general M by N matrix A using the row and
  column scaling factors in the vectors R and C.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M by N matrix A.
          On exit, the equilibrated matrix.  See EQUED for the form of

          the equilibrated matrix.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

  R       (input) REAL array, dimension (M)
          The row scale factors for A.

  C       (input) REAL array, dimension (N)
          The column scale factors for A.

  ROWCND  (input) REAL
          Ratio of the smallest R(i) to the largest R(i).

  COLCND  (input) REAL
          Ratio of the smallest C(i) to the largest C(i).

  AMAX    (input) REAL
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration
          = 'R':  Row equilibration, i.e., A has been premultiplied by

                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if row or column scaling
  should be done based on the ratio of the row or column scaling
  factors.  If ROWCND < THRESH, row scaling is done, and if
  COLCND < THRESH, column scaling is done.

  LARGE and SMALL are threshold values used to decide if row scaling
  should be done based on the absolute size of the largest matrix
  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        r -
        c -
        rowcnd -
        colcnd -
        amax -
        equed -

      • slaqp2

        public abstract void slaqp2(int m,
                            int n,
                            int offset,
                            float[] a,
                            int lda,
                            int[] jpvt,
                            float[] tau,
                            float[] vn1,
                            float[] vn2,
                            float[] work)
        
     ..

  Purpose
  =======

  SLAQP2 computes a QR factorization with column pivoting of
  the block A(OFFSET+1:M,1:N).
  The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0.

  OFFSET  (input) INTEGER
          The number of rows of the matrix A that must be pivoted
          but no factorized. OFFSET >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is 
          the triangular factor obtained; the elements in block 
          A(OFFSET+1:M,1:N) below the diagonal, together with the 
          array TAU, represent the orthogonal matrix Q as a product of

          elementary reflectors. Block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

  TAU     (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

  VN1     (input/output) REAL array, dimension (N)
          The vector with the partial column norms.

  VN2     (input/output) REAL array, dimension (N)
          The vector with the exact column norms.

  WORK    (workspace) REAL array, dimension (N)

  Further Details
  ===============

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.
  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        offset -
        a -
        lda -
        jpvt -
        tau -
        vn1 -
        vn2 -
        work -

      • slaqps

        public abstract void slaqps(int m,
                            int n,
                            int offset,
                            int nb,
                            org.netlib.util.intW kb,
                            float[] a,
                            int lda,
                            int[] jpvt,
                            float[] tau,
                            float[] vn1,
                            float[] vn2,
                            float[] auxv,
                            float[] f,
                            int ldf)
        
     ..

  Purpose
  =======

  SLAQPS computes a step of QR factorization with column pivoting
  of a real M-by-N matrix A by using Blas-3.  It tries to factorize
  NB columns from A starting from the row OFFSET+1, and updates all
  of the matrix with Blas-3 xGEMM.

  In some cases, due to catastrophic cancellations, it cannot
  factorize NB columns.  Hence, the actual number of factorized
  columns is returned in KB.

  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A. N >= 0

  OFFSET  (input) INTEGER
          The number of rows of A that have been factorized in
          previous steps.

  NB      (input) INTEGER
          The number of columns to factorize.

  KB      (output) INTEGER
          The number of columns actually factorized.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, block A(OFFSET+1:M,1:KB) is the triangular
          factor obtained and block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.
          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
          been updated.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  JPVT    (input/output) INTEGER array, dimension (N)
          JPVT(I) = K <==> Column K of the full matrix A has been
          permuted into position I in AP.

  TAU     (output) REAL array, dimension (KB)
          The scalar factors of the elementary reflectors.

  VN1     (input/output) REAL array, dimension (N)
          The vector with the partial column norms.

  VN2     (input/output) REAL array, dimension (N)
          The vector with the exact column norms.

  AUXV    (input/output) REAL array, dimension (NB)
          Auxiliar vector.

  F       (input/output) REAL array, dimension (LDF,NB)
          Matrix F' = L*Y'*A.

  LDF     (input) INTEGER
          The leading dimension of the array F. LDF >= max(1,N).

  Further Details
  ===============

  Based on contributions by
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    X. Sun, Computer Science Dept., Duke University, USA

  Partial column norm updating strategy modified by
    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    University of Zagreb, Croatia.
    June 2006.
  For more details see LAPACK Working Note 176.
  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        offset -
        nb -
        kb -
        a -
        lda -
        jpvt -
        tau -
        vn1 -
        vn2 -
        auxv -
        f -
        ldf -

      • slaqr0

        public void slaqr0(boolean wantt,
                   boolean wantz,
                   int n,
                   int ilo,
                   int ihi,
                   float[] h,
                   int ldh,
                   float[] wr,
                   float[] wi,
                   int iloz,
                   int ihiz,
                   float[] z,
                   int ldz,
                   float[] work,
                   int lwork,
                   org.netlib.util.intW info)
        
     ..

     Purpose
     =======

     SLAQR0 computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to SGEBAL, and then passed to SGEHRD when the

           matrix output by SGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) REAL array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO.GT.0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) REAL array, dimension (IHI)
     WI    (output) REAL array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed

           eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     ILOZ     (input) INTEGER
     IHIZ     (input) INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

     Z     (input/output) REAL array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) REAL array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then SLAQR0 does a workspace query.
           In this case, SLAQR0 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .GT. 0:  if INFO = i, SLAQR0 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
                accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    SLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== Exceptional deflation windows:  try to cure rare
     .    slow convergence by increasing the size of the
     .    deflation window after KEXNW iterations. =====

     ==== Exceptional shifts: try to cure rare slow convergence
     .    with ad-hoc exceptional shifts every KEXSH iterations.
     .    The constants WILK1 and WILK2 are used to form the
     .    exceptional shifts. ====
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        work -
        lwork -
        info -

      • slaqr1

        public void slaqr1(int n,
                   float[] h,
                   int ldh,
                   float sr1,
                   float si1,
                   float sr2,
                   float si2,
                   float[] v)
        
     ..

       Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a
       scalar multiple of the first column of the product

       (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)

       scaling to avoid overflows and most underflows. It
       is assumed that either

               1) sr1 = sr2 and si1 = -si2
           or
               2) si1 = si2 = 0.

       This is useful for starting double implicit shift bulges
       in the QR algorithm.


       N      (input) integer
              Order of the matrix H. N must be either 2 or 3.

       H      (input) REAL array of dimension (LDH,N)
              The 2-by-2 or 3-by-3 matrix H in (*).

       LDH    (input) integer
              The leading dimension of H as declared in
              the calling procedure.  LDH.GE.N

       SR1    (input) REAL
       SI1    The shifts in (*).
       SR2
       SI2

       V      (output) REAL array of dimension N
              A scalar multiple of the first column of the
              matrix K in (*).

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================

     .. Parameters ..
        Parameters:
        n -
        h -
        ldh -
        sr1 -
        si1 -
        sr2 -
        si2 -
        v -

      • slaqr2

        public void slaqr2(boolean wantt,
                   boolean wantz,
                   int n,
                   int ktop,
                   int kbot,
                   int nw,
                   float[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   float[] z,
                   int ldz,
                   org.netlib.util.intW ns,
                   org.netlib.util.intW nd,
                   float[] sr,
                   float[] si,
                   float[] v,
                   int ldv,
                   int nh,
                   float[] t,
                   int ldt,
                   int nv,
                   float[] wv,
                   int ldwv,
                   float[] work,
                   int lwork)
        
     ..

     This subroutine is identical to SLAQR3 except that it avoids
     recursion by calling SLAHQR instead of SLAQR4.



     Aggressive early deflation:

     This subroutine accepts as input an upper Hessenberg matrix
     H and performs an orthogonal similarity transformation
     designed to detect and deflate fully converged eigenvalues from
     a trailing principal submatrix.  On output H has been over-
     written by a new Hessenberg matrix that is a perturbation of
     an orthogonal similarity transformation of H.  It is to be
     hoped that the final version of H has many zero subdiagonal
     entries.


     WANTT   (input) LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

     WANTZ   (input) LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

     N       (input) INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

     KTOP    (input) INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

     KBOT    (input) INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

     NW      (input) INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

     H       (input/output) REAL array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

     LDH     (input) integer
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

     Z       (input/output) REAL array, dimension (LDZ,IHI)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

     LDZ     (input) integer
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

     NS      (output) integer
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

     ND      (output) integer
          The number of converged eigenvalues uncovered by this
          subroutine.

     SR      (output) REAL array, dimension KBOT
     SI      (output) REAL array, dimension KBOT
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

     V       (workspace) REAL array, dimension (LDV,NW)
          An NW-by-NW work array.

     LDV     (input) integer scalar
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

     NH      (input) integer scalar
          The number of columns of T.  NH.GE.NW.

     T       (workspace) REAL array, dimension (LDT,NW)

     LDT     (input) integer
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

     NV      (input) integer
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

     WV      (workspace) REAL array, dimension (LDWV,NW)

     LDWV    (input) integer
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

     WORK    (workspace) REAL array, dimension LWORK.
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

     LWORK   (input) integer
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; SLAQR2
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ==================================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ktop -
        kbot -
        nw -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        ns -
        nd -
        sr -
        si -
        v -
        ldv -
        nh -
        t -
        ldt -
        nv -
        wv -
        ldwv -
        work -
        lwork -

      • slaqr3

        public void slaqr3(boolean wantt,
                   boolean wantz,
                   int n,
                   int ktop,
                   int kbot,
                   int nw,
                   float[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   float[] z,
                   int ldz,
                   org.netlib.util.intW ns,
                   org.netlib.util.intW nd,
                   float[] sr,
                   float[] si,
                   float[] v,
                   int ldv,
                   int nh,
                   float[] t,
                   int ldt,
                   int nv,
                   float[] wv,
                   int ldwv,
                   float[] work,
                   int lwork)
        
     ..


     Aggressive early deflation:

     This subroutine accepts as input an upper Hessenberg matrix
     H and performs an orthogonal similarity transformation
     designed to detect and deflate fully converged eigenvalues from
     a trailing principal submatrix.  On output H has been over-
     written by a new Hessenberg matrix that is a perturbation of
     an orthogonal similarity transformation of H.  It is to be
     hoped that the final version of H has many zero subdiagonal
     entries.


     WANTT   (input) LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

     WANTZ   (input) LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

     N       (input) INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

     KTOP    (input) INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

     KBOT    (input) INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

     NW      (input) INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

     H       (input/output) REAL array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

     LDH     (input) integer
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

     ILOZ    (input) INTEGER
     IHIZ    (input) INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

     Z       (input/output) REAL array, dimension (LDZ,IHI)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

     LDZ     (input) integer
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

     NS      (output) integer
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

     ND      (output) integer
          The number of converged eigenvalues uncovered by this
          subroutine.

     SR      (output) REAL array, dimension KBOT
     SI      (output) REAL array, dimension KBOT
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

     V       (workspace) REAL array, dimension (LDV,NW)
          An NW-by-NW work array.

     LDV     (input) integer scalar
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

     NH      (input) integer scalar
          The number of columns of T.  NH.GE.NW.

     T       (workspace) REAL array, dimension (LDT,NW)

     LDT     (input) integer
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

     NV      (input) integer
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

     WV      (workspace) REAL array, dimension (LDWV,NW)

     LDWV    (input) integer
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

     WORK    (workspace) REAL array, dimension LWORK.
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

     LWORK   (input) integer
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; SLAQR3
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ==================================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        n -
        ktop -
        kbot -
        nw -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        ns -
        nd -
        sr -
        si -
        v -
        ldv -
        nh -
        t -
        ldt -
        nv -
        wv -
        ldwv -
        work -
        lwork -

      • slaqr4

        public void slaqr4(boolean wantt,
                   boolean wantz,
                   int n,
                   int ilo,
                   int ihi,
                   float[] h,
                   int ldh,
                   float[] wr,
                   float[] wi,
                   int iloz,
                   int ihiz,
                   float[] z,
                   int ldz,
                   float[] work,
                   int lwork,
                   org.netlib.util.intW info)
        
     ..

     This subroutine implements one level of recursion for SLAQR0.
     It is a complete implementation of the small bulge multi-shift
     QR algorithm.  It may be called by SLAQR0 and, for large enough
     deflation window size, it may be called by SLAQR3.  This
     subroutine is identical to SLAQR0 except that it calls SLAQR2
     instead of SLAQR3.

     Purpose
     =======

     SLAQR4 computes the eigenvalues of a Hessenberg matrix H
     and, optionally, the matrices T and Z from the Schur decomposition
     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
     Schur form), and Z is the orthogonal matrix of Schur vectors.

     Optionally Z may be postmultiplied into an input orthogonal
     matrix Q so that this routine can give the Schur factorization
     of a matrix A which has been reduced to the Hessenberg form H
     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

     Arguments
     =========

     WANTT   (input) LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

     WANTZ   (input) LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

     N     (input) INTEGER
           The order of the matrix H.  N .GE. 0.

     ILO   (input) INTEGER
     IHI   (input) INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to SGEBAL, and then passed to SGEHRD when the

           matrix output by SGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

     H     (input/output) REAL array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO.GT.0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

     LDH   (input) INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

     WR    (output) REAL array, dimension (IHI)
     WI    (output) REAL array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed

           eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

     ILOZ     (input) INTEGER
     IHIZ     (input) INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

     Z     (input/output) REAL array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

     LDZ   (input) INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

     WORK  (workspace/output) REAL array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

     LWORK (input) INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then SLAQR4 does a workspace query.
           In this case, SLAQR4 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.


     INFO  (output) INTEGER
             =  0:  successful exit
           .GT. 0:  if INFO = i, SLAQR4 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO .GT. 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO .GT. 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.

                If INFO .GT. 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
                accessed.

     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ================================================================
     References:
       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
       929--947, 2002.

       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
       of Matrix Analysis, volume 23, pages 948--973, 2002.

     ================================================================
     .. Parameters ..

     ==== Matrices of order NTINY or smaller must be processed by
     .    SLAHQR because of insufficient subdiagonal scratch space.
     .    (This is a hard limit.) ====

     ==== Exceptional deflation windows:  try to cure rare
     .    slow convergence by increasing the size of the
     .    deflation window after KEXNW iterations. =====

     ==== Exceptional shifts: try to cure rare slow convergence
     .    with ad-hoc exceptional shifts every KEXSH iterations.
     .    The constants WILK1 and WILK2 are used to form the
     .    exceptional shifts. ====
        Parameters:
        wantt -
        wantz -
        n -
        ilo -
        ihi -
        h -
        ldh -
        wr -
        wi -
        iloz -
        ihiz -
        z -
        ldz -
        work -
        lwork -
        info -

      • slaqr5

        public void slaqr5(boolean wantt,
                   boolean wantz,
                   int kacc22,
                   int n,
                   int ktop,
                   int kbot,
                   int nshfts,
                   float[] sr,
                   float[] si,
                   float[] h,
                   int ldh,
                   int iloz,
                   int ihiz,
                   float[] z,
                   int ldz,
                   float[] v,
                   int ldv,
                   float[] u,
                   int ldu,
                   int nv,
                   float[] wv,
                   int ldwv,
                   int nh,
                   float[] wh,
                   int ldwh)
        
     ..

     This auxiliary subroutine called by SLAQR0 performs a
     single small-bulge multi-shift QR sweep.

      WANTT  (input) logical scalar
             WANTT = .true. if the quasi-triangular Schur factor
             is being computed.  WANTT is set to .false. otherwise.

      WANTZ  (input) logical scalar
             WANTZ = .true. if the orthogonal Schur factor is being
             computed.  WANTZ is set to .false. otherwise.

      KACC22 (input) integer with value 0, 1, or 2.
             Specifies the computation mode of far-from-diagonal
             orthogonal updates.
        = 0: SLAQR5 does not accumulate reflections and does not
             use matrix-matrix multiply to update far-from-diagonal
             matrix entries.
        = 1: SLAQR5 accumulates reflections and uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries.
        = 2: SLAQR5 accumulates reflections, uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries,
             and takes advantage of 2-by-2 block structure during
             matrix multiplies.

      N      (input) integer scalar
             N is the order of the Hessenberg matrix H upon which this

             subroutine operates.

      KTOP   (input) integer scalar
      KBOT   (input) integer scalar
             These are the first and last rows and columns of an
             isolated diagonal block upon which the QR sweep is to be
             applied. It is assumed without a check that
                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0
             and
                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

      NSHFTS (input) integer scalar
             NSHFTS gives the number of simultaneous shifts.  NSHFTS
             must be positive and even.

      SR     (input) REAL array of size (NSHFTS)
      SI     (input) REAL array of size (NSHFTS)
             SR contains the real parts and SI contains the imaginary
             parts of the NSHFTS shifts of origin that define the
             multi-shift QR sweep.

      H      (input/output) REAL array of size (LDH,N)
             On input H contains a Hessenberg matrix.  On output a
             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied

             to the isolated diagonal block in rows and columns KTOP
             through KBOT.

      LDH    (input) integer scalar
             LDH is the leading dimension of H just as declared in the

             calling procedure.  LDH.GE.MAX(1,N).

      ILOZ   (input) INTEGER
      IHIZ   (input) INTEGER
             Specify the rows of Z to which transformations must be
             applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N

      Z      (input/output) REAL array of size (LDZ,IHI)
             If WANTZ = .TRUE., then the QR Sweep orthogonal
             similarity transformation is accumulated into
             Z(ILOZ:IHIZ,ILO:IHI) from the right.
             If WANTZ = .FALSE., then Z is unreferenced.

      LDZ    (input) integer scalar
             LDA is the leading dimension of Z just as declared in
             the calling procedure. LDZ.GE.N.

      V      (workspace) REAL array of size (LDV,NSHFTS/2)

      LDV    (input) integer scalar
             LDV is the leading dimension of V as declared in the
             calling procedure.  LDV.GE.3.

      U      (workspace) REAL array of size
             (LDU,3*NSHFTS-3)

      LDU    (input) integer scalar
             LDU is the leading dimension of U just as declared in the

             in the calling subroutine.  LDU.GE.3*NSHFTS-3.

      NH     (input) integer scalar
             NH is the number of columns in array WH available for
             workspace. NH.GE.1.

      WH     (workspace) REAL array of size (LDWH,NH)

      LDWH   (input) integer scalar
             Leading dimension of WH just as declared in the
             calling procedure.  LDWH.GE.3*NSHFTS-3.

      NV     (input) integer scalar
             NV is the number of rows in WV agailable for workspace.
             NV.GE.1.

      WV     (workspace) REAL array of size
             (LDWV,3*NSHFTS-3)

      LDWV   (input) integer scalar
             LDWV is the leading dimension of WV as declared in the
             in the calling subroutine.  LDWV.GE.NV.


     ================================================================
     Based on contributions by
        Karen Braman and Ralph Byers, Department of Mathematics,
        University of Kansas, USA

     ============================================================
     Reference:

     K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
     Algorithm Part I: Maintaining Well Focused Shifts, and
     Level 3 Performance, SIAM Journal of Matrix Analysis,
     volume 23, pages 929--947, 2002.

     ============================================================
     .. Parameters ..
        Parameters:
        wantt -
        wantz -
        kacc22 -
        n -
        ktop -
        kbot -
        nshfts -
        sr -
        si -
        h -
        ldh -
        iloz -
        ihiz -
        z -
        ldz -
        v -
        ldv -
        u -
        ldu -
        nv -
        wv -
        ldwv -
        nh -
        wh -
        ldwh -

      • slaqsb

        public abstract void slaqsb(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] s,
                            float scond,
                            float amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  SLAQSB equilibrates a symmetric band matrix A using the scaling
  factors in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U'*U or A = L*L' of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  S       (input) REAL array, dimension (N)
          The scale factors for A.

  SCOND   (input) REAL
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) REAL
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        s -
        scond -
        amax -
        equed -

      • slaqsp

        public abstract void slaqsp(java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] s,
                            float scond,
                            float amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  SLAQSP equilibrates a symmetric matrix A using the scaling factors
  in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in

          the same storage format as A.

  S       (input) REAL array, dimension (N)
          The scale factors for A.

  SCOND   (input) REAL
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) REAL
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        s -
        scond -
        amax -
        equed -

      • slaqsy

        public abstract void slaqsy(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] s,
                            float scond,
                            float amax,
                            org.netlib.util.StringW equed)
        
     ..

  Purpose
  =======

  SLAQSY equilibrates a symmetric matrix A using the scaling factors
  in the vector S.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if EQUED = 'Y', the equilibrated matrix:
          diag(S) * A * diag(S).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

  S       (input) REAL array, dimension (N)
          The scale factors for A.

  SCOND   (input) REAL
          Ratio of the smallest S(i) to the largest S(i).

  AMAX    (input) REAL
          Absolute value of largest matrix entry.

  EQUED   (output) CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).

  Internal Parameters
  ===================

  THRESH is a threshold value used to decide if scaling should be done

  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        s -
        scond -
        amax -
        equed -

      • slaqtr

        public abstract void slaqtr(boolean ltran,
                            boolean lreal,
                            int n,
                            float[] t,
                            int ldt,
                            float[] b,
                            float w,
                            org.netlib.util.floatW scale,
                            float[] x,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAQTR solves the real quasi-triangular system

               op(T)*p = scale*c,               if LREAL = .TRUE.

  or the complex quasi-triangular systems

             op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.

  in real arithmetic, where T is upper quasi-triangular.
  If LREAL = .FALSE., then the first diagonal block of T must be
  1 by 1, B is the specially structured matrix

                 B = [ b(1) b(2) ... b(n) ]
                     [       w            ]
                     [           w        ]
                     [              .     ]
                     [                 w  ]

  op(A) = A or A', A' denotes the conjugate transpose of
  matrix A.

  On input, X = [ c ].  On output, X = [ p ].
                [ d ]                  [ q ]

  This subroutine is designed for the condition number estimation
  in routine STRSNA.

  Arguments
  =========

  LTRAN   (input) LOGICAL
          On entry, LTRAN specifies the option of conjugate transpose:

             = .FALSE.,    op(T+i*B) = T+i*B,
             = .TRUE.,     op(T+i*B) = (T+i*B)'.

  LREAL   (input) LOGICAL
          On entry, LREAL specifies the input matrix structure:
             = .FALSE.,    the input is complex
             = .TRUE.,     the input is real

  N       (input) INTEGER
          On entry, N specifies the order of T+i*B. N >= 0.

  T       (input) REAL array, dimension (LDT,N)
          On entry, T contains a matrix in Schur canonical form.
          If LREAL = .FALSE., then the first diagonal block of T must
          be 1 by 1.

  LDT     (input) INTEGER
          The leading dimension of the matrix T. LDT >= max(1,N).

  B       (input) REAL array, dimension (N)
          On entry, B contains the elements to form the matrix
          B as described above.
          If LREAL = .TRUE., B is not referenced.

  W       (input) REAL
          On entry, W is the diagonal element of the matrix B.
          If LREAL = .TRUE., W is not referenced.

  SCALE   (output) REAL
          On exit, SCALE is the scale factor.

  X       (input/output) REAL array, dimension (2*N)
          On entry, X contains the right hand side of the system.
          On exit, X is overwritten by the solution.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          On exit, INFO is set to
             0: successful exit.
               1: the some diagonal 1 by 1 block has been perturbed by

                  a small number SMIN to keep nonsingularity.
               2: the some diagonal 2 by 2 block has been perturbed by

                  a small number in SLALN2 to keep nonsingularity.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltran -
        lreal -
        n -
        t -
        ldt -
        b -
        w -
        scale -
        x -
        work -
        info -

      • slar1v

        public void slar1v(int n,
                   int b1,
                   int bn,
                   float lambda,
                   float[] d,
                   float[] l,
                   float[] ld,
                   float[] lld,
                   float pivmin,
                   float gaptol,
                   float[] z,
                   boolean wantnc,
                   org.netlib.util.intW negcnt,
                   org.netlib.util.floatW ztz,
                   org.netlib.util.floatW mingma,
                   org.netlib.util.intW r,
                   int[] isuppz,
                   org.netlib.util.floatW nrminv,
                   org.netlib.util.floatW resid,
                   org.netlib.util.floatW rqcorr,
                   float[] work)
        
     ..

  Purpose
  =======

  SLAR1V computes the (scaled) r-th column of the inverse of
  the sumbmatrix in rows B1 through BN of the tridiagonal matrix
  L D L^T - sigma I. When sigma is close to an eigenvalue, the
  computed vector is an accurate eigenvector. Usually, r corresponds
  to the index where the eigenvector is largest in magnitude.
  The following steps accomplish this computation :
  (a) Stationary qd transform,  L D L^T - sigma I = L(+) D(+) L(+)^T,
  (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T,
  (c) Computation of the diagonal elements of the inverse of
      L D L^T - sigma I by combining the above transforms, and choosing
      r as the index where the diagonal of the inverse is (one of the)

      largest in magnitude.
  (d) Computation of the (scaled) r-th column of the inverse using the

      twisted factorization obtained by combining the top part of the
      the stationary and the bottom part of the progressive transform.


  Arguments
  =========

  N        (input) INTEGER
           The order of the matrix L D L^T.

  B1       (input) INTEGER
           First index of the submatrix of L D L^T.

  BN       (input) INTEGER
           Last index of the submatrix of L D L^T.

  LAMBDA    (input) REAL            
           The shift. In order to compute an accurate eigenvector,
           LAMBDA should be a good approximation to an eigenvalue
           of L D L^T.

  L        (input) REAL             array, dimension (N-1)
           The (n-1) subdiagonal elements of the unit bidiagonal matrix
           L, in elements 1 to N-1.

  D        (input) REAL             array, dimension (N)
           The n diagonal elements of the diagonal matrix D.

  LD       (input) REAL             array, dimension (N-1)
           The n-1 elements L(i)*D(i).

  LLD      (input) REAL             array, dimension (N-1)
           The n-1 elements L(i)*L(i)*D(i).

  PIVMIN   (input) REAL            
           The minimum pivot in the Sturm sequence.

  GAPTOL   (input) REAL            
           Tolerance that indicates when eigenvector entries are neglig
           w.r.t. their contribution to the residual.

  Z        (input/output) REAL             array, dimension (N)
           On input, all entries of Z must be set to 0.
           On output, Z contains the (scaled) r-th column of the
           inverse. The scaling is such that Z(R) equals 1.

  WANTNC   (input) LOGICAL
           Specifies whether NEGCNT has to be computed.

  NEGCNT   (output) INTEGER
           If WANTNC is .TRUE. then NEGCNT = the number of pivots < piv
           in the  matrix factorization L D L^T, and NEGCNT = -1 otherw

  ZTZ      (output) REAL            
           The square of the 2-norm of Z.

  MINGMA   (output) REAL            
           The reciprocal of the largest (in magnitude) diagonal
           element of the inverse of L D L^T - sigma I.

  R        (input/output) INTEGER
           The twist index for the twisted factorization used to
           compute Z.
           On input, 0 <= R <= N. If R is input as 0, R is set to
           the index where (L D L^T - sigma I)^{-1} is largest
           in magnitude. If 1 <= R <= N, R is unchanged.
           On output, R contains the twist index used to compute Z.
           Ideally, R designates the position of the maximum entry in t
           eigenvector.

  ISUPPZ   (output) INTEGER array, dimension (2)
           The support of the vector in Z, i.e., the vector Z is
           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

  NRMINV   (output) REAL            
           NRMINV = 1/SQRT( ZTZ )

  RESID    (output) REAL            
           The residual of the FP vector.
           RESID = ABS( MINGMA )/SQRT( ZTZ )

  RQCORR   (output) REAL            
           The Rayleigh Quotient correction to LAMBDA.
           RQCORR = MINGMA*TMP

  WORK     (workspace) REAL             array, dimension (4*N)

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        b1 -
        bn -
        lambda -
        d -
        l -
        ld -
        lld -
        pivmin -
        gaptol -
        z -
        wantnc -
        negcnt -
        ztz -
        mingma -
        r -
        isuppz -
        nrminv -
        resid -
        rqcorr -
        work -

      • slar2v

        public abstract void slar2v(int n,
                            float[] x,
                            float[] y,
                            float[] z,
                            int incx,
                            float[] c,
                            float[] s,
                            int incc)
        
     ..

  Purpose
  =======

  SLAR2V applies a vector of real plane rotations from both sides to
  a sequence of 2-by-2 real symmetric matrices, defined by the elements
  of the vectors x, y and z. For i = 1,2,...,n

     ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )
     ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be applied.

  X       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          The vector x.

  Y       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          The vector y.

  Z       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          The vector z.

  INCX    (input) INTEGER
          The increment between elements of X, Y and Z. INCX > 0.

  C       (input) REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  S       (input) REAL array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C and S. INCC > 0.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        x -
        y -
        z -
        incx -
        c -
        s -
        incc -

      • slarf

        public abstract void slarf(java.lang.String side,
                           int m,
                           int n,
                           float[] v,
                           int incv,
                           float tau,
                           float[] c,
                           int Ldc,
                           float[] work)
        
     ..

  Purpose
  =======

  SLARF applies a real elementary reflector H to a real m by n matrix
  C, from either the left or the right. H is represented in the form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) REAL array, dimension
                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of H. V is not used if
          TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0.

  TAU     (input) REAL
          The value tau in the representation of H.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        incv -
        tau -
        c -
        Ldc -
        work -

      • slarfb

        public abstract void slarfb(java.lang.String side,
                            java.lang.String trans,
                            java.lang.String direct,
                            java.lang.String storev,
                            int m,
                            int n,
                            int k,
                            float[] v,
                            int ldv,
                            float[] t,
                            int ldt,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int ldwork)
        
     ..

  Purpose
  =======

  SLARFB applies a real block reflector H or its transpose H' to a
  real m by n matrix C, from either the left or the right.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply H or H' from the Left
          = 'R': apply H or H' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply H (No transpose)
          = 'T': apply H' (Transpose)

  DIRECT  (input) CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise
          = 'R': Rowwise

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  K       (input) INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

  V       (input) REAL array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'

                                (LDV,N) if STOREV = 'R' and SIDE = 'R'

          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
          if STOREV = 'R', LDV >= K.

  T       (input) REAL array, dimension (LDT,K)
          The triangular k by k matrix T in the representation of the
          block reflector.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by H*C or H'*C or C*H or C*H'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDA >= max(1,M).

  WORK    (workspace) REAL array, dimension (LDWORK,K)

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        direct -
        storev -
        m -
        n -
        k -
        v -
        ldv -
        t -
        ldt -
        c -
        Ldc -
        work -
        ldwork -

      • slarfg

        public abstract void slarfg(int n,
                            org.netlib.util.floatW alpha,
                            float[] x,
                            int incx,
                            org.netlib.util.floatW tau)
        
     ..

  Purpose
  =======

  SLARFG generates a real elementary reflector H of order n, such
  that

        H * ( alpha ) = ( beta ),   H' * H = I.
            (   x   )   (   0  )

  where alpha and beta are scalars, and x is an (n-1)-element real
  vector. H is represented in the form

        H = I - tau * ( 1 ) * ( 1 v' ) ,
                      ( v )

  where tau is a real scalar and v is a real (n-1)-element
  vector.

  If the elements of x are all zero, then tau = 0 and H is taken to be

  the unit matrix.

  Otherwise  1 <= tau <= 2.

  Arguments
  =========

  N       (input) INTEGER
          The order of the elementary reflector.

  ALPHA   (input/output) REAL
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

  X       (input/output) REAL array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  TAU     (output) REAL
          The value tau.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        alpha -
        x -
        incx -
        tau -

      • slarft

        public abstract void slarft(java.lang.String direct,
                            java.lang.String storev,
                            int n,
                            int k,
                            float[] v,
                            int ldv,
                            float[] tau,
                            float[] t,
                            int ldt)
        
     ..

  Purpose
  =======

  SLARFT forms the triangular factor T of a real block reflector H
  of order n, which is defined as a product of k elementary reflectors.

  If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;


  If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.


  If STOREV = 'C', the vector which defines the elementary reflector
  H(i) is stored in the i-th column of the array V, and

     H  =  I - V * T * V'

  If STOREV = 'R', the vector which defines the elementary reflector
  H(i) is stored in the i-th row of the array V, and

     H  =  I - V' * T * V

  Arguments
  =========

  DIRECT  (input) CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise
          = 'R': rowwise

  N       (input) INTEGER
          The order of the block reflector H. N >= 0.

  K       (input) INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

  V       (input/output) REAL array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.


  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

  T       (output) REAL array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  Further Details
  ===============

  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and

  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

               V = (  1       )                 V = (  1 v1 v1 v1 v1 )

                   ( v1  1    )                     (     1 v2 v2 v2 )

                   ( v1 v2  1 )                     (        1 v3 v3 )

                   ( v1 v2 v3 )
                   ( v1 v2 v3 )

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )

                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )

                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )

                   (     1 v3 )
                   (        1 )

  =====================================================================

     .. Parameters ..
        Parameters:
        direct -
        storev -
        n -
        k -
        v -
        ldv -
        tau -
        t -
        ldt -

      • slarfx

        public abstract void slarfx(java.lang.String side,
                            int m,
                            int n,
                            float[] v,
                            float tau,
                            float[] c,
                            int Ldc,
                            float[] work)
        
     ..

  Purpose
  =======

  SLARFX applies a real elementary reflector H to a real m by n
  matrix C, from either the left or the right. H is represented in the

  form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix

  This version uses inline code if H has order < 11.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) REAL array, dimension (M) if SIDE = 'L'
                                     or (N) if SIDE = 'R'
          The vector v in the representation of H.

  TAU     (input) REAL
          The value tau in the representation of H.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDA >= (1,M).

  WORK    (workspace) REAL array, dimension
                      (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'
          WORK is not referenced if H has order < 11.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        tau -
        c -
        Ldc -
        work -

      • slargv

        public abstract void slargv(int n,
                            float[] x,
                            int incx,
                            float[] y,
                            int incy,
                            float[] c,
                            int incc)
        
     ..

  Purpose
  =======

  SLARGV generates a vector of real plane rotations, determined by
  elements of the real vectors x and y. For i = 1,2,...,n

     (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
     ( -s(i)  c(i) ) ( y(i) ) = (   0  )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be generated.

  X       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          On entry, the vector x.
          On exit, x(i) is overwritten by a(i), for i = 1,...,n.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  Y       (input/output) REAL array,
                         dimension (1+(N-1)*INCY)
          On entry, the vector y.
          On exit, the sines of the plane rotations.

  INCY    (input) INTEGER
          The increment between elements of Y. INCY > 0.

  C       (output) REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C. INCC > 0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        c -
        incc -

      • slarnv

        public abstract void slarnv(int idist,
                            int[] iseed,
                            int n,
                            float[] x)
        
     ..

  Purpose
  =======

  SLARNV returns a vector of n random real numbers from a uniform or
  normal distribution.

  Arguments
  =========

  IDIST   (input) INTEGER
          Specifies the distribution of the random numbers:
          = 1:  uniform (0,1)
          = 2:  uniform (-1,1)
          = 3:  normal (0,1)

  ISEED   (input/output) INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array

          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

  N       (input) INTEGER
          The number of random numbers to be generated.

  X       (output) REAL array, dimension (N)
          The generated random numbers.

  Further Details
  ===============

  This routine calls the auxiliary routine SLARUV to generate random
  real numbers from a uniform (0,1) distribution, in batches of up to
  128 using vectorisable code. The Box-Muller method is used to
  transform numbers from a uniform to a normal distribution.

  =====================================================================

     .. Parameters ..
        Parameters:
        idist -
        iseed -
        n -
        x -

      • slarra

        public void slarra(int n,
                   float[] d,
                   float[] e,
                   float[] e2,
                   float spltol,
                   float tnrm,
                   org.netlib.util.intW nsplit,
                   int[] isplit,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Compute the splitting points with threshold SPLTOL.
  SLARRA sets any "small" off-diagonal elements to zero.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix. N > 0.

  D       (input) REAL             array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal
          matrix T.

  E       (input/output) REAL             array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) need not be set.
          On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT,
          are set to zero, the other entries of E are untouched.

  E2      (input/output) REAL             array, dimension (N)
          On entry, the first (N-1) entries contain the SQUARES of the

          subdiagonal elements of the tridiagonal matrix T;
          E2(N) need not be set.
          On exit, the entries E2( ISPLIT( I ) ),
          1 <= I <= NSPLIT, have been set to zero

  SPLTOL (input) REAL            
          The threshold for splitting. Two criteria can be used:
          SPLTOL<0 : criterion based on absolute off-diagonal value
          SPLTOL>0 : criterion that preserves relative accuracy

  TNRM (input) REAL            
          The norm of the matrix.

  NSPLIT  (output) INTEGER
          The number of blocks T splits into. 1 <= NSPLIT <= N.

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to ISPLIT(1),
          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.


  INFO    (output) INTEGER
          = 0:  successful exit

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        e2 -
        spltol -
        tnrm -
        nsplit -
        isplit -
        info -

      • slarrb

        public void slarrb(int n,
                   float[] d,
                   float[] lld,
                   int ifirst,
                   int ilast,
                   float rtol1,
                   float rtol2,
                   int offset,
                   float[] w,
                   float[] wgap,
                   float[] werr,
                   float[] work,
                   int[] iwork,
                   float pivmin,
                   float spdiam,
                   int twist,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the relatively robust representation(RRR) L D L^T, SLARRB
  does "limited" bisection to refine the eigenvalues of L D L^T,
  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initi
  guesses for these eigenvalues are input in W, the corresponding estim
  of the error in these guesses and their gaps are input in WERR
  and WGAP, respectively. During bisection, intervals
  [left, right] are maintained by storing their mid-points and
  semi-widths in the arrays W and WERR respectively.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) REAL             array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  LLD     (input) REAL             array, dimension (N-1)
          The (N-1) elements L(i)*L(i)*D(i).

  IFIRST  (input) INTEGER
          The index of the first eigenvalue to be computed.

  ILAST   (input) INTEGER
          The index of the last eigenvalue to be computed.

  RTOL1   (input) REAL            
  RTOL2   (input) REAL            
          Tolerance for the convergence of the bisection intervals.
          An interval [LEFT,RIGHT] has converged if
          RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
          where GAP is the (estimated) distance to the nearest
          eigenvalue.

  OFFSET  (input) INTEGER
          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFS
          through ILAST-OFFSET elements of these arrays are to be used.

  W       (input/output) REAL             array, dimension (N)
          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
          estimates of the eigenvalues of L D L^T indexed IFIRST throug
          ILAST.
          On output, these estimates are refined.

  WGAP    (input/output) REAL             array, dimension (N-1)
          On input, the (estimated) gaps between consecutive
          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap betwe
          eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
          then WGAP(IFIRST-OFFSET) must be set to ZERO.
          On output, these gaps are refined.

  WERR    (input/output) REAL             array, dimension (N)
          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) 
          the errors in the estimates of the corresponding elements in 
          On output, these errors are refined.

  WORK    (workspace) REAL             array, dimension (2*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (2*N)
          Workspace.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence.

  SPDIAM  (input) DOUBLE PRECISION
          The spectral diameter of the matrix.

  TWIST   (input) INTEGER
          The twist index for the twisted factorization that is used
          for the negcount.
          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L
          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U
          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(

  INFO    (output) INTEGER
          Error flag.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        lld -
        ifirst -
        ilast -
        rtol1 -
        rtol2 -
        offset -
        w -
        wgap -
        werr -
        work -
        iwork -
        pivmin -
        spdiam -
        twist -
        info -

      • slarrc

        public void slarrc(java.lang.String jobt,
                   int n,
                   float vl,
                   float vu,
                   float[] d,
                   float[] e,
                   float pivmin,
                   org.netlib.util.intW eigcnt,
                   org.netlib.util.intW lcnt,
                   org.netlib.util.intW rcnt,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Find the number of eigenvalues of the symmetric tridiagonal matrix T

  that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T
  if JOBT = 'L'.

  Arguments
  =========

  JOBT    (input) CHARACTER*1
          = 'T':  Compute Sturm count for matrix T.
          = 'L':  Compute Sturm count for matrix L D L^T.

  N       (input) INTEGER
          The order of the matrix. N > 0.

  VL      (input) DOUBLE PRECISION
  VU      (input) DOUBLE PRECISION
          The lower and upper bounds for the eigenvalues.

  D       (input) DOUBLE PRECISION array, dimension (N)
          JOBT = 'T': The N diagonal elements of the tridiagonal matrix
          JOBT = 'L': The N diagonal elements of the diagonal matrix D.

  E       (input) DOUBLE PRECISION array, dimension (N)
          JOBT = 'T': The N-1 offdiagonal elements of the matrix T.
          JOBT = 'L': The N-1 offdiagonal elements of the matrix L.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  EIGCNT  (output) INTEGER
          The number of eigenvalues of the symmetric tridiagonal matrix
          that are in the interval (VL,VU]

  LCNT    (output) INTEGER
  RCNT    (output) INTEGER
          The left and right negcounts of the interval.

  INFO    (output) INTEGER

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobt -
        n -
        vl -
        vu -
        d -
        e -
        pivmin -
        eigcnt -
        lcnt -
        rcnt -
        info -

      • slarrd

        public void slarrd(java.lang.String range,
                   java.lang.String order,
                   int n,
                   float vl,
                   float vu,
                   int il,
                   int iu,
                   float[] gers,
                   float reltol,
                   float[] d,
                   float[] e,
                   float[] e2,
                   float pivmin,
                   int nsplit,
                   int[] isplit,
                   org.netlib.util.intW m,
                   float[] w,
                   float[] werr,
                   org.netlib.util.floatW wl,
                   org.netlib.util.floatW wu,
                   int[] iblock,
                   int[] indexw,
                   float[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLARRD computes the eigenvalues of a symmetric tridiagonal
  matrix T to suitable accuracy. This is an auxiliary code to be
  called from SSTEMR.
  The user may ask for all eigenvalues, all eigenvalues
  in the half-open interval (VL, VU], or the IL-th through IU-th
  eigenvalues.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  RANGE   (input) CHARACTER
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  ORDER   (input) CHARACTER
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and

                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  VL      (input) REAL            
  VU      (input) REAL            
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal

          to VL, or greater than VU, will not be returned.  VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  GERS    (input) REAL             array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)).

  RELTOL  (input) REAL            
          The minimum relative width of an interval.  When an interval

          is narrower than RELTOL times the larger (in
          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  D       (input) REAL             array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) REAL             array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.


  E2      (input) REAL             array, dimension (N-1)
          The (n-1) squared off-diagonal elements of the tridiagonal ma

  PIVMIN  (input) REAL            
          The minimum pivot allowed in the Sturm sequence for T.

  NSPLIT  (input) INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to ISPLIT(1),

          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

  M       (output) INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

  W       (output) REAL             array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalue approximations. SLARRD computes an interval
          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
          approximation is given as the interval midpoint
          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
          WERR(j) = abs( a_j - b_j)/2

  WERR    (output) REAL             array, dimension (N)
          The error bound on the corresponding eigenvalue approximation
          in W.

  WL      (output) REAL            
  WU      (output) REAL            
          The interval (WL, WU] contains all the wanted eigenvalues.
          If RANGE='V', then WL=VL and WU=VU.
          If RANGE='A', then WL and WU are the global Gerschgorin bound
                        on the spectrum.
          If RANGE='I', then WL and WU are computed by SLAEBZ from the

                        index range specified.

  IBLOCK  (output) INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (SLARRD may use the remaining N-M elements as
          workspace.)

  INDEXW  (output) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
          i-th eigenvalue W(i) is the j-th eigenvalue in block k.

  WORK    (workspace) REAL             array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the

                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick

                                out eigenvalues IL:IU.  In some cases,

                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  Internal Parameters
  ===================

  FUDGE   REAL            , default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

  Based on contributions by
     W. Kahan, University of California, Berkeley, USA
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        order -
        n -
        vl -
        vu -
        il -
        iu -
        gers -
        reltol -
        d -
        e -
        e2 -
        pivmin -
        nsplit -
        isplit -
        m -
        w -
        werr -
        wl -
        wu -
        iblock -
        indexw -
        work -
        iwork -
        info -

      • slarre

        public void slarre(java.lang.String range,
                   int n,
                   org.netlib.util.floatW vl,
                   org.netlib.util.floatW vu,
                   int il,
                   int iu,
                   float[] d,
                   float[] e,
                   float[] e2,
                   float rtol1,
                   float rtol2,
                   float spltol,
                   org.netlib.util.intW nsplit,
                   int[] isplit,
                   org.netlib.util.intW m,
                   float[] w,
                   float[] werr,
                   float[] wgap,
                   int[] iblock,
                   int[] indexw,
                   float[] gers,
                   org.netlib.util.floatW pivmin,
                   float[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  To find the desired eigenvalues of a given real symmetric
  tridiagonal matrix T, SLARRE sets any "small" off-diagonal
  elements to zero, and for each unreduced block T_i, it finds
  (a) a suitable shift at one end of the block's spectrum,
  (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
  (c) eigenvalues of each L_i D_i L_i^T.
  The representations and eigenvalues found are then used by
  SSTEMR to compute the eigenvectors of T.
  The accuracy varies depending on whether bisection is used to
  find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to
  conpute all and then discard any unwanted one.
  As an added benefit, SLARRE also outputs the n
  Gerschgorin intervals for the matrices L_i D_i L_i^T.

  Arguments
  =========

  RANGE   (input) CHARACTER
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  N       (input) INTEGER
          The order of the matrix. N > 0.

  VL      (input/output) REAL            
  VU      (input/output) REAL            
          If RANGE='V', the lower and upper bounds for the eigenvalues.
          Eigenvalues less than or equal to VL, or greater than VU,
          will not be returned.  VL < VU.
          If RANGE='I' or ='A', SLARRE computes bounds on the desired
          part of the spectrum.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N.

  D       (input/output) REAL             array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal
          matrix T.
          On exit, the N diagonal elements of the diagonal
          matrices D_i.

  E       (input/output) REAL             array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) need not be set.
          On exit, E contains the subdiagonal elements of the unit
          bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
          1 <= I <= NSPLIT, contain the base points sigma_i on output.


  E2      (input/output) REAL             array, dimension (N)
          On entry, the first (N-1) entries contain the SQUARES of the

          subdiagonal elements of the tridiagonal matrix T;
          E2(N) need not be set.
          On exit, the entries E2( ISPLIT( I ) ),
          1 <= I <= NSPLIT, have been set to zero

  RTOL1   (input) REAL            
  RTOL2   (input) REAL            
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

  SPLTOL (input) REAL            
          The threshold for splitting.

  NSPLIT  (output) INTEGER
          The number of blocks T splits into. 1 <= NSPLIT <= N.

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to ISPLIT(1),
          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

  M       (output) INTEGER
          The total number of eigenvalues (of all L_i D_i L_i^T)
          found.

  W       (output) REAL             array, dimension (N)
          The first M elements contain the eigenvalues. The
          eigenvalues of each of the blocks, L_i D_i L_i^T, are
          sorted in ascending order ( SLARRE may use the
          remaining N-M elements as workspace).

  WERR    (output) REAL             array, dimension (N)
          The error bound on the corresponding eigenvalue in W.

  WGAP    (output) REAL             array, dimension (N)
          The separation from the right neighbor eigenvalue in W.
          The gap is only with respect to the eigenvalues of the same b
          as each block has its own representation tree.
          Exception: at the right end of a block we store the left gap


  IBLOCK  (output) INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

  INDEXW  (output) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in block 2

  GERS    (output) REAL             array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)).

  PIVMIN  (output) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  WORK    (workspace) REAL             array, dimension (6*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (5*N)
          Workspace.

  INFO    (output) INTEGER
          = 0:  successful exit
          > 0:  A problem occured in SLARRE.
          < 0:  One of the called subroutines signaled an internal prob
                Needs inspection of the corresponding parameter IINFO
                for further information.

          =-1:  Problem in SLARRD.
          = 2:  No base representation could be found in MAXTRY iterati
                Increasing MAXTRY and recompilation might be a remedy.

          =-3:  Problem in SLARRB when computing the refined root
                representation for SLASQ2.
          =-4:  Problem in SLARRB when preforming bisection on the
                desired part of the spectrum.
          =-5:  Problem in SLASQ2.
          =-6:  Problem in SLASQ2.

  Further Details
  The base representations are required to suffer very little
  element growth and consequently define all their eigenvalues to
  high relative accuracy.
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        n -
        vl -
        vu -
        il -
        iu -
        d -
        e -
        e2 -
        rtol1 -
        rtol2 -
        spltol -
        nsplit -
        isplit -
        m -
        w -
        werr -
        wgap -
        iblock -
        indexw -
        gers -
        pivmin -
        work -
        iwork -
        info -

      • slarrf

        public void slarrf(int n,
                   float[] d,
                   float[] l,
                   float[] ld,
                   int clstrt,
                   int clend,
                   float[] w,
                   float[] wgap,
                   float[] werr,
                   float spdiam,
                   float clgapl,
                   float clgapr,
                   float pivmin,
                   org.netlib.util.floatW sigma,
                   float[] dplus,
                   float[] lplus,
                   float[] work,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the initial representation L D L^T and its cluster of close
  eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...

  W( CLEND ), SLARRF finds a new relatively robust representation
  L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
  eigenvalues of L(+) D(+) L(+)^T is relatively isolated.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix (subblock, if the matrix splitted).

  D       (input) REAL             array, dimension (N)
          The N diagonal elements of the diagonal matrix D.

  L       (input) REAL             array, dimension (N-1)
          The (N-1) subdiagonal elements of the unit bidiagonal
          matrix L.

  LD      (input) REAL             array, dimension (N-1)
          The (N-1) elements L(i)*D(i).

  CLSTRT  (input) INTEGER
          The index of the first eigenvalue in the cluster.

  CLEND   (input) INTEGER
          The index of the last eigenvalue in the cluster.

  W       (input) REAL             array, dimension >=  (CLEND-CLSTRT+1
          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.

          W( CLSTRT ) through W( CLEND ) form the cluster of relatively
          close eigenalues.

  WGAP    (input/output) REAL             array, dimension >=  (CLEND-C
          The separation from the right neighbor eigenvalue in W.

  WERR    (input) REAL             array, dimension >=  (CLEND-CLSTRT+1
          WERR contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue APPROXIMATION in W

  SPDIAM (input) estimate of the spectral diameter obtained from the
          Gerschgorin intervals

  CLGAPL, CLGAPR (input) absolute gap on each end of the cluster.
          Set by the calling routine to protect against shifts too clos
          to eigenvalues outside the cluster.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

  SIGMA   (output) REAL            
          The shift used to form L(+) D(+) L(+)^T.

  DPLUS   (output) REAL             array, dimension (N)
          The N diagonal elements of the diagonal matrix D(+).

  LPLUS   (output) REAL             array, dimension (N-1)
          The first (N-1) elements of LPLUS contain the subdiagonal
          elements of the unit bidiagonal matrix L(+).

  WORK    (workspace) REAL             array, dimension (2*N)
          Workspace.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        l -
        ld -
        clstrt -
        clend -
        w -
        wgap -
        werr -
        spdiam -
        clgapl -
        clgapr -
        pivmin -
        sigma -
        dplus -
        lplus -
        work -
        info -

      • slarrj

        public void slarrj(int n,
                   float[] d,
                   float[] e2,
                   int ifirst,
                   int ilast,
                   float rtol,
                   int offset,
                   float[] w,
                   float[] werr,
                   float[] work,
                   int[] iwork,
                   float pivmin,
                   float spdiam,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Given the initial eigenvalue approximations of T, SLARRJ
  does  bisection to refine the eigenvalues of T,
  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initi
  guesses for these eigenvalues are input in W, the corresponding estim
  of the error in these guesses in WERR. During bisection, intervals
  [left, right] are maintained by storing their mid-points and
  semi-widths in the arrays W and WERR respectively.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.

  D       (input) REAL             array, dimension (N)
          The N diagonal elements of T.

  E2      (input) REAL             array, dimension (N-1)
          The Squares of the (N-1) subdiagonal elements of T.

  IFIRST  (input) INTEGER
          The index of the first eigenvalue to be computed.

  ILAST   (input) INTEGER
          The index of the last eigenvalue to be computed.

  RTOL   (input) REAL            
          Tolerance for the convergence of the bisection intervals.
          An interval [LEFT,RIGHT] has converged if
          RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).

  OFFSET  (input) INTEGER
          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
          through ILAST-OFFSET elements of these arrays are to be used.

  W       (input/output) REAL             array, dimension (N)
          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
          estimates of the eigenvalues of L D L^T indexed IFIRST throug
          ILAST.
          On output, these estimates are refined.

  WERR    (input/output) REAL             array, dimension (N)
          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) 
          the errors in the estimates of the corresponding elements in 
          On output, these errors are refined.

  WORK    (workspace) REAL             array, dimension (2*N)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (2*N)
          Workspace.

  PIVMIN  (input) DOUBLE PRECISION
          The minimum pivot in the Sturm sequence for T.

  SPDIAM  (input) DOUBLE PRECISION
          The spectral diameter of T.

  INFO    (output) INTEGER
          Error flag.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e2 -
        ifirst -
        ilast -
        rtol -
        offset -
        w -
        werr -
        work -
        iwork -
        pivmin -
        spdiam -
        info -

      • slarrk

        public void slarrk(int n,
                   int iw,
                   float gl,
                   float gu,
                   float[] d,
                   float[] e2,
                   float pivmin,
                   float reltol,
                   org.netlib.util.floatW w,
                   org.netlib.util.floatW werr,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLARRK computes one eigenvalue of a symmetric tridiagonal
  matrix T to suitable accuracy. This is an auxiliary code to be
  called from SSTEMR.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  IW      (input) INTEGER
          The index of the eigenvalues to be returned.

  GL      (input) REAL            
  GU      (input) REAL            
          An upper and a lower bound on the eigenvalue.

  D       (input) REAL             array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E2      (input) REAL             array, dimension (N-1)
          The (n-1) squared off-diagonal elements of the tridiagonal ma

  PIVMIN  (input) REAL            
          The minimum pivot allowed in the Sturm sequence for T.

  RELTOL  (input) REAL            
          The minimum relative width of an interval.  When an interval

          is narrower than RELTOL times the larger (in
          magnitude) endpoint, then it is considered to be
          sufficiently small, i.e., converged.  Note: this should
          always be at least radix*machine epsilon.

  W       (output) REAL            

  WERR    (output) REAL            
          The error bound on the corresponding eigenvalue approximation
          in W.

  INFO    (output) INTEGER
          = 0:       Eigenvalue converged
          = -1:      Eigenvalue did NOT converge

  Internal Parameters
  ===================

  FUDGE   REAL            , default = 2
          A "fudge factor" to widen the Gershgorin intervals.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        iw -
        gl -
        gu -
        d -
        e2 -
        pivmin -
        reltol -
        w -
        werr -
        info -

      • slarrr

        public void slarrr(int n,
                   float[] d,
                   float[] e,
                   org.netlib.util.intW info)
        
     ..


  Purpose
  =======

  Perform tests to decide whether the symmetric tridiagonal matrix T
  warrants expensive computations which guarantee high relative accurac
  in the eigenvalues.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix. N > 0.

  D       (input) REAL             array, dimension (N)
          The N diagonal elements of the tridiagonal matrix T.

  E       (input/output) REAL             array, dimension (N)
          On entry, the first (N-1) entries contain the subdiagonal
          elements of the tridiagonal matrix T; E(N) is set to ZERO.

  INFO    (output) INTEGER
          INFO = 0(default) : the matrix warrants computations preservi
                              relative accuracy.
          INFO = 1          : the matrix warrants computations guarante
                              only absolute accuracy.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • slarrv

        public void slarrv(int n,
                   float vl,
                   float vu,
                   float[] d,
                   float[] l,
                   float pivmin,
                   int[] isplit,
                   int m,
                   int dol,
                   int dou,
                   float minrgp,
                   org.netlib.util.floatW rtol1,
                   org.netlib.util.floatW rtol2,
                   float[] w,
                   float[] werr,
                   float[] wgap,
                   int[] iblock,
                   int[] indexw,
                   float[] gers,
                   float[] z,
                   int ldz,
                   int[] isuppz,
                   float[] work,
                   int[] iwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLARRV computes the eigenvectors of the tridiagonal matrix
  T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L
  The input eigenvalues should have been computed by SLARRE.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  VL      (input) REAL            
  VU      (input) REAL            
          Lower and upper bounds of the interval that contains the desi
          eigenvalues. VL < VU. Needed to compute gaps on the left or r
          end of the extremal eigenvalues in the desired RANGE.

  D       (input/output) REAL             array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D.
          On exit, D may be overwritten.

  L       (input/output) REAL             array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not splitted.) At the end of each block
          is stored the corresponding shift as given by SLARRE.
          On exit, L is overwritten.

  PIVMIN  (in) DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.

  M       (input) INTEGER
          The total number of input eigenvalues.  0 <= M <= N.

  DOL     (input) INTEGER
  DOU     (input) INTEGER
          If the user wants to compute only selected eigenvectors from 
          the eigenvalues supplied, he can specify an index range DOL:D
          Or else the setting DOL=1, DOU=M should be applied.
          Note that DOL and DOU refer to the order in which the eigenva
          are stored in W.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain
          computed eigenvectors. All other columns of Z are set to zero

  MINRGP  (input) REAL            

  RTOL1   (input) REAL            
  RTOL2   (input) REAL            
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

  W       (input/output) REAL             array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from SLARRE is expected here ). Furthermore, they are with

          respect to the shift of the corresponding root representation
          for their block. On exit, W holds the eigenvalues of the
          UNshifted matrix.

  WERR    (input/output) REAL             array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W

  WGAP    (input/output) REAL             array, dimension (N)
          The separation from the right neighbor eigenvalue in W.

  IBLOCK  (input) INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

  INDEXW  (input) INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second bl

  GERS    (input) REAL             array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
          be computed from the original UNshifted matrix.

  Z       (output) REAL             array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The I-th eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I ).

  WORK    (workspace) REAL             array, dimension (12*N)

  IWORK   (workspace) INTEGER array, dimension (7*N)

  INFO    (output) INTEGER
          = 0:  successful exit

          > 0:  A problem occured in SLARRV.
          < 0:  One of the called subroutines signaled an internal prob
                Needs inspection of the corresponding parameter IINFO
                for further information.

          =-1:  Problem in SLARRB when refining a child's eigenvalues.

          =-2:  Problem in SLARRF when computing the RRR of a child.
                When a child is inside a tight cluster, it can be diffi
                to find an RRR. A partial remedy from the user's point 
                view is to make the parameter MINRGP smaller and recomp
                However, as the orthogonality of the computed vectors i
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINR
          =-3:  Problem in SLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps.

  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        vl -
        vu -
        d -
        l -
        pivmin -
        isplit -
        m -
        dol -
        dou -
        minrgp -
        rtol1 -
        rtol2 -
        w -
        werr -
        wgap -
        iblock -
        indexw -
        gers -
        z -
        ldz -
        isuppz -
        work -
        iwork -
        info -

      • slartg

        public abstract void slartg(float f,
                            float g,
                            org.netlib.util.floatW cs,
                            org.netlib.util.floatW sn,
                            org.netlib.util.floatW r)
        
     ..

  Purpose
  =======

  SLARTG generate a plane rotation so that

     [  CS  SN  ]  .  [ F ]  =  [ R ]   where CS**2 + SN**2 = 1.
     [ -SN  CS  ]     [ G ]     [ 0 ]

  This is a slower, more accurate version of the BLAS1 routine SROTG,
  with the following other differences:
     F and G are unchanged on return.
     If G=0, then CS=1 and SN=0.
     If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
        floating point operations (saves work in SBDSQR when
        there are zeros on the diagonal).

  If F exceeds G in magnitude, CS will be positive.

  Arguments
  =========

  F       (input) REAL
          The first component of vector to be rotated.

  G       (input) REAL
          The second component of vector to be rotated.

  CS      (output) REAL
          The cosine of the rotation.

  SN      (output) REAL
          The sine of the rotation.

  R       (output) REAL
          The nonzero component of the rotated vector.

  This version has a few statements commented out for thread safety
  (machine parameters are computed on each entry). 10 feb 03, SJH.

  =====================================================================

     .. Parameters ..
        Parameters:
        f -
        g -
        cs -
        sn -
        r -

      • slartv

        public abstract void slartv(int n,
                            float[] x,
                            int incx,
                            float[] y,
                            int incy,
                            float[] c,
                            float[] s,
                            int incc)
        
     ..

  Purpose
  =======

  SLARTV applies a vector of real plane rotations to elements of the
  real vectors x and y. For i = 1,2,...,n

     ( x(i) ) := (  c(i)  s(i) ) ( x(i) )
     ( y(i) )    ( -s(i)  c(i) ) ( y(i) )

  Arguments
  =========

  N       (input) INTEGER
          The number of plane rotations to be applied.

  X       (input/output) REAL array,
                         dimension (1+(N-1)*INCX)
          The vector x.

  INCX    (input) INTEGER
          The increment between elements of X. INCX > 0.

  Y       (input/output) REAL array,
                         dimension (1+(N-1)*INCY)
          The vector y.

  INCY    (input) INTEGER
          The increment between elements of Y. INCY > 0.

  C       (input) REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

  S       (input) REAL array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

  INCC    (input) INTEGER
          The increment between elements of C and S. INCC > 0.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        x -
        incx -
        y -
        incy -
        c -
        s -
        incc -

      • slaruv

        public abstract void slaruv(int[] iseed,
                            int n,
                            float[] x)
        
     ..

  Purpose
  =======

  SLARUV returns a vector of n random real numbers from a uniform (0,1)
  distribution (n <= 128).

  This is an auxiliary routine called by SLARNV and CLARNV.

  Arguments
  =========

  ISEED   (input/output) INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array

          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

  N       (input) INTEGER
          The number of random numbers to be generated. N <= 128.

  X       (output) REAL array, dimension (N)
          The generated random numbers.

  Further Details
  ===============

  This routine uses a multiplicative congruential method with modulus
  2**48 and multiplier 33952834046453 (see G.S.Fishman,
  'Multiplicative congruential random number generators with modulus
  2**b: an exhaustive analysis for b = 32 and a partial analysis for
  b = 48', Math. Comp. 189, pp 331-344, 1990).

  48-bit integers are stored in 4 integer array elements with 12 bits
  per element. Hence the routine is portable across machines with
  integers of 32 bits or more.

  =====================================================================

     .. Parameters ..
        Parameters:
        iseed -
        n -
        x -

      • slarz

        public abstract void slarz(java.lang.String side,
                           int m,
                           int n,
                           int l,
                           float[] v,
                           int incv,
                           float tau,
                           float[] c,
                           int Ldc,
                           float[] work)
        
     ..

  Purpose
  =======

  SLARZ applies a real elementary reflector H to a real M-by-N
  matrix C, from either the left or the right. H is represented in the

  form

        H = I - tau * v * v'

  where tau is a real scalar and v is a real vector.

  If tau = 0, then H is taken to be the unit matrix.


  H is a product of k elementary reflectors as returned by STZRZF.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  L       (input) INTEGER
          The number of entries of the vector V containing
          the meaningful part of the Householder vectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  V       (input) REAL array, dimension (1+(L-1)*abs(INCV))
          The vector v in the representation of H as returned by
          STZRZF. V is not used if TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0.

  TAU     (input) REAL
          The value tau in the representation of H.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',

          or C * H if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        l -
        v -
        incv -
        tau -
        c -
        Ldc -
        work -

      • slarzb

        public abstract void slarzb(java.lang.String side,
                            java.lang.String trans,
                            java.lang.String direct,
                            java.lang.String storev,
                            int m,
                            int n,
                            int k,
                            int l,
                            float[] v,
                            int ldv,
                            float[] t,
                            int ldt,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int ldwork)
        
     ..

  Purpose
  =======

  SLARZB applies a real block reflector H or its transpose H**T to
  a real distributed M-by-N  C from the left or the right.

  Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply H or H' from the Left
          = 'R': apply H or H' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H' (Transpose)

  DIRECT  (input) CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)

          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise                        (not supported yet)

          = 'R': Rowwise

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  K       (input) INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

  L       (input) INTEGER
          The number of columns of the matrix V containing the
          meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  V       (input) REAL array, dimension (LDV,NV).
          If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.

  T       (input) REAL array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by H*C or H'*C or C*H or C*H'.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension (LDWORK,K)

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        direct -
        storev -
        m -
        n -
        k -
        l -
        v -
        ldv -
        t -
        ldt -
        c -
        Ldc -
        work -
        ldwork -

      • slarzt

        public abstract void slarzt(java.lang.String direct,
                            java.lang.String storev,
                            int n,
                            int k,
                            float[] v,
                            int ldv,
                            float[] tau,
                            float[] t,
                            int ldt)
        
     ..

  Purpose
  =======

  SLARZT forms the triangular factor T of a real block reflector
  H of order > n, which is defined as a product of k elementary
  reflectors.

  If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;


  If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.


  If STOREV = 'C', the vector which defines the elementary reflector
  H(i) is stored in the i-th column of the array V, and

     H  =  I - V * T * V'

  If STOREV = 'R', the vector which defines the elementary reflector
  H(i) is stored in the i-th row of the array V, and

     H  =  I - V' * T * V

  Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

  Arguments
  =========

  DIRECT  (input) CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)

          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise                        (not supported yet)

          = 'R': rowwise

  N       (input) INTEGER
          The order of the block reflector H. N >= 0.

  K       (input) INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

  V       (input/output) REAL array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

  LDV     (input) INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.


  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

  T       (output) REAL array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= K.

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and

  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

                                              ______V_____
         ( v1 v2 v3 )                        /            \
         ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
     V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
         ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
         ( v1 v2 v3 )
            .  .  .
            .  .  .
            1  .  .
               1  .
                  1

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

                                                        ______V_____
            1                                          /            \
            .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
            .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
            .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
            .  .  .
         ( v1 v2 v3 )
         ( v1 v2 v3 )
     V = ( v1 v2 v3 )
         ( v1 v2 v3 )
         ( v1 v2 v3 )

  =====================================================================

     .. Parameters ..
        Parameters:
        direct -
        storev -
        n -
        k -
        v -
        ldv -
        tau -
        t -
        ldt -

      • slas2

        public abstract void slas2(float f,
                           float g,
                           float h,
                           org.netlib.util.floatW ssmin,
                           org.netlib.util.floatW ssmax)
        
     ..

  Purpose
  =======

  SLAS2  computes the singular values of the 2-by-2 matrix
     [  F   G  ]
     [  0   H  ].
  On return, SSMIN is the smaller singular value and SSMAX is the
  larger singular value.

  Arguments
  =========

  F       (input) REAL
          The (1,1) element of the 2-by-2 matrix.

  G       (input) REAL
          The (1,2) element of the 2-by-2 matrix.

  H       (input) REAL
          The (2,2) element of the 2-by-2 matrix.

  SSMIN   (output) REAL
          The smaller singular value.

  SSMAX   (output) REAL
          The larger singular value.

  Further Details
  ===============

  Barring over/underflow, all output quantities are correct to within
  a few units in the last place (ulps), even in the absence of a guard

  digit in addition/subtraction.

  In IEEE arithmetic, the code works correctly if one matrix element is
  infinite.

  Overflow will not occur unless the largest singular value itself
  overflows, or is within a few ulps of overflow. (On machines with
  partial overflow, like the Cray, overflow may occur if the largest
  singular value is within a factor of 2 of overflow.)

  Underflow is harmless if underflow is gradual. Otherwise, results
  may correspond to a matrix modified by perturbations of size near
  the underflow threshold.

  ====================================================================


     .. Parameters ..
        Parameters:
        f -
        g -
        h -
        ssmin -
        ssmax -

      • slascl

        public abstract void slascl(java.lang.String type,
                            int kl,
                            int ku,
                            float cfrom,
                            float cto,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASCL multiplies the M by N real matrix A by the real scalar
  CTO/CFROM.  This is done without over/underflow as long as the final

  result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that

  A may be full, upper triangular, lower triangular, upper Hessenberg,

  or banded.

  Arguments
  =========

  TYPE    (input) CHARACTER*1
          TYPE indices the storage type of the input matrix.
          = 'G':  A is a full matrix.
          = 'L':  A is a lower triangular matrix.
          = 'U':  A is an upper triangular matrix.
          = 'H':  A is an upper Hessenberg matrix.
          = 'B':  A is a symmetric band matrix with lower bandwidth KL

                  and upper bandwidth KU and with the only the lower
                  half stored.
          = 'Q':  A is a symmetric band matrix with lower bandwidth KL

                  and upper bandwidth KU and with the only the upper
                  half stored.
          = 'Z':  A is a band matrix with lower bandwidth KL and upper

                  bandwidth KU.

  KL      (input) INTEGER
          The lower bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

  KU      (input) INTEGER
          The upper bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

  CFROM   (input) REAL
  CTO     (input) REAL
          The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
          without over/underflow if the final result CTO*A(I,J)/CFROM
          can be represented without over/underflow.  CFROM must be
          nonzero.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          The matrix to be multiplied by CTO/CFROM.  See TYPE for the
          storage type.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  INFO    (output) INTEGER
          0  - successful exit
          <0 - if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        type -
        kl -
        ku -
        cfrom -
        cto -
        m -
        n -
        a -
        lda -
        info -

      • slasd0

        public abstract void slasd0(int n,
                            int sqre,
                            float[] d,
                            float[] e,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            int smlsiz,
                            int[] iwork,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Using a divide and conquer approach, SLASD0 computes the singular
  value decomposition (SVD) of a real upper bidiagonal N-by-M
  matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
  The algorithm computes orthogonal matrices U and VT such that
  B = U * S * VT. The singular values S are overwritten on D.

  A related subroutine, SLASDA, computes only the singular values,
  and optionally, the singular vectors in compact form.

  Arguments
  =========

  N      (input) INTEGER
         On entry, the row dimension of the upper bidiagonal matrix.
         This is also the dimension of the main diagonal array D.

  SQRE   (input) INTEGER
         Specifies the column dimension of the bidiagonal matrix.
         = 0: The bidiagonal matrix has column dimension M = N;
         = 1: The bidiagonal matrix has column dimension M = N+1;

  D      (input/output) REAL array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix.
         On exit D, if INFO = 0, contains its singular values.

  E      (input) REAL array, dimension (M-1)
         Contains the subdiagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

  U      (output) REAL array, dimension at least (LDQ, N)
         On exit, U contains the left singular vectors.

  LDU    (input) INTEGER
         On entry, leading dimension of U.

  VT     (output) REAL array, dimension at least (LDVT, M)
         On exit, VT' contains the right singular vectors.

  LDVT   (input) INTEGER
         On entry, leading dimension of VT.

  SMLSIZ (input) INTEGER
         On entry, maximum size of the subproblems at the
         bottom of the computation tree.

  IWORK  (workspace) INTEGER array, dimension (8*N)

  WORK   (workspace) REAL array, dimension (3*M**2+2*M)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        sqre -
        d -
        e -
        u -
        ldu -
        vt -
        ldvt -
        smlsiz -
        iwork -
        work -
        info -

      • slasd1

        public abstract void slasd1(int nl,
                            int nr,
                            int sqre,
                            float[] d,
                            org.netlib.util.floatW alpha,
                            org.netlib.util.floatW beta,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            int[] idxq,
                            int[] iwork,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
  where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

  A related subroutine SLASD7 handles the case in which the singular
  values (and the singular vectors in factored form) are desired.

  SLASD1 computes the SVD as follows:

                ( D1(in)  0    0     0 )
    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                (   0     0   D2(in) 0 )

      = U(out) * ( D(out) 0) * VT(out)

  where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
  elsewhere; and the entry b is empty if SQRE = 0.

  The left singular vectors of the original matrix are stored in U, and
  the transpose of the right singular vectors are stored in VT, and the
  singular values are in D.  The algorithm consists of three stages:

     The first stage consists of deflating the size of the problem
     when there are multiple singular values or when there are zeros in
     the Z vector.  For each such occurence the dimension of the
     secular equation problem is reduced by one.  This stage is
     performed by the routine SLASD2.

     The second stage consists of calculating the updated
     singular values. This is done by finding the square roots of the
     roots of the secular equation via the routine SLASD4 (as called
     by SLASD3). This routine also calculates the singular vectors of
     the current problem.

     The final stage consists of computing the updated singular vectors
     directly using the updated singular values.  The singular vectors

     for the current problem are multiplied with the singular vectors
     from the overall problem.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  D      (input/output) REAL array, dimension (NL+NR+1).
         N = NL+NR+1
         On entry D(1:NL,1:NL) contains the singular values of the
         upper block; and D(NL+2:N) contains the singular values of
         the lower block. On exit D(1:N) contains the singular values
         of the modified matrix.

  ALPHA  (input/output) REAL
         Contains the diagonal element associated with the added row.

  BETA   (input/output) REAL
         Contains the off-diagonal element associated with the added
         row.

  U      (input/output) REAL array, dimension (LDU,N)
         On entry U(1:NL, 1:NL) contains the left singular vectors of
         the upper block; U(NL+2:N, NL+2:N) contains the left singular

         vectors of the lower block. On exit U contains the left
         singular vectors of the bidiagonal matrix.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= max( 1, N ).

  VT     (input/output) REAL array, dimension (LDVT,M)
         where M = N + SQRE.
         On entry VT(1:NL+1, 1:NL+1)' contains the right singular
         vectors of the upper block; VT(NL+2:M, NL+2:M)' contains
         the right singular vectors of the lower block. On exit
         VT' contains the right singular vectors of the
         bidiagonal matrix.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= max( 1, M ).

  IDXQ  (output) INTEGER array, dimension (N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order, i.e.
         D( IDXQ( I = 1, N ) ) will be in ascending order.

  IWORK  (workspace) INTEGER array, dimension (4*N)

  WORK   (workspace) REAL array, dimension (3*M**2+2*M)

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        d -
        alpha -
        beta -
        u -
        ldu -
        vt -
        ldvt -
        idxq -
        iwork -
        work -
        info -

      • slasd2

        public abstract void slasd2(int nl,
                            int nr,
                            int sqre,
                            org.netlib.util.intW k,
                            float[] d,
                            float[] z,
                            float alpha,
                            float beta,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int ldvt,
                            float[] dsigma,
                            float[] u2,
                            int ldu2,
                            float[] vt2,
                            int ldvt2,
                            int[] idxp,
                            int[] idx,
                            int[] idxc,
                            int[] idxq,
                            int[] coltyp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD2 merges the two sets of singular values together into a single

  sorted set.  Then it tries to deflate the size of the problem.
  There are two ways in which deflation can occur:  when two or more
  singular values are close together or if there is a tiny entry in the
  Z vector.  For each such occurrence the order of the related secular

  equation problem is reduced by one.

  SLASD2 is called from SLASD1.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  D      (input/output) REAL array, dimension (N)
         On entry D contains the singular values of the two submatrices
         to be combined.  On exit D contains the trailing (N-K) updated
         singular values (those which were deflated) sorted into
         increasing order.

  Z      (output) REAL array, dimension (N)
         On exit Z contains the updating row vector in the secular
         equation.

  ALPHA  (input) REAL
         Contains the diagonal element associated with the added row.

  BETA   (input) REAL
         Contains the off-diagonal element associated with the added
         row.

  U      (input/output) REAL array, dimension (LDU,N)
         On entry U contains the left singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL, NL), and (NL+2, NL+2), (N,N).
         On exit U contains the trailing (N-K) updated left singular
         vectors (those which were deflated) in its last N-K columns.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= N.

  VT     (input/output) REAL array, dimension (LDVT,M)
         On entry VT' contains the right singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
         On exit VT' contains the trailing (N-K) updated right singular
         vectors (those which were deflated) in its last N-K columns.
         In case SQRE =1, the last row of VT spans the right null
         space.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= M.

  DSIGMA (output) REAL array, dimension (N)
         Contains a copy of the diagonal elements (K-1 singular values

         and one zero) in the secular equation.

  U2     (output) REAL array, dimension (LDU2,N)
         Contains a copy of the first K-1 left singular vectors which
         will be used by SLASD3 in a matrix multiply (SGEMM) to solve
         for the new left singular vectors. U2 is arranged into four
         blocks. The first block contains a column with 1 at NL+1 and
         zero everywhere else; the second block contains non-zero
         entries only at and above NL; the third contains non-zero
         entries only below NL+1; and the fourth is dense.

  LDU2   (input) INTEGER
         The leading dimension of the array U2.  LDU2 >= N.

  VT2    (output) REAL array, dimension (LDVT2,N)
         VT2' contains a copy of the first K right singular vectors
         which will be used by SLASD3 in a matrix multiply (SGEMM) to
         solve for the new right singular vectors. VT2 is arranged into
         three blocks. The first block contains a row that corresponds

         to the special 0 diagonal element in SIGMA; the second block
         contains non-zeros only at and before NL +1; the third block
         contains non-zeros only at and after  NL +2.

  LDVT2  (input) INTEGER
         The leading dimension of the array VT2.  LDVT2 >= M.

  IDXP   (workspace) INTEGER array, dimension (N)
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output IDXP(2:K)
         points to the nondeflated D-values and IDXP(K+1:N)
         points to the deflated singular values.

  IDX    (workspace) INTEGER array, dimension (N)
         This will contain the permutation used to sort the contents of
         D into ascending order.

  IDXC   (output) INTEGER array, dimension (N)
         This will contain the permutation used to arrange the columns

         of the deflated U matrix into three groups:  the first group
         contains non-zero entries only at and above NL, the second
         contains non-zero entries only below NL+2, and the third is
         dense.

  IDXQ   (input/output) INTEGER array, dimension (N)
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that entries in

         the first hlaf of this permutation must first be moved one
         position backward; and entries in the second half
         must first have NL+1 added to their values.

  COLTYP (workspace/output) INTEGER array, dimension (N)
         As workspace, this will contain a label which will indicate
         which of the following types a column in the U2 matrix or a
         row in the VT2 matrix is:
         1 : non-zero in the upper half only
         2 : non-zero in the lower half only
         3 : dense
         4 : deflated

         On exit, it is an array of dimension 4, with COLTYP(I) being
         the dimension of the I-th type columns.

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        k -
        d -
        z -
        alpha -
        beta -
        u -
        ldu -
        vt -
        ldvt -
        dsigma -
        u2 -
        ldu2 -
        vt2 -
        ldvt2 -
        idxp -
        idx -
        idxc -
        idxq -
        coltyp -
        info -

      • slasd3

        public abstract void slasd3(int nl,
                            int nr,
                            int sqre,
                            int k,
                            float[] d,
                            float[] q,
                            int ldq,
                            float[] dsigma,
                            float[] u,
                            int ldu,
                            float[] u2,
                            int ldu2,
                            float[] vt,
                            int ldvt,
                            float[] vt2,
                            int ldvt2,
                            int[] idxc,
                            int[] ctot,
                            float[] z,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD3 finds all the square roots of the roots of the secular
  equation, as defined by the values in D and Z.  It makes the
  appropriate calls to SLASD4 and then updates the singular
  vectors by matrix multiplication.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.

  SLASD3 is called from SLASD1.

  Arguments
  =========

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (input) INTEGER
         The size of the secular equation, 1 =< K = < N.

  D      (output) REAL array, dimension(K)
         On exit the square roots of the roots of the secular equation,
         in ascending order.

  Q      (workspace) REAL array,
                     dimension at least (LDQ,K).

  LDQ    (input) INTEGER
         The leading dimension of the array Q.  LDQ >= K.

  DSIGMA (input/output) REAL array, dimension(K)
         The first K elements of this array contain the old roots
         of the deflated updating problem.  These are the poles
         of the secular equation.

  U      (output) REAL array, dimension (LDU, N)
         The last N - K columns of this matrix contain the deflated
         left singular vectors.

  LDU    (input) INTEGER
         The leading dimension of the array U.  LDU >= N.

  U2     (input) REAL array, dimension (LDU2, N)
         The first K columns of this matrix contain the non-deflated
         left singular vectors for the split problem.

  LDU2   (input) INTEGER
         The leading dimension of the array U2.  LDU2 >= N.

  VT     (output) REAL array, dimension (LDVT, M)
         The last M - K columns of VT' contain the deflated
         right singular vectors.

  LDVT   (input) INTEGER
         The leading dimension of the array VT.  LDVT >= N.

  VT2    (input/output) REAL array, dimension (LDVT2, N)
         The first K columns of VT2' contain the non-deflated
         right singular vectors for the split problem.

  LDVT2  (input) INTEGER
         The leading dimension of the array VT2.  LDVT2 >= N.

  IDXC   (input) INTEGER array, dimension (N)
         The permutation used to arrange the columns of U (and rows of

         VT) into three groups:  the first group contains non-zero
         entries only at and above (or before) NL +1; the second
         contains non-zero entries only at and below (or after) NL+2;
         and the third is dense. The first column of U and the row of
         VT are treated separately, however.

         The rows of the singular vectors found by SLASD4
         must be likewise permuted before the matrix multiplies can
         take place.

  CTOT   (input) INTEGER array, dimension (4)
         A count of the total number of the various types of columns
         in U (or rows in VT), as described in IDXC. The fourth column

         type is any column which has been deflated.

  Z      (input/output) REAL array, dimension (K)
         The first K elements of this array contain the components
         of the deflation-adjusted updating row vector.

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        nl -
        nr -
        sqre -
        k -
        d -
        q -
        ldq -
        dsigma -
        u -
        ldu -
        u2 -
        ldu2 -
        vt -
        ldvt -
        vt2 -
        ldvt2 -
        idxc -
        ctot -
        z -
        info -

      • slasd4

        public abstract void slasd4(int n,
                            int i,
                            float[] d,
                            float[] z,
                            float[] delta,
                            float rho,
                            org.netlib.util.floatW sigma,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This subroutine computes the square root of the I-th updated
  eigenvalue of a positive symmetric rank-one modification to
  a positive diagonal matrix whose entries are given as the squares
  of the corresponding entries in the array d, and that

         0 <= D(i) < D(j)  for  i < j

  and that RHO > 0. This is arranged by the calling routine, and is
  no loss in generality.  The rank-one modified system is thus

         diag( D ) * diag( D ) +  RHO *  Z * Z_transpose.

  where we assume the Euclidean norm of Z is 1.

  The method consists of approximating the rational functions in the
  secular equation by simpler interpolating rational functions.

  Arguments
  =========

  N      (input) INTEGER
         The length of all arrays.

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

  D      (input) REAL array, dimension ( N )
         The original eigenvalues.  It is assumed that they are in
         order, 0 <= D(I) < D(J)  for I < J.

  Z      (input) REAL array, dimension (N)
         The components of the updating vector.

  DELTA  (output) REAL array, dimension (N)
         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
         contains the information necessary to construct the
         (singular) eigenvectors.

  RHO    (input) REAL
         The scalar in the symmetric updating formula.

  SIGMA  (output) REAL
         The computed sigma_I, the I-th updated eigenvalue.

  WORK   (workspace) REAL array, dimension (N)
         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
         component.  If N = 1, then WORK( 1 ) = 1.

  INFO   (output) INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Internal Parameters
  ===================

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.

            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1

  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
  if we are working with THREE poles!

  MAXIT is the maximum number of iterations allowed for each
  eigenvalue.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        i -
        d -
        z -
        delta -
        rho -
        sigma -
        work -
        info -

      • slasd5

        public abstract void slasd5(int i,
                            float[] d,
                            float[] z,
                            float[] delta,
                            float rho,
                            org.netlib.util.floatW dsigma,
                            float[] work)
        
     ..

  Purpose
  =======

  This subroutine computes the square root of the I-th eigenvalue
  of a positive symmetric rank-one modification of a 2-by-2 diagonal
  matrix

             diag( D ) * diag( D ) +  RHO *  Z * transpose(Z) .

  The diagonal entries in the array D are assumed to satisfy

             0 <= D(i) < D(j)  for  i < j .

  We also assume RHO > 0 and that the Euclidean norm of the vector
  Z is one.

  Arguments
  =========

  I      (input) INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

  D      (input) REAL array, dimension (2)
         The original eigenvalues.  We assume 0 <= D(1) < D(2).

  Z      (input) REAL array, dimension (2)
         The components of the updating vector.

  DELTA  (output) REAL array, dimension (2)
         Contains (D(j) - sigma_I) in its  j-th component.
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

  RHO    (input) REAL
         The scalar in the symmetric updating formula.

  DSIGMA (output) REAL
         The computed sigma_I, the I-th updated eigenvalue.

  WORK   (workspace) REAL array, dimension (2)
         WORK contains (D(j) + sigma_I) in its  j-th component.

  Further Details
  ===============

  Based on contributions by
     Ren-Cang Li, Computer Science Division, University of California
     at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        i -
        d -
        z -
        delta -
        rho -
        dsigma -
        work -

      • slasd6

        public abstract void slasd6(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            float[] d,
                            float[] vf,
                            float[] vl,
                            org.netlib.util.floatW alpha,
                            org.netlib.util.floatW beta,
                            int[] idxq,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            int ldgcol,
                            float[] givnum,
                            int ldgnum,
                            float[] poles,
                            float[] difl,
                            float[] difr,
                            float[] z,
                            org.netlib.util.intW k,
                            org.netlib.util.floatW c,
                            org.netlib.util.floatW s,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD6 computes the SVD of an updated upper bidiagonal matrix B
  obtained by merging two smaller ones by appending a row. This
  routine is used only for the problem which requires all singular
  values and optionally singular vector matrices in factored form.
  B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
  A related subroutine, SLASD1, handles the case in which all singular

  values and singular vectors of the bidiagonal matrix are desired.

  SLASD6 computes the SVD as follows:

                ( D1(in)  0    0     0 )
    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                (   0     0   D2(in) 0 )

      = U(out) * ( D(out) 0) * VT(out)

  where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
  elsewhere; and the entry b is empty if SQRE = 0.

  The singular values of B can be computed using D1, D2, the first
  components of all the right singular vectors of the lower block, and

  the last components of all the right singular vectors of the upper
  block. These components are stored and updated in VF and VL,
  respectively, in SLASD6. Hence U and VT are not explicitly
  referenced.

  The singular values are stored in D. The algorithm consists of two
  stages:

        The first stage consists of deflating the size of the problem
        when there are multiple singular values or if there is a zero
        in the Z vector. For each such occurence the dimension of the
        secular equation problem is reduced by one. This stage is
        performed by the routine SLASD7.

        The second stage consists of calculating the updated
        singular values. This is done by finding the roots of the
        secular equation via the routine SLASD4 (as called by SLASD8).

        This routine also updates VF and VL and computes the distances

        between the updated singular values and the old singular
        values.

  SLASD6 is called from SLASDA.

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Compute singular values only.
         = 1: Compute singular vectors in factored form as well.

  NL     (input) INTEGER
         The row dimension of the upper block.  NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block.  NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

  D      (input/output) REAL array, dimension (NL+NR+1).
         On entry D(1:NL,1:NL) contains the singular values of the
         upper block, and D(NL+2:N) contains the singular values
         of the lower block. On exit D(1:N) contains the singular
         values of the modified matrix.

  VF     (input/output) REAL array, dimension (M)
         On entry, VF(1:NL+1) contains the first components of all
         right singular vectors of the upper block; and VF(NL+2:M)
         contains the first components of all right singular vectors
         of the lower block. On exit, VF contains the first components

         of all right singular vectors of the bidiagonal matrix.

  VL     (input/output) REAL array, dimension (M)
         On entry, VL(1:NL+1) contains the  last components of all
         right singular vectors of the upper block; and VL(NL+2:M)
         contains the last components of all right singular vectors of

         the lower block. On exit, VL contains the last components of
         all right singular vectors of the bidiagonal matrix.

  ALPHA  (input/output) REAL
         Contains the diagonal element associated with the added row.

  BETA   (input/output) REAL
         Contains the off-diagonal element associated with the added
         row.

  IDXQ   (output) INTEGER array, dimension (N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order, i.e.
         D( IDXQ( I = 1, N ) ) will be in ascending order.

  PERM   (output) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) to be applied
         to each block. Not referenced if ICOMPQ = 0.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem. Not referenced if ICOMPQ = 0.

  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation. Not referenced if ICOMPQ = 0.

  LDGCOL (input) INTEGER
         leading dimension of GIVCOL, must be at least N.

  GIVNUM (output) REAL array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value to be used in the
         corresponding Givens rotation. Not referenced if ICOMPQ = 0.

  LDGNUM (input) INTEGER
         The leading dimension of GIVNUM and POLES, must be at least N.

  POLES  (output) REAL array, dimension ( LDGNUM, 2 )
         On exit, POLES(1,*) is an array containing the new singular
         values obtained from solving the secular equation, and
         POLES(2,*) is an array containing the poles in the secular
         equation. Not referenced if ICOMPQ = 0.

  DIFL   (output) REAL array, dimension ( N )
         On exit, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

  DIFR   (output) REAL array,
                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         On exit, DIFR(I, 1) is the distance between I-th updated
         (undeflated) singular value and the I+1-th (undeflated) old
         singular value.

         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
         normalizing factors for the right singular vector matrix.

         See SLASD8 for details on DIFL and DIFR.

  Z      (output) REAL array, dimension ( M )
         The first elements of this array contain the components
         of the deflation-adjusted updating row vector.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

  C      (output) REAL
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (output) REAL
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  WORK   (workspace) REAL array, dimension ( 4 * M )

  IWORK  (workspace) INTEGER array, dimension ( 3 * N )

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        d -
        vf -
        vl -
        alpha -
        beta -
        idxq -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        poles -
        difl -
        difr -
        z -
        k -
        c -
        s -
        work -
        iwork -
        info -

      • slasd7

        public abstract void slasd7(int icompq,
                            int nl,
                            int nr,
                            int sqre,
                            org.netlib.util.intW k,
                            float[] d,
                            float[] z,
                            float[] zw,
                            float[] vf,
                            float[] vfw,
                            float[] vl,
                            float[] vlw,
                            float alpha,
                            float beta,
                            float[] dsigma,
                            int[] idx,
                            int[] idxp,
                            int[] idxq,
                            int[] perm,
                            org.netlib.util.intW givptr,
                            int[] givcol,
                            int ldgcol,
                            float[] givnum,
                            int ldgnum,
                            org.netlib.util.floatW c,
                            org.netlib.util.floatW s,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD7 merges the two sets of singular values together into a single

  sorted set. Then it tries to deflate the size of the problem. There
  are two ways in which deflation can occur:  when two or more singular
  values are close together or if there is a tiny entry in the Z
  vector. For each such occurrence the order of the related
  secular equation problem is reduced by one.

  SLASD7 is called from SLASD6.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          Specifies whether singular vectors are to be computed
          in compact form, as follows:
          = 0: Compute singular values only.
          = 1: Compute singular vectors of upper
               bidiagonal matrix in compact form.

  NL     (input) INTEGER
         The row dimension of the upper block. NL >= 1.

  NR     (input) INTEGER
         The row dimension of the lower block. NR >= 1.

  SQRE   (input) INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has
         N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.

  K      (output) INTEGER
         Contains the dimension of the non-deflated matrix, this is
         the order of the related secular equation. 1 <= K <=N.

  D      (input/output) REAL array, dimension ( N )
         On entry D contains the singular values of the two submatrices
         to be combined. On exit D contains the trailing (N-K) updated

         singular values (those which were deflated) sorted into
         increasing order.

  Z      (output) REAL array, dimension ( M )
         On exit Z contains the updating row vector in the secular
         equation.

  ZW     (workspace) REAL array, dimension ( M )
         Workspace for Z.

  VF     (input/output) REAL array, dimension ( M )
         On entry, VF(1:NL+1) contains the first components of all
         right singular vectors of the upper block; and VF(NL+2:M)
         contains the first components of all right singular vectors
         of the lower block. On exit, VF contains the first components

         of all right singular vectors of the bidiagonal matrix.

  VFW    (workspace) REAL array, dimension ( M )
         Workspace for VF.

  VL     (input/output) REAL array, dimension ( M )
         On entry, VL(1:NL+1) contains the  last components of all
         right singular vectors of the upper block; and VL(NL+2:M)
         contains the last components of all right singular vectors
         of the lower block. On exit, VL contains the last components
         of all right singular vectors of the bidiagonal matrix.

  VLW    (workspace) REAL array, dimension ( M )
         Workspace for VL.

  ALPHA  (input) REAL
         Contains the diagonal element associated with the added row.

  BETA   (input) REAL
         Contains the off-diagonal element associated with the added
         row.

  DSIGMA (output) REAL array, dimension ( N )
         Contains a copy of the diagonal elements (K-1 singular values

         and one zero) in the secular equation.

  IDX    (workspace) INTEGER array, dimension ( N )
         This will contain the permutation used to sort the contents of
         D into ascending order.

  IDXP   (workspace) INTEGER array, dimension ( N )
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output IDXP(2:K)
         points to the nondeflated D-values and IDXP(K+1:N)
         points to the deflated singular values.

  IDXQ   (input) INTEGER array, dimension ( N )
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that entries in

         the first half of this permutation must first be moved one
         position backward; and entries in the second half
         must first have NL+1 added to their values.

  PERM   (output) INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) to be applied
         to each singular block. Not referenced if ICOMPQ = 0.

  GIVPTR (output) INTEGER
         The number of Givens rotations which took place in this
         subproblem. Not referenced if ICOMPQ = 0.

  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation. Not referenced if ICOMPQ = 0.

  LDGCOL (input) INTEGER
         The leading dimension of GIVCOL, must be at least N.

  GIVNUM (output) REAL array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value to be used in the
         corresponding Givens rotation. Not referenced if ICOMPQ = 0.

  LDGNUM (input) INTEGER
         The leading dimension of GIVNUM, must be at least N.

  C      (output) REAL
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

  S      (output) REAL
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        nl -
        nr -
        sqre -
        k -
        d -
        z -
        zw -
        vf -
        vfw -
        vl -
        vlw -
        alpha -
        beta -
        dsigma -
        idx -
        idxp -
        idxq -
        perm -
        givptr -
        givcol -
        ldgcol -
        givnum -
        ldgnum -
        c -
        s -
        info -

      • slasd8

        public abstract void slasd8(int icompq,
                            int k,
                            float[] d,
                            float[] z,
                            float[] vf,
                            float[] vl,
                            float[] difl,
                            float[] difr,
                            int lddifr,
                            float[] dsigma,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASD8 finds the square roots of the roots of the secular equation,
  as defined by the values in DSIGMA and Z. It makes the appropriate
  calls to SLASD4, and stores, for each  element in D, the distance
  to its two nearest poles (elements in DSIGMA). It also updates
  the arrays VF and VL, the first and last components of all the
  right singular vectors of the original bidiagonal matrix.

  SLASD8 is called from SLASD6.

  Arguments
  =========

  ICOMPQ  (input) INTEGER
          Specifies whether singular vectors are to be computed in
          factored form in the calling routine:
          = 0: Compute singular values only.
          = 1: Compute singular vectors in factored form as well.

  K       (input) INTEGER
          The number of terms in the rational function to be solved
          by SLASD4.  K >= 1.

  D       (output) REAL array, dimension ( K )
          On output, D contains the updated singular values.

  Z       (input) REAL array, dimension ( K )
          The first K elements of this array contain the components
          of the deflation-adjusted updating row vector.

  VF      (input/output) REAL array, dimension ( K )
          On entry, VF contains  information passed through DBEDE8.
          On exit, VF contains the first K components of the first
          components of all right singular vectors of the bidiagonal
          matrix.

  VL      (input/output) REAL array, dimension ( K )
          On entry, VL contains  information passed through DBEDE8.
          On exit, VL contains the first K components of the last
          components of all right singular vectors of the bidiagonal
          matrix.

  DIFL    (output) REAL array, dimension ( K )
          On exit, DIFL(I) = D(I) - DSIGMA(I).

  DIFR    (output) REAL array,
                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
                   dimension ( K ) if ICOMPQ = 0.
          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
          defined and will not be referenced.

          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
          normalizing factors for the right singular vector matrix.

  LDDIFR  (input) INTEGER
          The leading dimension of DIFR, must be at least K.

  DSIGMA  (input) REAL array, dimension ( K )
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

  WORK    (workspace) REAL array, dimension at least 3 * K

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        k -
        d -
        z -
        vf -
        vl -
        difl -
        difr -
        lddifr -
        dsigma -
        work -
        info -

      • slasda

        public abstract void slasda(int icompq,
                            int smlsiz,
                            int n,
                            int sqre,
                            float[] d,
                            float[] e,
                            float[] u,
                            int ldu,
                            float[] vt,
                            int[] k,
                            float[] difl,
                            float[] difr,
                            float[] z,
                            float[] poles,
                            int[] givptr,
                            int[] givcol,
                            int ldgcol,
                            int[] perm,
                            float[] givnum,
                            float[] c,
                            float[] s,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Using a divide and conquer approach, SLASDA computes the singular
  value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
  B with diagonal D and offdiagonal E, where M = N + SQRE. The
  algorithm computes the singular values in the SVD B = U * S * VT.
  The orthogonal matrices U and VT are optionally computed in
  compact form.

  A related subroutine, SLASD0, computes the singular values and
  the singular vectors in explicit form.

  Arguments
  =========

  ICOMPQ (input) INTEGER
         Specifies whether singular vectors are to be computed
         in compact form, as follows
         = 0: Compute singular values only.
         = 1: Compute singular vectors of upper bidiagonal
              matrix in compact form.

  SMLSIZ (input) INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

  N      (input) INTEGER
         The row dimension of the upper bidiagonal matrix. This is
         also the dimension of the main diagonal array D.

  SQRE   (input) INTEGER
         Specifies the column dimension of the bidiagonal matrix.
         = 0: The bidiagonal matrix has column dimension M = N;
         = 1: The bidiagonal matrix has column dimension M = N + 1.

  D      (input/output) REAL array, dimension ( N )
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit D, if INFO = 0, contains its singular values.


  E      (input) REAL array, dimension ( M-1 )
         Contains the subdiagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

  U      (output) REAL array,
         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
         singular vector matrices of all subproblems at the bottom
         level.

  LDU    (input) INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
         GIVNUM, and Z.

  VT     (output) REAL array,
         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced

         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right

         singular vector matrices of all subproblems at the bottom
         level.

  K      (output) INTEGER array, dimension ( N ) 
         if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
         secular equation on the computation tree.

  DIFL   (output) REAL array, dimension ( LDU, NLVL ),
         where NLVL = floor(log_2 (N/SMLSIZ))).

  DIFR   (output) REAL array,
                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)

         record distances between singular values on the I-th
         level and singular values on the (I -1)-th level, and
         DIFR(1:N, 2 * I ) contains the normalizing factors for
         the right singular vector matrix. See SLASD8 for details.

  Z      (output) REAL array,
                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         The first K elements of Z(1, I) contain the components of
         the deflation-adjusted updating row vector for subproblems
         on the I-th level.

  POLES  (output) REAL array,
         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced

         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
         POLES(1, 2*I) contain  the new and old singular values
         involved in the secular equations on the I-th level.

  GIVPTR (output) INTEGER array,
         dimension ( N ) if ICOMPQ = 1, and not referenced if
         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
         the number of Givens rotations performed on the I-th
         problem on the computation tree.

  GIVCOL (output) INTEGER array,
         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
         of Givens rotations performed on the I-th level on the
         computation tree.

  LDGCOL (input) INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

  PERM   (output) INTEGER array, dimension ( LDGCOL, NLVL ) 
         if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
         permutations done on the I-th level of the computation tree.

  GIVNUM (output) REAL array,
         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
         values of Givens rotations performed on the I-th level on
         the computation tree.

  C      (output) REAL array,
         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.

         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,

         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  S      (output) REAL array, dimension ( N ) if
         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
         and the I-th subproblem is not square, on exit, S( I )
         contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

  WORK   (workspace) REAL array, dimension
         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

  IWORK  (workspace) INTEGER array, dimension (7*N).

  INFO   (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an singular value did not converge

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        icompq -
        smlsiz -
        n -
        sqre -
        d -
        e -
        u -
        ldu -
        vt -
        k -
        difl -
        difr -
        z -
        poles -
        givptr -
        givcol -
        ldgcol -
        perm -
        givnum -
        c -
        s -
        work -
        iwork -
        info -

      • slasdq

        public abstract void slasdq(java.lang.String uplo,
                            int sqre,
                            int n,
                            int ncvt,
                            int nru,
                            int ncc,
                            float[] d,
                            float[] e,
                            float[] vt,
                            int ldvt,
                            float[] u,
                            int ldu,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASDQ computes the singular value decomposition (SVD) of a real
  (upper or lower) bidiagonal matrix with diagonal D and offdiagonal
  E, accumulating the transformations if desired. Letting B denote
  the input bidiagonal matrix, the algorithm computes orthogonal
  matrices Q and P such that B = Q * S * P' (P' denotes the transpose
  of P). The singular values S are overwritten on D.

  The input matrix U  is changed to U  * Q  if desired.
  The input matrix VT is changed to P' * VT if desired.
  The input matrix C  is changed to Q' * C  if desired.

  See "Computing  Small Singular Values of Bidiagonal Matrices With
  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
  LAPACK Working Note #3, for a detailed description of the algorithm.


  Arguments
  =========

  UPLO  (input) CHARACTER*1
        On entry, UPLO specifies whether the input bidiagonal matrix
        is upper or lower bidiagonal, and wether it is square are
        not.
           UPLO = 'U' or 'u'   B is upper bidiagonal.
           UPLO = 'L' or 'l'   B is lower bidiagonal.

  SQRE  (input) INTEGER
        = 0: then the input matrix is N-by-N.
        = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and
             (N+1)-by-N if UPLU = 'L'.

        The bidiagonal matrix has
        N = NL + NR + 1 rows and
        M = N + SQRE >= N columns.

  N     (input) INTEGER
        On entry, N specifies the number of rows and columns
        in the matrix. N must be at least 0.

  NCVT  (input) INTEGER
        On entry, NCVT specifies the number of columns of
        the matrix VT. NCVT must be at least 0.

  NRU   (input) INTEGER
        On entry, NRU specifies the number of rows of
        the matrix U. NRU must be at least 0.

  NCC   (input) INTEGER
        On entry, NCC specifies the number of columns of
        the matrix C. NCC must be at least 0.

  D     (input/output) REAL array, dimension (N)
        On entry, D contains the diagonal entries of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in ascending order.

  E     (input/output) REAL array.
        dimension is (N-1) if SQRE = 0 and N if SQRE = 1.
        On entry, the entries of E contain the offdiagonal entries
        of the bidiagonal matrix whose SVD is desired. On normal
        exit, E will contain 0. If the algorithm does not converge,
        D and E will contain the diagonal and superdiagonal entries
        of a bidiagonal matrix orthogonally equivalent to the one
        given as input.

  VT    (input/output) REAL array, dimension (LDVT, NCVT)
        On entry, contains a matrix which on exit has been
        premultiplied by P', dimension N-by-NCVT if SQRE = 0
        and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).

  LDVT  (input) INTEGER
        On entry, LDVT specifies the leading dimension of VT as
        declared in the calling (sub) program. LDVT must be at
        least 1. If NCVT is nonzero LDVT must also be at least N.

  U     (input/output) REAL array, dimension (LDU, N)
        On entry, contains a  matrix which on exit has been
        postmultiplied by Q, dimension NRU-by-N if SQRE = 0
        and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).

  LDU   (input) INTEGER
        On entry, LDU  specifies the leading dimension of U as
        declared in the calling (sub) program. LDU must be at
        least max( 1, NRU ) .

  C     (input/output) REAL array, dimension (LDC, NCC)
        On entry, contains an N-by-NCC matrix which on exit
        has been premultiplied by Q'  dimension N-by-NCC if SQRE = 0
        and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).

  LDC   (input) INTEGER
        On entry, LDC  specifies the leading dimension of C as
        declared in the calling (sub) program. LDC must be at
        least 1. If NCC is nonzero, LDC must also be at least N.

  WORK  (workspace) REAL array, dimension (4*N)
        Workspace. Only referenced if one of NCVT, NRU, or NCC is
        nonzero, and if N is at least 2.

  INFO  (output) INTEGER
        On exit, a value of 0 indicates a successful exit.
        If INFO < 0, argument number -INFO is illegal.
        If INFO > 0, the algorithm did not converge, and INFO
        specifies how many superdiagonals did not converge.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        sqre -
        n -
        ncvt -
        nru -
        ncc -
        d -
        e -
        vt -
        ldvt -
        u -
        ldu -
        c -
        Ldc -
        work -
        info -

      • slasdt

        public abstract void slasdt(int n,
                            org.netlib.util.intW lvl,
                            org.netlib.util.intW nd,
                            int[] inode,
                            int[] ndiml,
                            int[] ndimr,
                            int msub)
        
     ..

  Purpose
  =======

  SLASDT creates a tree of subproblems for bidiagonal divide and
  conquer.

  Arguments
  =========

   N      (input) INTEGER
          On entry, the number of diagonal elements of the
          bidiagonal matrix.

   LVL    (output) INTEGER
          On exit, the number of levels on the computation tree.

   ND     (output) INTEGER
          On exit, the number of nodes on the tree.

   INODE  (output) INTEGER array, dimension ( N )
          On exit, centers of subproblems.

   NDIML  (output) INTEGER array, dimension ( N )
          On exit, row dimensions of left children.

   NDIMR  (output) INTEGER array, dimension ( N )
          On exit, row dimensions of right children.

   MSUB   (input) INTEGER.
          On entry, the maximum row dimension each subproblem at the
          bottom of the tree can be of.

  Further Details
  ===============

  Based on contributions by
     Ming Gu and Huan Ren, Computer Science Division, University of
     California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        lvl -
        nd -
        inode -
        ndiml -
        ndimr -
        msub -

      • slaset

        public abstract void slaset(java.lang.String uplo,
                            int m,
                            int n,
                            float alpha,
                            float beta,
                            float[] a,
                            int lda)
        
     ..

  Purpose
  =======

  SLASET initializes an m-by-n matrix A to BETA on the diagonal and
  ALPHA on the offdiagonals.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies the part of the matrix A to be set.
          = 'U':      Upper triangular part is set; the strictly lower

                      triangular part of A is not changed.
          = 'L':      Lower triangular part is set; the strictly upper

                      triangular part of A is not changed.
          Otherwise:  All of the matrix A is set.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  ALPHA   (input) REAL
          The constant to which the offdiagonal elements are to be set.

  BETA    (input) REAL
          The constant to which the diagonal elements are to be set.

  A       (input/output) REAL array, dimension (LDA,N)
          On exit, the leading m-by-n submatrix of A is set as follows:

          if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
          if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
          otherwise,     A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,

          and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

 =====================================================================


     .. Local Scalars ..
        Parameters:
        uplo -
        m -
        n -
        alpha -
        beta -
        a -
        lda -

      • slasq1

        public abstract void slasq1(int n,
                            float[] d,
                            float[] e,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASQ1 computes the singular values of a real N-by-N bidiagonal
  matrix with diagonal D and off-diagonal E. The singular values
  are computed to high relative accuracy, in the absence of
  denormalization, underflow and overflow. The algorithm was first
  presented in

  "Accurate singular values and differential qd algorithms" by K. V.
  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
  1994,

  and the present implementation is described in "An implementation of

  the dqds Algorithm (Positive Case)", LAPACK Working Note.

  Arguments
  =========

  N     (input) INTEGER
        The number of rows and columns in the matrix. N >= 0.

  D     (input/output) REAL array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in decreasing order.

  E     (input/output) REAL array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired.
        On exit, E is overwritten.

  WORK  (workspace) REAL array, dimension (4*N)

  INFO  (output) INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 30*N
                  iterations (in inner while loop)
             = 3, termination criterion of outer while loop not met 
                  (program created more than N unreduced blocks)

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        work -
        info -

      • slasq2

        public abstract void slasq2(int n,
                            float[] z,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASQ2 computes all the eigenvalues of the symmetric positive 
  definite tridiagonal matrix associated with the qd array Z to high
  relative accuracy are computed to high relative accuracy, in the
  absence of denormalization, underflow and overflow.

  To see the relation of Z to the tridiagonal matrix, let L be a
  unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
  let U be an upper bidiagonal matrix with 1's above and diagonal
  Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
  symmetric tridiagonal to which it is similar.

  Note : SLASQ2 defines a logical variable, IEEE, which is true
  on machines which follow ieee-754 floating-point standard in their
  handling of infinities and NaNs, and false otherwise. This variable
  is passed to SLAZQ3.

  Arguments
  =========

  N     (input) INTEGER
        The number of rows and columns in the matrix. N >= 0.

  Z     (workspace) REAL array, dimension (4*N)
        On entry Z holds the qd array. On exit, entries 1 to N hold
        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
        shifts that failed.

  INFO  (output) INTEGER
        = 0: successful exit
        < 0: if the i-th argument is a scalar and had an illegal
             value, then INFO = -i, if the i-th argument is an
             array and the j-entry had an illegal value, then
             INFO = -(i*100+j)
        > 0: the algorithm failed
              = 1, a split was marked by a positive value in E
              = 2, current block of Z not diagonalized after 30*N
                   iterations (in inner while loop)
              = 3, termination criterion of outer while loop not met 
                   (program created more than N unreduced blocks)

  Further Details
  ===============
  Local Variables: I0:N0 defines a current unreduced segment of Z.
  The shifts are accumulated in SIGMA. Iteration count is in ITER.
  Ping-pong is controlled by PP (alternates between 0 and 1).

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        z -
        info -

      • slasq3

        public abstract void slasq3(int i0,
                            org.netlib.util.intW n0,
                            float[] z,
                            int pp,
                            org.netlib.util.floatW dmin,
                            org.netlib.util.floatW sigma,
                            org.netlib.util.floatW desig,
                            org.netlib.util.floatW qmax,
                            org.netlib.util.intW nfail,
                            org.netlib.util.intW iter,
                            org.netlib.util.intW ndiv,
                            boolean ieee)
        
     ..

  Purpose
  =======

  SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
  In case of failure it changes shifts, and tries again until output
  is positive.

  Arguments
  =========

  I0     (input) INTEGER
         First index.

  N0     (input) INTEGER
         Last index.

  Z      (input) REAL array, dimension ( 4*N )
         Z holds the qd array.

  PP     (input) INTEGER
         PP=0 for ping, PP=1 for pong.

  DMIN   (output) REAL
         Minimum value of d.

  SIGMA  (output) REAL
         Sum of shifts used in current segment.

  DESIG  (input/output) REAL
         Lower order part of SIGMA

  QMAX   (input) REAL
         Maximum value of q.

  NFAIL  (output) INTEGER
         Number of times shift was too big.

  ITER   (output) INTEGER
         Number of iterations.

  NDIV   (output) INTEGER
         Number of divisions.

  TTYPE  (output) INTEGER
         Shift type.

  IEEE   (input) LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        sigma -
        desig -
        qmax -
        nfail -
        iter -
        ndiv -
        ieee -

      • slasq4

        public abstract void slasq4(int i0,
                            int n0,
                            float[] z,
                            int pp,
                            int n0in,
                            float dmin,
                            float dmin1,
                            float dmin2,
                            float dn,
                            float dn1,
                            float dn2,
                            org.netlib.util.floatW tau,
                            org.netlib.util.intW ttype)
        
     ..

  Purpose
  =======

  SLASQ4 computes an approximation TAU to the smallest eigenvalue 
  using values of d from the previous transform.

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) REAL array, dimension ( 4*N )
        Z holds the qd array.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  N0IN  (input) INTEGER
        The value of N0 at start of EIGTEST.

  DMIN  (input) REAL
        Minimum value of d.

  DMIN1 (input) REAL
        Minimum value of d, excluding D( N0 ).

  DMIN2 (input) REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (input) REAL
        d(N)

  DN1   (input) REAL
        d(N-1)

  DN2   (input) REAL
        d(N-2)

  TAU   (output) REAL
        This is the shift.

  TTYPE (output) INTEGER
        Shift type.

  Further Details
  ===============
  CNST1 = 9/16

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        n0in -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -
        ttype -

      • slasq5

        public abstract void slasq5(int i0,
                            int n0,
                            float[] z,
                            int pp,
                            float tau,
                            org.netlib.util.floatW dmin,
                            org.netlib.util.floatW dmin1,
                            org.netlib.util.floatW dmin2,
                            org.netlib.util.floatW dn,
                            org.netlib.util.floatW dnm1,
                            org.netlib.util.floatW dnm2,
                            boolean ieee)
        
     ..

  Purpose
  =======

  SLASQ5 computes one dqds transform in ping-pong form, one
  version for IEEE machines another for non IEEE machines.

  Arguments
  =========

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) REAL array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  TAU   (input) REAL
        This is the shift.

  DMIN  (output) REAL
        Minimum value of d.

  DMIN1 (output) REAL
        Minimum value of d, excluding D( N0 ).

  DMIN2 (output) REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (output) REAL
        d(N0), the last value of d.

  DNM1  (output) REAL
        d(N0-1).

  DNM2  (output) REAL
        d(N0-2).

  IEEE  (input) LOGICAL
        Flag for IEEE or non IEEE arithmetic.

  =====================================================================

     .. Parameter ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        tau -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dnm1 -
        dnm2 -
        ieee -

      • slasq6

        public abstract void slasq6(int i0,
                            int n0,
                            float[] z,
                            int pp,
                            org.netlib.util.floatW dmin,
                            org.netlib.util.floatW dmin1,
                            org.netlib.util.floatW dmin2,
                            org.netlib.util.floatW dn,
                            org.netlib.util.floatW dnm1,
                            org.netlib.util.floatW dnm2)
        
     ..

  Purpose
  =======

  SLASQ6 computes one dqd (shift equal to zero) transform in
  ping-pong form, with protection against underflow and overflow.

  Arguments
  =========

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) REAL array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  DMIN  (output) REAL
        Minimum value of d.

  DMIN1 (output) REAL
        Minimum value of d, excluding D( N0 ).

  DMIN2 (output) REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (output) REAL
        d(N0), the last value of d.

  DNM1  (output) REAL
        d(N0-1).

  DNM2  (output) REAL
        d(N0-2).

  =====================================================================

     .. Parameter ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dnm1 -
        dnm2 -

      • slasr

        public abstract void slasr(java.lang.String side,
                           java.lang.String pivot,
                           java.lang.String direct,
                           int m,
                           int n,
                           float[] c,
                           float[] s,
                           float[] a,
                           int lda)
        
     ..

  Purpose
  =======

  SLASR applies a sequence of plane rotations to a real matrix A,
  from either the left or the right.
  
  When SIDE = 'L', the transformation takes the form
  
     A := P*A
  
  and when SIDE = 'R', the transformation takes the form
  
     A := A*P**T
  
  where P is an orthogonal matrix consisting of a sequence of z plane
  rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
  and P**T is the transpose of P.
  
  When DIRECT = 'F' (Forward sequence), then
  
     P = P(z-1) * ... * P(2) * P(1)
  
  and when DIRECT = 'B' (Backward sequence), then
  
     P = P(1) * P(2) * ... * P(z-1)
  
  where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

  
     R(k) = (  c(k)  s(k) )
          = ( -s(k)  c(k) ).
  
  When PIVOT = 'V' (Variable pivot), the rotation is performed
  for the plane (k,k+1), i.e., P(k) has the form
  
     P(k) = (  1                                            )
            (       ...                                     )
            (              1                                )
            (                   c(k)  s(k)                  )
            (                  -s(k)  c(k)                  )
            (                                1              )
            (                                     ...       )
            (                                            1  )
  
  where R(k) appears as a rank-2 modification to the identity matrix in
  rows and columns k and k+1.
  
  When PIVOT = 'T' (Top pivot), the rotation is performed for the
  plane (1,k+1), so P(k) has the form
  
     P(k) = (  c(k)                    s(k)                 )
            (         1                                     )
            (              ...                              )
            (                     1                         )
            ( -s(k)                    c(k)                 )
            (                                 1             )
            (                                      ...      )
            (                                             1 )
  
  where R(k) appears in rows and columns 1 and k+1.
  
  Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
  performed for the plane (k,z), giving P(k) the form
  
     P(k) = ( 1                                             )
            (      ...                                      )
            (             1                                 )
            (                  c(k)                    s(k) )
            (                         1                     )
            (                              ...              )
            (                                     1         )
            (                 -s(k)                    c(k) )
  
  where R(k) appears in rows and columns k and z.  The rotations are
  performed without ever forming P(k) explicitly.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          Specifies whether the plane rotation matrix P is applied to
          A on the left or the right.
          = 'L':  Left, compute A := P*A
          = 'R':  Right, compute A:= A*P**T

  PIVOT   (input) CHARACTER*1
          Specifies the plane for which P(k) is a plane rotation
          matrix.
          = 'V':  Variable pivot, the plane (k,k+1)
          = 'T':  Top pivot, the plane (1,k+1)
          = 'B':  Bottom pivot, the plane (k,z)

  DIRECT  (input) CHARACTER*1
          Specifies whether P is a forward or backward sequence of
          plane rotations.
          = 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
          = 'B':  Backward, P = P(1)*P(2)*...*P(z-1)

  M       (input) INTEGER
          The number of rows of the matrix A.  If m <= 1, an immediate

          return is effected.

  N       (input) INTEGER
          The number of columns of the matrix A.  If n <= 1, an
          immediate return is effected.

  C       (input) REAL array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The cosines c(k) of the plane rotations.

  S       (input) REAL array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The sines s(k) of the plane rotations.  The 2-by-2 plane
          rotation part of the matrix P(k), R(k), has the form
          R(k) = (  c(k)  s(k) )
                 ( -s(k)  c(k) ).

  A       (input/output) REAL array, dimension (LDA,N)
          The M-by-N matrix A.  On exit, A is overwritten by P*A if
          SIDE = 'R' or by A*P**T if SIDE = 'L'.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        pivot -
        direct -
        m -
        n -
        c -
        s -
        a -
        lda -

      • slasrt

        public abstract void slasrt(java.lang.String id,
                            int n,
                            float[] d,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  Sort the numbers in D in increasing order (if ID = 'I') or
  in decreasing order (if ID = 'D' ).

  Use Quick Sort, reverting to Insertion sort on arrays of
  size <= 20. Dimension of STACK limits N to about 2**32.

  Arguments
  =========

  ID      (input) CHARACTER*1
          = 'I': sort D in increasing order;
          = 'D': sort D in decreasing order.

  N       (input) INTEGER
          The length of the array D.

  D       (input/output) REAL array, dimension (N)
          On entry, the array to be sorted.
          On exit, D has been sorted into increasing order
          (D(1) <= ... <= D(N) ) or into decreasing order
          (D(1) >= ... >= D(N) ), depending on ID.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        id -
        n -
        d -
        info -

      • slassq

        public abstract void slassq(int n,
                            float[] x,
                            int incx,
                            org.netlib.util.floatW scale,
                            org.netlib.util.floatW sumsq)
        
     ..

  Purpose
  =======

  SLASSQ  returns the values  scl  and  smsq  such that

     ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,


  where  x( i ) = X( 1 + ( i - 1 )*INCX ). The value of  sumsq  is
  assumed to be non-negative and  scl  returns the value

     scl = max( scale, abs( x( i ) ) ).

  scale and sumsq must be supplied in SCALE and SUMSQ and
  scl and smsq are overwritten on SCALE and SUMSQ respectively.

  The routine makes only one pass through the vector x.

  Arguments
  =========

  N       (input) INTEGER
          The number of elements to be used from the vector X.

  X       (input) REAL array, dimension (N)
          The vector for which a scaled sum of squares is computed.
             x( i )  = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.

  INCX    (input) INTEGER
          The increment between successive values of the vector X.
          INCX > 0.

  SCALE   (input/output) REAL
          On entry, the value  scale  in the equation above.
          On exit, SCALE is overwritten with  scl , the scaling factor

          for the sum of squares.

  SUMSQ   (input/output) REAL
          On entry, the value  sumsq  in the equation above.
          On exit, SUMSQ is overwritten with  smsq , the basic sum of
          squares from which  scl  has been factored out.

 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        x -
        incx -
        scale -
        sumsq -

      • slasv2

        public abstract void slasv2(float f,
                            float g,
                            float h,
                            org.netlib.util.floatW ssmin,
                            org.netlib.util.floatW ssmax,
                            org.netlib.util.floatW snr,
                            org.netlib.util.floatW csr,
                            org.netlib.util.floatW snl,
                            org.netlib.util.floatW csl)
        
     ..

  Purpose
  =======

  SLASV2 computes the singular value decomposition of a 2-by-2
  triangular matrix
     [  F   G  ]
     [  0   H  ].
  On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
  smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and

  right singular vectors for abs(SSMAX), giving the decomposition

     [ CSL  SNL ] [  F   G  ] [ CSR -SNR ]  =  [ SSMAX   0   ]
     [-SNL  CSL ] [  0   H  ] [ SNR  CSR ]     [  0    SSMIN ].

  Arguments
  =========

  F       (input) REAL
          The (1,1) element of the 2-by-2 matrix.

  G       (input) REAL
          The (1,2) element of the 2-by-2 matrix.

  H       (input) REAL
          The (2,2) element of the 2-by-2 matrix.

  SSMIN   (output) REAL
          abs(SSMIN) is the smaller singular value.

  SSMAX   (output) REAL
          abs(SSMAX) is the larger singular value.

  SNL     (output) REAL
  CSL     (output) REAL
          The vector (CSL, SNL) is a unit left singular vector for the

          singular value abs(SSMAX).

  SNR     (output) REAL
  CSR     (output) REAL
          The vector (CSR, SNR) is a unit right singular vector for the
          singular value abs(SSMAX).

  Further Details
  ===============

  Any input parameter may be aliased with any output parameter.

  Barring over/underflow and assuming a guard digit in subtraction, all
  output quantities are correct to within a few units in the last
  place (ulps).

  In IEEE arithmetic, the code works correctly if one matrix element is
  infinite.

  Overflow will not occur unless the largest singular value itself
  overflows or is within a few ulps of overflow. (On machines with
  partial overflow, like the Cray, overflow may occur if the largest
  singular value is within a factor of 2 of overflow.)

  Underflow is harmless if underflow is gradual. Otherwise, results
  may correspond to a matrix modified by perturbations of size near
  the underflow threshold.

 =====================================================================


     .. Parameters ..
        Parameters:
        f -
        g -
        h -
        ssmin -
        ssmax -
        snr -
        csr -
        snl -
        csl -

      • slaswp

        public abstract void slaswp(int n,
                            float[] a,
                            int lda,
                            int k1,
                            int k2,
                            int[] ipiv,
                            int incx)
        
     ..

  Purpose
  =======

  SLASWP performs a series of row interchanges on the matrix A.
  One row interchange is initiated for each of rows K1 through K2 of A.

  Arguments
  =========

  N       (input) INTEGER
          The number of columns of the matrix A.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the matrix of column dimension N to which the row
          interchanges will be applied.
          On exit, the permuted matrix.

  LDA     (input) INTEGER
          The leading dimension of the array A.

  K1      (input) INTEGER
          The first element of IPIV for which a row interchange will
          be done.

  K2      (input) INTEGER
          The last element of IPIV for which a row interchange will
          be done.

  IPIV    (input) INTEGER array, dimension (K2*abs(INCX))
          The vector of pivot indices.  Only the elements in positions

          K1 through K2 of IPIV are accessed.
          IPIV(K) = L implies rows K and L are to be interchanged.

  INCX    (input) INTEGER
          The increment between successive values of IPIV.  If IPIV
          is negative, the pivots are applied in reverse order.

  Further Details
  ===============

  Modified by
   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 =====================================================================


     .. Local Scalars ..
        Parameters:
        n -
        a -
        lda -
        k1 -
        k2 -
        ipiv -
        incx -

      • slasy2

        public abstract void slasy2(boolean ltranl,
                            boolean ltranr,
                            int isgn,
                            int n1,
                            int n2,
                            float[] tl,
                            int ldtl,
                            float[] tr,
                            int ldtr,
                            float[] b,
                            int ldb,
                            org.netlib.util.floatW scale,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW xnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in

         op(TL)*X + ISGN*X*op(TR) = SCALE*B,

  where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or

  -1.  op(T) = T or T', where T' denotes the transpose of T.

  Arguments
  =========

  LTRANL  (input) LOGICAL
          On entry, LTRANL specifies the op(TL):
             = .FALSE., op(TL) = TL,
             = .TRUE., op(TL) = TL'.

  LTRANR  (input) LOGICAL
          On entry, LTRANR specifies the op(TR):
            = .FALSE., op(TR) = TR,
            = .TRUE., op(TR) = TR'.

  ISGN    (input) INTEGER
          On entry, ISGN specifies the sign of the equation
          as described before. ISGN may only be 1 or -1.

  N1      (input) INTEGER
          On entry, N1 specifies the order of matrix TL.
          N1 may only be 0, 1 or 2.

  N2      (input) INTEGER
          On entry, N2 specifies the order of matrix TR.
          N2 may only be 0, 1 or 2.

  TL      (input) REAL array, dimension (LDTL,2)
          On entry, TL contains an N1 by N1 matrix.

  LDTL    (input) INTEGER
          The leading dimension of the matrix TL. LDTL >= max(1,N1).

  TR      (input) REAL array, dimension (LDTR,2)
          On entry, TR contains an N2 by N2 matrix.

  LDTR    (input) INTEGER
          The leading dimension of the matrix TR. LDTR >= max(1,N2).

  B       (input) REAL array, dimension (LDB,2)
          On entry, the N1 by N2 matrix B contains the right-hand
          side of the equation.

  LDB     (input) INTEGER
          The leading dimension of the matrix B. LDB >= max(1,N1).

  SCALE   (output) REAL
          On exit, SCALE contains the scale factor. SCALE is chosen
          less than or equal to 1 to prevent the solution overflowing.


  X       (output) REAL array, dimension (LDX,2)
          On exit, X contains the N1 by N2 solution.

  LDX     (input) INTEGER
          The leading dimension of the matrix X. LDX >= max(1,N1).

  XNORM   (output) REAL
          On exit, XNORM is the infinity-norm of the solution.

  INFO    (output) INTEGER
          On exit, INFO is set to
             0: successful exit.
             1: TL and TR have too close eigenvalues, so TL or
                TR is perturbed to get a nonsingular equation.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

 =====================================================================


     .. Parameters ..
        Parameters:
        ltranl -
        ltranr -
        isgn -
        n1 -
        n2 -
        tl -
        ldtl -
        tr -
        ldtr -
        b -
        ldb -
        scale -
        x -
        ldx -
        xnorm -
        info -

      • slasyf

        public abstract void slasyf(java.lang.String uplo,
                            int n,
                            int nb,
                            org.netlib.util.intW kb,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] w,
                            int ldw,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLASYF computes a partial factorization of a real symmetric matrix A

  using the Bunch-Kaufman diagonal pivoting method. The partial
  factorization has the form:

  A  =  ( I  U12 ) ( A11  0  ) (  I    0   )  if UPLO = 'U', or:
        ( 0  U22 ) (  0   D  ) ( U12' U22' )

  A  =  ( L11  0 ) (  D   0  ) ( L11' L21' )  if UPLO = 'L'
        ( L21  I ) (  0  A22 ) (  0    I   )

  where the order of D is at most NB. The actual order is returned in
  the argument KB, and is either NB or NB-1, or N if N <= NB.

  SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code
  (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
  A22 (if UPLO = 'L').

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NB      (input) INTEGER
          The maximum number of columns of the matrix A that should be

          factored.  NB should be at least 2 to allow for 2-by-2 pivot

          blocks.

  KB      (output) INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, A contains details of the partial factorization.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If UPLO = 'U', only the last KB elements of IPIV are set;
          if UPLO = 'L', only the first KB elements are set.

          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  W       (workspace) REAL array, dimension (LDW,NB)

  LDW     (input) INTEGER
          The leading dimension of the array W.  LDW >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nb -
        kb -
        a -
        lda -
        ipiv -
        w -
        ldw -
        info -

      • slatbs

        public abstract void slatbs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] x,
                            org.netlib.util.floatW scale,
                            float[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLATBS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow, where A is an upper or lower
  triangular band matrix.  Here A' denotes the transpose of A, x and b

  are n-element vectors, and s is a scaling factor, usually less than
  or equal to 1, chosen so that the components of x will be less than
  the overflow threshold.  If the unscaled problem will not cause
  overflow, the Level 2 BLAS routine STBSV is called.  If the matrix A

  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
  non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  X       (input/output) REAL array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) REAL
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, STBSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STBSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call STBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        kd -
        ab -
        ldab -
        x -
        scale -
        cnorm -
        info -

      • slatdf

        public abstract void slatdf(int ijob,
                            int n,
                            float[] z,
                            int ldz,
                            float[] rhs,
                            org.netlib.util.floatW rdsum,
                            org.netlib.util.floatW rdscal,
                            int[] ipiv,
                            int[] jpiv)
        
     ..

  Purpose
  =======

  SLATDF uses the LU factorization of the n-by-n matrix Z computed by
  SGETC2 and computes a contribution to the reciprocal Dif-estimate
  by solving Z * x = b for x, and choosing the r.h.s. b such that
  the norm of x is as large as possible. On entry RHS = b holds the
  contribution from earlier solved sub-systems, and on return RHS = x.


  The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q,
  where P and Q are permutation matrices. L is lower triangular with
  unit diagonal elements and U is upper triangular.

  Arguments
  =========

  IJOB    (input) INTEGER
          IJOB = 2: First compute an approximative null-vector e
              of Z using SGECON, e is normalized and solve for
              Zx = +-e - f with the sign giving the greater value
              of 2-norm(x). About 5 times as expensive as Default.
          IJOB .ne. 2: Local look ahead strategy where all entries of
              the r.h.s. b is choosen as either +1 or -1 (Default).

  N       (input) INTEGER
          The number of columns of the matrix Z.

  Z       (input) REAL array, dimension (LDZ, N)
          On entry, the LU part of the factorization of the n-by-n
          matrix Z computed by SGETC2:  Z = P * L * U * Q

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDA >= max(1, N).

  RHS     (input/output) REAL array, dimension N.
          On entry, RHS contains contributions from other subsystems.
          On exit, RHS contains the solution of the subsystem with
          entries acoording to the value of IJOB (see above).

  RDSUM   (input/output) REAL
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by STGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

  RDSCAL  (input/output) REAL
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when STGSY2 is called by
                STGSYL.

  IPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

  JPIV    (input) INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  This routine is a further developed implementation of algorithm
  BSOLVE in [1] using complete pivoting in the LU factorization.

  [1] Bo Kagstrom and Lars Westin,
      Generalized Schur Methods with Condition Estimators for
      Solving the Generalized Sylvester Equation, IEEE Transactions
      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

  [2] Peter Poromaa,
      On Efficient and Robust Estimators for the Separation
      between two Regular Matrix Pairs with Applications in
      Condition Estimation. Report IMINF-95.05, Departement of
      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.


  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        n -
        z -
        ldz -
        rhs -
        rdsum -
        rdscal -
        ipiv -
        jpiv -

      • slatps

        public abstract void slatps(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            float[] ap,
                            float[] x,
                            org.netlib.util.floatW scale,
                            float[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLATPS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow, where A is an upper or lower
  triangular matrix stored in packed form.  Here A' denotes the
  transpose of A, x and b are n-element vectors, and s is a scaling
  factor, usually less than or equal to 1, chosen so that the
  components of x will be less than the overflow threshold.  If the
  unscaled problem will not cause overflow, the Level 2 BLAS routine
  STPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
  then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  X       (input/output) REAL array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) REAL
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, STPSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STPSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call STPSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        ap -
        x -
        scale -
        cnorm -
        info -

      • slatrd

        public abstract void slatrd(java.lang.String uplo,
                            int n,
                            int nb,
                            float[] a,
                            int lda,
                            float[] e,
                            float[] tau,
                            float[] w,
                            int ldw)
        
     ..

  Purpose
  =======

  SLATRD reduces NB rows and columns of a real symmetric matrix A to
  symmetric tridiagonal form by an orthogonal similarity
  transformation Q' * A * Q, and returns the matrices V and W which are
  needed to apply the transformation to the unreduced part of A.

  If UPLO = 'U', SLATRD reduces the last NB rows and columns of a
  matrix, of which the upper triangle is supplied;
  if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
  matrix, of which the lower triangle is supplied.

  This is an auxiliary routine called by SSYTRD.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U': Upper triangular
          = 'L': Lower triangular

  N       (input) INTEGER
          The order of the matrix A.

  NB      (input) INTEGER
          The number of rows and columns to be reduced.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit:
          if UPLO = 'U', the last NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the orthogonal matrix Q as a

            product of elementary reflectors;
          if UPLO = 'L', the first NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements below the diagonal
            with the array TAU, represent the  orthogonal matrix Q as a
            product of elementary reflectors.
          See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= (1,N).

  E       (output) REAL array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          elements of the last NB columns of the reduced matrix;
          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
          the first NB columns of the reduced matrix.

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors, stored in
          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.

          See Further Details.

  W       (output) REAL array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.

  LDW     (input) INTEGER
          The leading dimension of the array W. LDW >= max(1,N).

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n) H(n-1) . . . H(n-nb+1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),

  and tau in TAU(i-1).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),

  and tau in TAU(i).

  The elements of the vectors v together form the n-by-nb matrix V
  which is needed, with W, to apply the transformation to the unreduced
  part of the matrix, using a symmetric rank-2k update of the form:
  A := A - V*W' - W*V'.

  The contents of A on exit are illustrated by the following examples
  with n = 5 and nb = 2:

  if UPLO = 'U':                       if UPLO = 'L':

    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )

  where d denotes a diagonal element of the reduced matrix, a denotes
  an element of the original matrix that is unchanged, and vi denotes
  an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nb -
        a -
        lda -
        e -
        tau -
        w -
        ldw -

      • slatrs

        public abstract void slatrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            java.lang.String normin,
                            int n,
                            float[] a,
                            int lda,
                            float[] x,
                            org.netlib.util.floatW scale,
                            float[] cnorm,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLATRS solves one of the triangular systems

     A *x = s*b  or  A'*x = s*b

  with scaling to prevent overflow.  Here A is an upper or lower
  triangular matrix, A' denotes the transpose of A, x and b are
  n-element vectors, and s is a scaling factor, usually less than
  or equal to 1, chosen so that the components of x will be less than
  the overflow threshold.  If the unscaled problem will not cause
  overflow, the Level 2 BLAS routine STRSV is called.  If the matrix A

  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
  non-trivial solution to A*x = 0 is returned.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  TRANS   (input) CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A'* x = s*b  (Transpose)
          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)


  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  NORMIN  (input) CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading n by n
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading n by n lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max (1,N).

  X       (input/output) REAL array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

  SCALE   (output) REAL
          The scaling factor s for the triangular system
             A * x = s*b  or  A'* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.


  CNORM   (input or output) REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value

  Further Details
  ======= =======

  A rough bound on x is computed; if that is less than overflow, STRSV

  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:

     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )

                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to

  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
  algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        normin -
        n -
        a -
        lda -
        x -
        scale -
        cnorm -
        info -

      • slatrz

        public abstract void slatrz(int m,
                            int n,
                            int l,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work)
        
     ..

  Purpose
  =======

  SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
  [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
  of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
  matrix and, R and A1 are M-by-M upper triangular matrices.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing the
          meaningful part of the Householder vectors. N-M >= L >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements N-L+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (M)
          The scalar factors of the elementary reflectors.

  WORK    (workspace) REAL array, dimension (M)

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an l element vector. tau and z( k )
  are chosen to annihilate the elements of the kth row of A2.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A2, such that the elements of z( k ) are
  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A1.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        l -
        a -
        lda -
        tau -
        work -

      • slatzm

        public abstract void slatzm(java.lang.String side,
                            int m,
                            int n,
                            float[] v,
                            int incv,
                            float tau,
                            float[] c1,
                            float[] c2,
                            int Ldc,
                            float[] work)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine SORMRZ.

  SLATZM applies a Householder matrix generated by STZRQF to a matrix.


  Let P = I - tau*u*u',   u = ( 1 ),
                              ( v )
  where v is an (m-1) vector if SIDE = 'L', or a (n-1) vector if
  SIDE = 'R'.

  If SIDE equals 'L', let
         C = [ C1 ] 1
             [ C2 ] m-1
               n
  Then C is overwritten by P*C.

  If SIDE equals 'R', let
         C = [ C1, C2 ] m
                1  n-1
  Then C is overwritten by C*P.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': form P * C
          = 'R': form C * P

  M       (input) INTEGER
          The number of rows of the matrix C.

  N       (input) INTEGER
          The number of columns of the matrix C.

  V       (input) REAL array, dimension
                  (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of P. V is not used
          if TAU = 0.

  INCV    (input) INTEGER
          The increment between elements of v. INCV <> 0

  TAU     (input) REAL
          The value tau in the representation of P.

  C1      (input/output) REAL array, dimension
                         (LDC,N) if SIDE = 'L'
                         (M,1)   if SIDE = 'R'
          On entry, the n-vector C1 if SIDE = 'L', or the m-vector C1
          if SIDE = 'R'.

          On exit, the first row of P*C if SIDE = 'L', or the first
          column of C*P if SIDE = 'R'.

  C2      (input/output) REAL array, dimension
                         (LDC, N)   if SIDE = 'L'
                         (LDC, N-1) if SIDE = 'R'
          On entry, the (m - 1) x n matrix C2 if SIDE = 'L', or the
          m x (n - 1) matrix C2 if SIDE = 'R'.

          On exit, rows 2:m of P*C if SIDE = 'L', or columns 2:m of C*P
          if SIDE = 'R'.

  LDC     (input) INTEGER
          The leading dimension of the arrays C1 and C2. LDC >= (1,M).


  WORK    (workspace) REAL array, dimension
                      (N) if SIDE = 'L'
                      (M) if SIDE = 'R'

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        m -
        n -
        v -
        incv -
        tau -
        c1 -
        c2 -
        Ldc -
        work -

      • slauu2

        public abstract void slauu2(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAUU2 computes the product U * U' or L' * L, where the triangular
  factor U or L is stored in the upper or lower triangular part of
  the array A.

  If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
  overwriting the factor U in A.
  If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
  overwriting the factor L in A.

  This is the unblocked form of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the triangular factor U or L.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U';
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L' * L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • slauum

        public abstract void slauum(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SLAUUM computes the product U * U' or L' * L, where the triangular
  factor U or L is stored in the upper or lower triangular part of
  the array A.

  If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
  overwriting the factor U in A.
  If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
  overwriting the factor L in A.

  This is the blocked form of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the triangular factor U or L.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U';
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L' * L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • slazq3

        public void slazq3(int i0,
                   org.netlib.util.intW n0,
                   float[] z,
                   int pp,
                   org.netlib.util.floatW dmin,
                   org.netlib.util.floatW sigma,
                   org.netlib.util.floatW desig,
                   org.netlib.util.floatW qmax,
                   org.netlib.util.intW nfail,
                   org.netlib.util.intW iter,
                   org.netlib.util.intW ndiv,
                   boolean ieee,
                   org.netlib.util.intW ttype,
                   org.netlib.util.floatW dmin1,
                   org.netlib.util.floatW dmin2,
                   org.netlib.util.floatW dn,
                   org.netlib.util.floatW dn1,
                   org.netlib.util.floatW dn2,
                   org.netlib.util.floatW tau)
        
     ..

  Purpose
  =======

  SLAZQ3 checks for deflation, computes a shift (TAU) and calls dqds.
  In case of failure it changes shifts, and tries again until output
  is positive.

  Arguments
  =========

  I0     (input) INTEGER
         First index.

  N0     (input) INTEGER
         Last index.

  Z      (input) REAL array, dimension ( 4*N )
         Z holds the qd array.

  PP     (input) INTEGER
         PP=0 for ping, PP=1 for pong.

  DMIN   (output) REAL
         Minimum value of d.

  SIGMA  (output) REAL
         Sum of shifts used in current segment.

  DESIG  (input/output) REAL
         Lower order part of SIGMA

  QMAX   (input) REAL
         Maximum value of q.

  NFAIL  (output) INTEGER
         Number of times shift was too big.

  ITER   (output) INTEGER
         Number of iterations.

  NDIV   (output) INTEGER
         Number of divisions.

  IEEE   (input) LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).

  TTYPE  (input/output) INTEGER
         Shift type.  TTYPE is passed as an argument in order to save
         its value between calls to SLAZQ3

  DMIN1  (input/output) REAL
  DMIN2  (input/output) REAL
  DN     (input/output) REAL
  DN1    (input/output) REAL
  DN2    (input/output) REAL
  TAU    (input/output) REAL
         These are passed as arguments in order to save their values
         between calls to SLAZQ3

  This is a thread safe version of SLASQ3, which passes TTYPE, DMIN1,
  DMIN2, DN, DN1. DN2 and TAU through the argument list in place of
  declaring them in a SAVE statment.

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        dmin -
        sigma -
        desig -
        qmax -
        nfail -
        iter -
        ndiv -
        ieee -
        ttype -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -

      • slazq4

        public void slazq4(int i0,
                   int n0,
                   float[] z,
                   int pp,
                   int n0in,
                   float dmin,
                   float dmin1,
                   float dmin2,
                   float dn,
                   float dn1,
                   float dn2,
                   org.netlib.util.floatW tau,
                   org.netlib.util.intW ttype,
                   org.netlib.util.floatW g)
        
     ..

  Purpose
  =======

  SLAZQ4 computes an approximation TAU to the smallest eigenvalue 
  using values of d from the previous transform.

  I0    (input) INTEGER
        First index.

  N0    (input) INTEGER
        Last index.

  Z     (input) REAL array, dimension ( 4*N )
        Z holds the qd array.

  PP    (input) INTEGER
        PP=0 for ping, PP=1 for pong.

  N0IN  (input) INTEGER
        The value of N0 at start of EIGTEST.

  DMIN  (input) REAL
        Minimum value of d.

  DMIN1 (input) REAL
        Minimum value of d, excluding D( N0 ).

  DMIN2 (input) REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

  DN    (input) REAL
        d(N)

  DN1   (input) REAL
        d(N-1)

  DN2   (input) REAL
        d(N-2)

  TAU   (output) REAL
        This is the shift.

  TTYPE (output) INTEGER
        Shift type.

  G     (input/output) REAL
        G is passed as an argument in order to save its value between
        calls to SLAZQ4

  Further Details
  ===============
  CNST1 = 9/16

  This is a thread safe version of SLASQ4, which passes G through the
  argument list in place of declaring G in a SAVE statment.

  =====================================================================

     .. Parameters ..
        Parameters:
        i0 -
        n0 -
        z -
        pp -
        n0in -
        dmin -
        dmin1 -
        dmin2 -
        dn -
        dn1 -
        dn2 -
        tau -
        ttype -
        g -

      • sopgtr

        public abstract void sopgtr(java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] tau,
                            float[] q,
                            int ldq,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SOPGTR generates a real orthogonal matrix Q which is defined as the
  product of n-1 elementary reflectors H(i) of order n, as returned by

  SSPTRD using packed storage:

  if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),

  if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to SSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to SSPTRD.

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The vectors which define the elementary reflectors, as
          returned by SSPTRD.

  TAU     (input) REAL array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SSPTRD.

  Q       (output) REAL array, dimension (LDQ,N)
          The N-by-N orthogonal matrix Q.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N).

  WORK    (workspace) REAL array, dimension (N-1)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        tau -
        q -
        ldq -
        work -
        info -

      • sopmtr

        public abstract void sopmtr(java.lang.String side,
                            java.lang.String uplo,
                            java.lang.String trans,
                            int m,
                            int n,
                            float[] ap,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SOPMTR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  nq-1 elementary reflectors, as returned by SSPTRD using packed
  storage:

  if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);

  if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to SSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to SSPTRD.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  AP      (input) REAL array, dimension
                               (M*(M+1)/2) if SIDE = 'L'
                               (N*(N+1)/2) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by SSPTRD.  AP is modified by the routine but
          restored on exit.

  TAU     (input) REAL array, dimension (M-1) if SIDE = 'L'
                                     or (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SSPTRD.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L'
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        uplo -
        trans -
        m -
        n -
        ap -
        tau -
        c -
        Ldc -
        work -
        info -

      • sorg2l

        public abstract void sorg2l(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORG2L generates an m by n real matrix Q with orthonormal columns,
  which is defined as the last n columns of a product of k elementary
  reflectors of order m

        Q  =  H(k) . . . H(2) H(1)

  as returned by SGEQLF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGEQLF in the last k columns of its array
          argument A.
          On exit, the m by n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQLF.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • sorg2r

        public abstract void sorg2r(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORG2R generates an m by n real matrix Q with orthonormal columns,
  which is defined as the first n columns of a product of k elementary

  reflectors of order m

        Q  =  H(1) H(2) . . . H(k)

  as returned by SGEQRF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGEQRF in the first k columns of its array
          argument A.
          On exit, the m-by-n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQRF.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • sorgbr

        public abstract void sorgbr(java.lang.String vect,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGBR generates one of the real orthogonal matrices Q or P**T
  determined by SGEBRD when reducing a real matrix A to bidiagonal
  form: A = Q * B * P**T.  Q and P**T are defined as products of
  elementary reflectors H(i) or G(i) respectively.

  If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
  is of order M:
  if m >= k, Q = H(1) H(2) . . . H(k) and SORGBR returns the first n
  columns of Q, where m >= n >= k;
  if m < k, Q = H(1) H(2) . . . H(m-1) and SORGBR returns Q as an
  M-by-M matrix.

  If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
  is of order N:
  if k < n, P**T = G(k) . . . G(2) G(1) and SORGBR returns the first m

  rows of P**T, where n >= m >= k;
  if k >= n, P**T = G(n-1) . . . G(2) G(1) and SORGBR returns P**T as
  an N-by-N matrix.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          Specifies whether the matrix Q or the matrix P**T is
          required, as defined in the transformation applied by SGEBRD:
          = 'Q':  generate Q;
          = 'P':  generate P**T.

  M       (input) INTEGER
          The number of rows of the matrix Q or P**T to be returned.
          M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q or P**T to be returned.
          N >= 0.
          If VECT = 'Q', M >= N >= min(M,K);
          if VECT = 'P', N >= M >= min(N,K).

  K       (input) INTEGER
          If VECT = 'Q', the number of columns in the original M-by-K
          matrix reduced by SGEBRD.
          If VECT = 'P', the number of rows in the original K-by-N
          matrix reduced by SGEBRD.
          K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by SGEBRD.
          On exit, the M-by-N matrix Q or P**T.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension
                                (min(M,K)) if VECT = 'Q'
                                (min(N,K)) if VECT = 'P'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i), which determines Q or P**T, as
          returned by SGEBRD in its array argument TAUQ or TAUP.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,min(M,N)).
          For optimum performance LWORK >= min(M,N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorghr

        public abstract void sorghr(int n,
                            int ilo,
                            int ihi,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGHR generates a real orthogonal matrix Q which is defined as the
  product of IHI-ILO elementary reflectors of order N, as returned by
  SGEHRD:

  Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI must have the same values as in the previous call
          of SGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by SGEHRD.
          On exit, the N-by-N orthogonal matrix Q.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAU     (input) REAL array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEHRD.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= IHI-ILO.
          For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorgl2

        public abstract void sorgl2(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGL2 generates an m by n real matrix Q with orthonormal rows,
  which is defined as the first m rows of a product of k elementary
  reflectors of order n

        Q  =  H(k) . . . H(2) H(1)

  as returned by SGELQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines

          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by SGELQF in the first k rows of its array argument A.
          On exit, the m-by-n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGELQF.

  WORK    (workspace) REAL array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • sorglq

        public abstract void sorglq(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGLQ generates an M-by-N real matrix Q with orthonormal rows,
  which is defined as the first M rows of a product of K elementary
  reflectors of order N

        Q  =  H(k) . . . H(2) H(1)

  as returned by SGELQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines

          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by SGELQF in the first k rows of its array argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGELQF.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorgql

        public abstract void sorgql(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGQL generates an M-by-N real matrix Q with orthonormal columns,
  which is defined as the last N columns of a product of K elementary
  reflectors of order M

        Q  =  H(k) . . . H(2) H(1)

  as returned by SGEQLF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGEQLF in the last k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQLF.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorgqr

        public abstract void sorgqr(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGQR generates an M-by-N real matrix Q with orthonormal columns,
  which is defined as the first N columns of a product of K elementary

  reflectors of order M

        Q  =  H(1) H(2) . . . H(k)

  as returned by SGEQRF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGEQRF in the first k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQRF.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorgr2

        public abstract void sorgr2(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGR2 generates an m by n real matrix Q with orthonormal rows,
  which is defined as the last m rows of a product of k elementary
  reflectors of order n

        Q  =  H(1) H(2) . . . H(k)

  as returned by SGERQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGERQF in the last k rows of its array argument
          A.
          On exit, the m by n matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGERQF.

  WORK    (workspace) REAL array, dimension (M)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        info -

      • sorgrq

        public abstract void sorgrq(int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGRQ generates an M-by-N real matrix Q with orthonormal rows,
  which is defined as the last M rows of a product of K elementary
  reflectors of order N

        Q  =  H(1) H(2) . . . H(k)

  as returned by SGERQF.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix Q. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix Q. N >= M.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as

          returned by SGERQF in the last k rows of its array argument
          A.
          On exit, the M-by-N matrix Q.

  LDA     (input) INTEGER
          The first dimension of the array A. LDA >= max(1,M).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGERQF.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        k -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorgtr

        public abstract void sorgtr(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORGTR generates a real orthogonal matrix Q which is defined as the
  product of n-1 elementary reflectors of order N, as returned by
  SSYTRD:

  if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),

  if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from SSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from SSYTRD.

  N       (input) INTEGER
          The order of the matrix Q. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by SSYTRD.
          On exit, the N-by-N orthogonal matrix Q.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  TAU     (input) REAL array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SSYTRD.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N-1).
          For optimum performance LWORK >= (N-1)*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • sorm2l

        public abstract void sorm2l(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORM2L overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by SGEQLF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQLF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • sorm2r

        public abstract void sorm2r(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORM2R overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by SGEQRF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQRF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • sormbr

        public abstract void sormbr(java.lang.String vect,
                            java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  If VECT = 'Q', SORMBR overwrites the general real M-by-N matrix C
  with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  If VECT = 'P', SORMBR overwrites the general real M-by-N matrix C
  with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      P * C          C * P
  TRANS = 'T':      P**T * C       C * P**T

  Here Q and P**T are the orthogonal matrices determined by SGEBRD when
  reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and

  P**T are defined as products of elementary reflectors H(i) and G(i)
  respectively.

  Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
  order of the orthogonal matrix Q or P**T that is applied.

  If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
  if nq >= k, Q = H(1) H(2) . . . H(k);
  if nq < k, Q = H(1) H(2) . . . H(nq-1).

  If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
  if k < nq, P = G(1) G(2) . . . G(k);
  if k >= nq, P = G(1) G(2) . . . G(nq-1).

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'Q': apply Q or Q**T;
          = 'P': apply P or P**T.

  SIDE    (input) CHARACTER*1
          = 'L': apply Q, Q**T, P or P**T from the Left;
          = 'R': apply Q, Q**T, P or P**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q  or P;
          = 'T':  Transpose, apply Q**T or P**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          If VECT = 'Q', the number of columns in the original
          matrix reduced by SGEBRD.
          If VECT = 'P', the number of rows in the original
          matrix reduced by SGEBRD.
          K >= 0.

  A       (input) REAL array, dimension
                                (LDA,min(nq,K)) if VECT = 'Q'
                                (LDA,nq)        if VECT = 'P'
          The vectors which define the elementary reflectors H(i) and
          G(i), whose products determine the matrices Q and P, as
          returned by SGEBRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If VECT = 'Q', LDA >= max(1,nq);
          if VECT = 'P', LDA >= max(1,min(nq,K)).

  TAU     (input) REAL array, dimension (min(nq,K))
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i) which determines Q or P, as returned
          by SGEBRD in the array argument TAUQ or TAUP.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
          or P*C or P**T*C or C*P or C*P**T.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        vect -
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormhr

        public abstract void sormhr(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int ilo,
                            int ihi,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMHR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  IHI-ILO elementary reflectors, as returned by SGEHRD:

  Q = H(ilo) H(ilo+1) . . . H(ihi-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          ILO and IHI must have the same values as in the previous call
          of SGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
          ILO = 1 and IHI = 0, if M = 0;
          if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
          ILO = 1 and IHI = 0, if N = 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by SGEHRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

  TAU     (input) REAL array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEHRD.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        trans -
        m -
        n -
        ilo -
        ihi -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sorml2

        public abstract void sorml2(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORML2 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by SGELQF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGELQF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • sormlq

        public abstract void sormlq(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMLQ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by SGELQF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGELQF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormql

        public abstract void sormql(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMQL overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(k) . . . H(2) H(1)

  as returned by SGEQLF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQLF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormqr

        public abstract void sormqr(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMQR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by SGEQRF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGEQRF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormr2

        public abstract void sormr2(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMR2 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by SGERQF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGERQF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • sormr3

        public abstract void sormr3(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            int l,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMR3 overwrites the general real m by n matrix C with

        Q * C  if SIDE = 'L' and TRANS = 'N', or

        Q'* C  if SIDE = 'L' and TRANS = 'T', or

        C * Q  if SIDE = 'R' and TRANS = 'N', or

        C * Q' if SIDE = 'R' and TRANS = 'T',

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by STZRZF. Q is of order m if SIDE = 'L' and of order n
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q' from the Left
          = 'R': apply Q or Q' from the Right

  TRANS   (input) CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          STZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by STZRZF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace) REAL array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        l -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        info -

      • sormrq

        public abstract void sormrq(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMRQ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by SGERQF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          SGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SGERQF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormrz

        public abstract void sormrz(java.lang.String side,
                            java.lang.String trans,
                            int m,
                            int n,
                            int k,
                            int l,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMRZ overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix defined as the product of k
  elementary reflectors

        Q = H(1) H(2) . . . H(k)

  as returned by STZRZF. Q is of order M if SIDE = 'L' and of order N
  if SIDE = 'R'.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

  L       (input) INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by

          STZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

  TAU     (input) REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by STZRZF.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        trans -
        m -
        n -
        k -
        l -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • sormtr

        public abstract void sormtr(java.lang.String side,
                            java.lang.String uplo,
                            java.lang.String trans,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] c,
                            int Ldc,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SORMTR overwrites the general real M-by-N matrix C with

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T

  where Q is a real orthogonal matrix of order nq, with nq = m if
  SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
  nq-1 elementary reflectors, as returned by SSYTRD:

  if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);

  if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

  UPLO    (input) CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from SSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from SSYTRD.

  TRANS   (input) CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

  M       (input) INTEGER
          The number of rows of the matrix C. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix C. N >= 0.

  A       (input) REAL array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by SSYTRD.

  LDA     (input) INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

  TAU     (input) REAL array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by SSYTRD.

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.


  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        side -
        uplo -
        trans -
        m -
        n -
        a -
        lda -
        tau -
        c -
        Ldc -
        work -
        lwork -
        info -

      • spbcon

        public abstract void spbcon(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite band matrix using the
  Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A, stored in the

          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).


  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  ANORM   (input) REAL
          The 1-norm (or infinity-norm) of the symmetric band matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • spbequ

        public abstract void spbequ(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] s,
                            org.netlib.util.floatW scond,
                            org.netlib.util.floatW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite band matrix A and reduce its condition
  number (with respect to the two-norm).  S contains the scale factors,
  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
  choice of S puts the condition number of B within a factor N of the
  smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular of A is stored;
          = 'L':  Lower triangular of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column

          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB     (input) INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

  S       (output) REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        s -
        scond -
        amax -
        info -

      • spbrfs

        public abstract void spbrfs(java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] afb,
                            int ldafb,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and banded, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column

          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  AFB     (input) REAL array, dimension (LDAFB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A as computed by

          SPBTRF, in the same storage format as A (see AB).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SPBTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • spbstf

        public abstract void spbstf(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBSTF computes a split Cholesky factorization of a real
  symmetric positive definite band matrix A.

  This routine is designed to be used in conjunction with SSBGST.

  The factorization has the form  A = S**T*S  where S is a band matrix

  of the same bandwidth as A and the following structure:

    S = ( U    )
        ( M  L )

  where U is upper triangular of order m = (n+kd)/2, and L is lower
  triangular of order n-m.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first kd+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the factor S from the split Cholesky
          factorization A = S**T*S. See Further Details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the factorization could not be completed,
               because the updated element a(i,i) was negative; the
               matrix A is not positive definite.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 7, KD = 2:

  S = ( s11  s12  s13                     )
      (      s22  s23  s24                )
      (           s33  s34                )
      (                s44                )
      (           s53  s54  s55           )
      (                s64  s65  s66      )
      (                     s75  s76  s77 )

  If UPLO = 'U', the array AB holds:

  on entry:                          on exit:

    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75

   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76

  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77


  If UPLO = 'L', the array AB holds:

  on entry:                          on exit:

  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • spbsv

        public abstract void spbsv(java.lang.String uplo,
                           int n,
                           int kd,
                           int nrhs,
                           float[] ab,
                           int ldab,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite band matrix and X
  and B are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T * U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular band matrix, and L is a lower
  triangular band matrix, with the same number of superdiagonals or
  subdiagonals as A.  The factored form of A is then used to solve the

  system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
          See below for further details.

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • spbsvx

        public abstract void spbsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] afb,
                            int ldafb,
                            org.netlib.util.StringW equed,
                            float[] s,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite band matrix and X
  and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T * U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular band matrix, and L is a lower
     triangular band matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFB contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  AB and AFB will not
                  be modified.
          = 'N':  The matrix A will be copied to AFB and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFB and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right-hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array, except

          if FACT = 'F' and EQUED = 'Y', then A must contain the
          equilibrated matrix diag(S)*A*diag(S).  The j-th column of A

          is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
          See below for further details.

          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  LDAB    (input) INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

  AFB     (input or output) REAL array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the band matrix
          A, in the same storage format as A (see AB).  If EQUED = 'Y',
          then AFB is the factored form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

          If FACT = 'E', then AFB is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the equilibrated
          matrix A (see the description of A for the form of the
          equilibrated matrix).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) REAL array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45  a46
                         a55  a56
     (aij=conjg(aji))         a66

  Band storage of the upper triangle of A:

    *   a13  a24  a35  a46
   a12  a23  a34  a45  a56
     a11  a22  a33  a44  a55  a66

  Similarly, if UPLO = 'L' the format of A is as follows:

     a11  a22  a33  a44  a55  a66
     a21  a32  a43  a54  a65   *
     a31  a42  a53  a64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        afb -
        ldafb -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • spbtf2

        public abstract void spbtf2(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBTF2 computes the Cholesky factorization of a real symmetric
  positive definite band matrix A.

  The factorization has the form
     A = U' * U ,  if UPLO = 'U', or
     A = L  * L',  if UPLO = 'L',
  where U is an upper triangular matrix, U' is the transpose of U, and

  L is lower triangular.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',

          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U'*U or A = L*L' of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • spbtrf

        public abstract void spbtrf(java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBTRF computes the Cholesky factorization of a real symmetric
  positive definite band matrix A.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  Further Details
  ===============

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':

  On entry:                       On exit:

    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

  Similarly, if UPLO = 'L' the format of A is as follows:

  On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

  Array elements marked * are not used by the routine.

  Contributed by
  Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        kd -
        ab -
        ldab -
        info -

      • spbtrs

        public abstract void spbtrs(java.lang.String uplo,
                            int n,
                            int kd,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPBTRS solves a system of linear equations A*X = B with a symmetric
  positive definite band matrix A using the Cholesky factorization
  A = U**T*U or A = L*L**T computed by SPBTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T of the band matrix A, stored in the

          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).


  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • spocon

        public abstract void spocon(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOCON estimates the reciprocal of the condition number (in the 
  1-norm) of a real symmetric positive definite matrix using the
  Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by SPOTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  ANORM   (input) REAL
          The 1-norm (or infinity-norm) of the symmetric matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • spoequ

        public abstract void spoequ(int n,
                            float[] a,
                            int lda,
                            float[] s,
                            org.netlib.util.floatW scond,
                            org.netlib.util.floatW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite matrix A and reduce its condition number

  (with respect to the two-norm).  S contains the scale factors,
  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
  choice of S puts the condition number of B within a factor N of the
  smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The N-by-N symmetric positive definite matrix whose scaling
          factors are to be computed.  Only the diagonal elements of A

          are referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  S       (output) REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        a -
        lda -
        s -
        scond -
        amax -
        info -

      • sporfs

        public abstract void sporfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPORFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite,
  and provides error bounds and backward error estimates for the
  solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) REAL array, dimension (LDAF,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by SPOTRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SPOTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sposv

        public abstract void sposv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix and X and B
  are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T* U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is a lower triangular
  matrix.  The factored form of A is then used to solve the system of
  equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • sposvx

        public abstract void sposvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            org.netlib.util.StringW equed,
                            float[] s,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix and X and B
  are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T* U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is a lower triangular

     matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AF contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  A and AF will not
                  be modified.
          = 'N':  The matrix A will be copied to AF and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AF and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A, except if FACT = 'F' and
          EQUED = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.  A is not modified if
          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) REAL array, dimension (LDAF,N)
          If FACT = 'F', then AF is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, in the same storage
          format as A.  If EQUED .ne. 'N', then AF is the factored form
          of the equilibrated matrix diag(S)*A*diag(S).

          If FACT = 'N', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the original
          matrix A.

          If FACT = 'E', then AF is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T of the equilibrated
          matrix A (see the description of A for the form of the
          equilibrated matrix).

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) REAL array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • spotf2

        public abstract void spotf2(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOTF2 computes the Cholesky factorization of a real symmetric
  positive definite matrix A.

  The factorization has the form
     A = U' * U ,  if UPLO = 'U', or
     A = L  * L',  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U'*U  or A = L*L'.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • spotrf

        public abstract void spotrf(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOTRF computes the Cholesky factorization of a real symmetric
  positive definite matrix A.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  This is the block version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • spotri

        public abstract void spotri(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOTRI computes the inverse of a real symmetric positive definite
  matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
  computed by SPOTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, as computed by
          SPOTRF.
          On exit, the upper or lower triangle of the (symmetric)
          inverse of A, overwriting the input factor U or L.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is

                zero, and the inverse could not be computed.

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        info -

      • spotrs

        public abstract void spotrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPOTRS solves a system of linear equations A*X = B with a symmetric
  positive definite matrix A using the Cholesky factorization
  A = U**T*U or A = L*L**T computed by SPOTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by SPOTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • sppcon

        public abstract void sppcon(java.lang.String uplo,
                            int n,
                            float[] ap,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite packed matrix using
  the Cholesky factorization A = U**T*U or A = L*L**T computed by
  SPPTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

  ANORM   (input) REAL
          The 1-norm (or infinity-norm) of the symmetric matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • sppequ

        public abstract void sppequ(java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] s,
                            org.netlib.util.floatW scond,
                            org.netlib.util.floatW amax,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite matrix A in packed storage and reduce
  its condition number (with respect to the two-norm).  S contains the

  scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix

  B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
  This choice of S puts the condition number of B within a factor N of

  the smallest possible condition number over all possible diagonal
  scalings.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  S       (output) REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

  SCOND   (output) REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

  AMAX    (output) REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        s -
        scond -
        amax -
        info -

      • spprfs

        public abstract void spprfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] afp,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and packed, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

  AFP     (input) REAL array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, as computed by SPPTRF/CPPTRF,
          packed columnwise in a linear array in the same format as A
          (see AP).

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SPPTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        afp -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sppsv

        public abstract void sppsv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           float[] ap,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix stored in
  packed format and X and B are N-by-NRHS matrices.

  The Cholesky decomposition is used to factor A as
     A = U**T* U,  if UPLO = 'U', or
     A = L * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is a lower triangular
  matrix.  The factored form of A is then used to solve the system of
  equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.  

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, in the same storage
          format as A.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i of A is not
                positive definite, so the factorization could not be
                completed, and the solution has not been computed.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • sppsvx

        public abstract void sppsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] afp,
                            org.netlib.util.StringW equed,
                            float[] s,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
  compute the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric positive definite matrix stored in
  packed format and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') as
        A = U**T* U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is a lower triangular

     matrix.

  3. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFP contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  AP and AFP will not
                  be modified.
          = 'N':  The matrix A will be copied to AFP and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then

                  copied to AFP and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array, except if FACT = 'F'

          and EQUED = 'Y', then A must contain the equilibrated matrix

          diag(S)*A*diag(S).  The j-th column of A is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.  A is not modified if
          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

  AFP     (input or output) REAL array, dimension
                            (N*(N+1)/2)
          If FACT = 'F', then AFP is an input argument and on entry
          contains the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L', in the same storage
          format as A.  If EQUED .ne. 'N', then AFP is the factored
          form of the equilibrated matrix A.

          If FACT = 'N', then AFP is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L' of the original matrix A.


          If FACT = 'E', then AFP is an output argument and on exit
          returns the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L' of the equilibrated
          matrix A (see the description of AP for the form of the
          equilibrated matrix).

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by

                  diag(S) * A * diag(S).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  S       (input or output) REAL array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is
          an input argument if FACT = 'F'; otherwise, S is an output
          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
          must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
          B is overwritten by diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
          the original system of equations.  Note that if EQUED = 'Y',

          A and B are modified on exit, and the solution to the
          equilibrated system is inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        ap -
        afp -
        equed -
        s -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • spptrf

        public abstract void spptrf(java.lang.String uplo,
                            int n,
                            float[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPTRF computes the Cholesky factorization of a real symmetric
  positive definite matrix A stored in packed format.

  The factorization has the form
     A = U**T * U,  if UPLO = 'U', or
     A = L  * L**T,  if UPLO = 'L',
  where U is an upper triangular matrix and L is lower triangular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T, in the same

          storage format as A.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

  Further Details
  ======= =======

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        info -

      • spptri

        public abstract void spptri(java.lang.String uplo,
                            int n,
                            float[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPTRI computes the inverse of a real symmetric positive definite
  matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
  computed by SPPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangular factor is stored in AP;
          = 'L':  Lower triangular factor is stored in AP.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, packed columnwise as

          a linear array.  The j-th column of U or L is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

          On exit, the upper or lower triangle of the (symmetric)
          inverse of A, overwriting the input factor U or L.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is

                zero, and the inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        info -

      • spptrs

        public abstract void spptrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPPTRS solves a system of linear equations A*X = B with a symmetric
  positive definite matrix A in packed storage using the Cholesky
  factorization A = U**T*U or A = L*L**T computed by SPPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization

          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • sptcon

        public abstract void sptcon(int n,
                            float[] d,
                            float[] e,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTCON computes the reciprocal of the condition number (in the
  1-norm) of a real symmetric positive definite tridiagonal matrix
  using the factorization A = L*D*L**T or A = U**T*D*U computed by
  SPTTRF.

  Norm(inv(A)) is computed by a direct method, and the reciprocal of
  the condition number is computed as
               RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          factorization of A, as computed by SPTTRF.

  E       (input) REAL array, dimension (N-1)
          The (n-1) off-diagonal elements of the unit bidiagonal factor
          U or L from the factorization of A,  as computed by SPTTRF.

  ANORM   (input) REAL
          The 1-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
          1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The method used is described in Nicholas J. Higham, "Efficient
  Algorithms for Computing the Condition Number of a Tridiagonal
  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        anorm -
        rcond -
        work -
        info -

      • spteqr

        public abstract void spteqr(java.lang.String compz,
                            int n,
                            float[] d,
                            float[] e,
                            float[] z,
                            int ldz,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
  symmetric positive definite tridiagonal matrix by first factoring the
  matrix using SPTTRF, and then calling SBDSQR to compute the singular

  values of the bidiagonal factor.

  This routine computes the eigenvalues of the positive definite
  tridiagonal matrix to high relative accuracy.  This means that if the
  eigenvalues range over many orders of magnitude in size, then the
  small eigenvalues and corresponding eigenvectors will be computed
  more accurately than, for example, with the standard QR method.

  The eigenvectors of a full or band symmetric positive definite matrix
  can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
  reduce this matrix to tridiagonal form. (The reduction to tridiagonal
  form, however, may preclude the possibility of obtaining high
  relative accuracy in the small eigenvalues of the original matrix, if
  these eigenvalues range over many orders of magnitude.)

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvectors of original symmetric
                  matrix also.  Array Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal
          matrix.
          On normal exit, D contains the eigenvalues, in descending
          order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  Z       (input/output) REAL array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix used in the
          reduction to tridiagonal form.
          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the

          original symmetric matrix;
          if COMPZ = 'I', the orthonormal eigenvectors of the
          tridiagonal matrix.
          If INFO > 0 on exit, Z contains the eigenvectors associated
          with only the stored eigenvalues.
          If  COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          COMPZ = 'V' or 'I', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (4*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, and i is:
                <= N  the Cholesky factorization of the matrix could
                      not be performed because the i-th principal minor
                      was not positive definite.
                > N   the SVD algorithm failed to converge;
                      if INFO = N+i, i off-diagonal elements of the
                      bidiagonal factor did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • sptrfs

        public abstract void sptrfs(int n,
                            int nrhs,
                            float[] d,
                            float[] e,
                            float[] df,
                            float[] ef,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric positive definite

  and tridiagonal, and provides error bounds and backward error
  estimates for the solution.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.

  E       (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

  DF      (input) REAL array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          factorization computed by SPTTRF.

  EF      (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the factorization computed by SPTTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SPTTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        df -
        ef -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        info -

      • sptsv

        public abstract void sptsv(int n,
                           int nrhs,
                           float[] d,
                           float[] e,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTSV computes the solution to a real system of linear equations
  A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
  matrix, and X and B are N-by-NRHS matrices.

  A is factored as A = L*D*L**T, and the factored form of A is then
  used to solve the system of equations.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the factorization A = L*D*L**T.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L**T factorization of
          A.  (E can also be regarded as the superdiagonal of the unit

          bidiagonal factor U from the U**T*D*U factorization of A.)

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the solution has not been
                computed.  The factorization has not been completed
                unless i = N.

  =====================================================================

     .. External Subroutines ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        info -

      • sptsvx

        public abstract void sptsvx(java.lang.String fact,
                            int n,
                            int nrhs,
                            float[] d,
                            float[] e,
                            float[] df,
                            float[] ef,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTSVX uses the factorization A = L*D*L**T to compute the solution
  to a real system of linear equations A*X = B, where A is an N-by-N
  symmetric positive definite tridiagonal matrix and X and B are
  N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
     is a unit lower bidiagonal matrix and D is diagonal.  The
     factorization can also be regarded as having the form
     A = U**T*D*U.

  2. If the leading i-by-i principal minor is not positive definite,
     then the routine returns with INFO = i. Otherwise, the factored
     form of A is used to estimate the condition number of the matrix
     A.  If the reciprocal of the condition number is less than machine
     precision, INFO = N+1 is returned as a warning, but the routine
     still goes on to solve for X and compute error bounds as
     described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, DF and EF contain the factored form of A.
                  D, E, DF, and EF will not be modified.
          = 'N':  The matrix A will be copied to DF and EF and
                  factored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.

  E       (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

  DF      (input or output) REAL array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.
          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.

  EF      (input or output) REAL array, dimension (N-1)
          If FACT = 'F', then EF is an input argument and on entry
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.
          If FACT = 'N', then EF is an output argument and on exit
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.

  B       (input) REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The reciprocal condition number of the matrix A.  If RCOND
          is less than the machine precision (in particular, if
          RCOND = 0), the matrix is singular to working precision.
          This condition is indicated by a return code of INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in any
          element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        n -
        nrhs -
        d -
        e -
        df -
        ef -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        info -

      • spttrf

        public abstract void spttrf(int n,
                            float[] d,
                            float[] e,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTTRF computes the L*D*L' factorization of a real symmetric
  positive definite tridiagonal matrix A.  The factorization may also
  be regarded as having the form A = U'*D*U.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the L*D*L' factorization of A.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L' factorization of A.

          E can also be regarded as the superdiagonal of the unit
          bidiagonal factor U from the U'*D*U factorization of A.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite; if k < N, the factorization could not
               be completed, while if k = N, the factorization was
               completed, but D(N) <= 0.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • spttrs

        public abstract void spttrs(int n,
                            int nrhs,
                            float[] d,
                            float[] e,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SPTTRS solves a tridiagonal system of the form
     A * X = B
  using the L*D*L' factorization of A computed by SPTTRF.  D is a
  diagonal matrix specified in the vector D, L is a unit bidiagonal
  matrix whose subdiagonal is specified in the vector E, and X and B
  are N by NRHS matrices.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          L*D*L' factorization of A.

  E       (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the L*D*L' factorization of A.  E can also be regarded
          as the superdiagonal of the unit bidiagonal factor U from the
          factorization A = U'*D*U.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side vectors B for the system of
          linear equations.
          On exit, the solution vectors, X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -
        info -

      • sptts2

        public abstract void sptts2(int n,
                            int nrhs,
                            float[] d,
                            float[] e,
                            float[] b,
                            int ldb)
        
     ..

  Purpose
  =======

  SPTTS2 solves a tridiagonal system of the form
     A * X = B
  using the L*D*L' factorization of A computed by SPTTRF.  D is a
  diagonal matrix specified in the vector D, L is a unit bidiagonal
  matrix whose subdiagonal is specified in the vector E, and X and B
  are N by NRHS matrices.

  Arguments
  =========

  N       (input) INTEGER
          The order of the tridiagonal matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          L*D*L' factorization of A.

  E       (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the unit bidiagonal factor

          L from the L*D*L' factorization of A.  E can also be regarded
          as the superdiagonal of the unit bidiagonal factor U from the
          factorization A = U'*D*U.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side vectors B for the system of
          linear equations.
          On exit, the solution vectors, X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        n -
        nrhs -
        d -
        e -
        b -
        ldb -

      • srscl

        public abstract void srscl(int n,
                           float sa,
                           float[] sx,
                           int incx)
        
     ..

  Purpose
  =======

  SRSCL multiplies an n-element real vector x by the real scalar 1/a.
  This is done without overflow or underflow as long as
  the final result x/a does not overflow or underflow.

  Arguments
  =========

  N       (input) INTEGER
          The number of components of the vector x.

  SA      (input) REAL
          The scalar a which is used to divide each component of x.
          SA must be >= 0, or the subroutine will divide by zero.

  SX      (input/output) REAL array, dimension
                         (1+(N-1)*abs(INCX))
          The n-element vector x.

  INCX    (input) INTEGER
          The increment between successive values of the vector SX.
          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n


 =====================================================================


     .. Parameters ..
        Parameters:
        n -
        sa -
        sx -
        incx -

      • ssbev

        public abstract void ssbev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           int kd,
                           float[] ab,
                           int ldab,
                           float[] w,
                           float[] z,
                           int ldz,
                           float[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBEV computes all the eigenvalues and, optionally, eigenvectors of
  a real symmetric band matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (max(1,3*N-2))

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        kd -
        ab -
        ldab -
        w -
        z -
        ldz -
        work -
        info -

      • ssbevd

        public abstract void ssbevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBEVD computes all the eigenvalues and, optionally, eigenvectors of

  a real symmetric band matrix A. If eigenvectors are desired, it uses

  a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          IF N <= 1,                LWORK must be at least 1.
          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
          If JOBZ  = 'V' and N > 2, LWORK must be at least
                         ( 1 + 5*N + 2*N**2 ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array LIWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        kd -
        ab -
        ldab -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssbevx

        public abstract void ssbevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] q,
                            int ldq,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric band matrix A.  Eigenvalues and eigenvectors can

  be selected by specifying either a range of values or a range of
  indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found;
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found;
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.

  Q       (output) REAL array, dimension (LDQ, N)
          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                         reduction to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  If JOBZ = 'V', then
          LDQ >= max(1,N).

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing AB to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (7*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        kd -
        ab -
        ldab -
        q -
        ldq -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • ssbgst

        public abstract void ssbgst(java.lang.String vect,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            float[] ab,
                            int ldab,
                            float[] bb,
                            int ldbb,
                            float[] x,
                            int ldx,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBGST reduces a real symmetric-definite banded generalized
  eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
  such that C has the same bandwidth as A.

  B must have been previously factorized as S**T*S by SPBSTF, using a
  split Cholesky factorization. A is overwritten by C = X**T*A*X, where
  X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
  bandwidth of A.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.


  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the transformed matrix X**T*A*X, stored in the same

          format as A.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input) REAL array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of

          B, as returned by SPBSTF, stored in the first KB+1 rows of
          the array.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  X       (output) REAL array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X.
          If VECT = 'N', the array X is not referenced.

  LDX     (input) INTEGER
          The leading dimension of the array X.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

  WORK    (workspace) REAL array, dimension (2*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        x -
        ldx -
        work -
        info -

      • ssbgv

        public abstract void ssbgv(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           int ka,
                           int kb,
                           float[] ab,
                           int ldab,
                           float[] bb,
                           int ldbb,
                           float[] w,
                           float[] z,
                           int ldz,
                           float[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBGV computes all the eigenvalues, and optionally, the eigenvectors

  of a real generalized symmetric-definite banded eigenproblem, of
  the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
  and banded, and B is also positive definite.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KB >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) REAL array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by SPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i). The eigenvectors are
          normalized so that Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= N.

  WORK    (workspace) REAL array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobz -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        w -
        z -
        ldz -
        work -
        info -

      • ssbgvd

        public abstract void ssbgvd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            float[] ab,
                            int ldab,
                            float[] bb,
                            int ldbb,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite banded eigenproblem, of the

  form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and

  banded, and B is also positive definite.  If eigenvectors are
  desired, it uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) REAL array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by SPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 3*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.


  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssbgvx

        public abstract void ssbgvx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            int ka,
                            int kb,
                            float[] ab,
                            int ldab,
                            float[] bb,
                            int ldbb,
                            float[] q,
                            int ldq,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite banded eigenproblem, of
  the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric

  and banded, and B is also positive definite.  Eigenvalues and
  eigenvectors can be selected by specifying either all eigenvalues,
  a range of values or a range of indices for the desired eigenvalues.


  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  KA      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

  KB      (input) INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

  AB      (input/output) REAL array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

  BB      (input/output) REAL array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by SPBSTF.

  LDBB    (input) INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

  Q       (output) REAL array, dimension (LDQ, N)
          If JOBZ = 'V', the n-by-n matrix used in the reduction of
          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
          and consequently C to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  If JOBZ = 'N',
          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (7N)

  IWORK   (workspace/output) INTEGER array, dimension (5N)

  IFAIL   (output) INTEGER array, dimension (M)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvalues that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0 : successful exit
          < 0 : if INFO = -i, the i-th argument had an illegal value
          <= N: if INFO = i, then i eigenvectors failed to converge.
                  Their indices are stored in IFAIL.
          > N : SPBSTF returned an error code; i.e.,
                if INFO = N + i, for 1 <= i <= N, then the leading
                minor of order i of B is not positive definite.
                The factorization of B could not be completed and
                no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        ka -
        kb -
        ab -
        ldab -
        bb -
        ldbb -
        q -
        ldq -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • ssbtrd

        public abstract void ssbtrd(java.lang.String vect,
                            java.lang.String uplo,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            float[] d,
                            float[] e,
                            float[] q,
                            int ldq,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSBTRD reduces a real symmetric band matrix A to symmetric
  tridiagonal form T by an orthogonal similarity transformation:
  Q**T * A * Q = T.

  Arguments
  =========

  VECT    (input) CHARACTER*1
          = 'N':  do not form Q;
          = 'V':  form Q;
          = 'U':  update a matrix X, by forming X*Q.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by

          values generated during the reduction.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  D       (output) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (output) REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if VECT = 'U', then Q must contain an N-by-N
          matrix X; if VECT = 'N' or 'V', then Q need not be set.

          On exit:
          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
          if VECT = 'U', Q contains the product X*Q;
          if VECT = 'N', the array Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Modified by Linda Kaufman, Bell Labs.

  =====================================================================

     .. Parameters ..
        Parameters:
        vect -
        uplo -
        n -
        kd -
        ab -
        ldab -
        d -
        e -
        q -
        ldq -
        work -
        info -

      • sspcon

        public abstract void sspcon(java.lang.String uplo,
                            int n,
                            float[] ap,
                            int[] ipiv,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric packed matrix A using the factorization
  A = U*D*U**T or A = L*D*L**T computed by SSPTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by SSPTRF, stored as a
          packed triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSPTRF.

  ANORM   (input) REAL
          The 1-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (2*N)

  IWORK    (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • sspev

        public abstract void sspev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           float[] ap,
                           float[] w,
                           float[] z,
                           int ldz,
                           float[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPEV computes all the eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A in packed storage.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ap -
        w -
        z -
        ldz -
        work -
        info -

      • sspevd

        public abstract void sspevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPEVD computes all the eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A in packed storage. If eigenvectors are
  desired, it uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the required LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK must be at least 1.
          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
          If JOBZ = 'V' and N > 1, LWORK must be at least
                                                 1 + 6*N + N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        ap -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sspevx

        public abstract void sspevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            float[] ap,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A in packed storage.  Eigenvalues/vectors

  can be selected by specifying either a range of values or a range of

  indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found;
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found;
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing AP to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the selected eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (8*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        ap -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • sspgst

        public abstract void sspgst(int itype,
                            java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] bp,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPGST reduces a real symmetric-definite generalized eigenproblem
  to standard form, using packed storage.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

  B must have been previously factorized as U**T*U or L*L**T by SPPTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
          = 2 or 3: compute U*A*U**T or L**T*A*L.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**T*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**T.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  BP      (input) REAL array, dimension (N*(N+1)/2)
          The triangular factor from the Cholesky factorization of B,
          stored in the same format as A, as returned by SPPTRF.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        ap -
        bp -
        info -

      • sspgv

        public abstract void sspgv(int itype,
                           java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           float[] ap,
                           float[] bp,
                           float[] w,
                           float[] z,
                           int ldz,
                           float[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPGV computes all the eigenvalues and, optionally, the eigenvectors

  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
  Here A and B are assumed to be symmetric, stored in packed format,
  and B is also positive definite.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) REAL array, dimension
                            (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPPTRF or SSPEV returned an error code:
             <= N:  if INFO = i, SSPEV failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero.
             > N:   if INFO = n + i, for 1 <= i <= n, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        ap -
        bp -
        w -
        z -
        ldz -
        work -
        info -

      • sspgvd

        public abstract void sspgvd(int itype,
                            java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] bp,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
  B are assumed to be symmetric, stored in packed format, and B is also
  positive definite.
  If eigenvectors are desired, it uses a divide and conquer algorithm.


  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the required LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPPTRF or SSPEVD returned an error code:
             <= N:  if INFO = i, SSPEVD failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        ap -
        bp -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sspgvx

        public abstract void sspgvx(int itype,
                            java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] bp,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
  and B are assumed to be symmetric, stored in packed storage, and B
  is also positive definite.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of indices
  for the desired eigenvalues.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A and B are stored;
          = 'L':  Lower triangle of A and B are stored.

  N       (input) INTEGER
          The order of the matrix pencil (A,B).  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the contents of AP are destroyed.

  BP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.


          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'N', then Z is not referenced.
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.

          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (8*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPPTRF or SSPEVX returned an error code:
             <= N:  if INFO = i, SSPEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 =====================================================================


     .. Local Scalars ..
        Parameters:
        itype -
        jobz -
        range -
        uplo -
        n -
        ap -
        bp -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • ssprfs

        public abstract void ssprfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] afp,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric indefinite
  and packed, and provides error bounds and backward error estimates
  for the solution.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


  AFP     (input) REAL array, dimension (N*(N+1)/2)
          The factored form of the matrix A.  AFP contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by SSPTRF, stored as a packed
          triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSPTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SSPTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        afp -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • sspsv

        public abstract void sspsv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           float[] ap,
                           int[] ipiv,
                           float[] b,
                           int ldb,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric matrix stored in packed format and X
  and B are N-by-NRHS matrices.

  The diagonal pivoting method is used to factor A as
     A = U * D * U**T,  if UPLO = 'U', or
     A = L * D * L**T,  if UPLO = 'L',
  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, D is symmetric and block diagonal with 1-by-1
  and 2-by-2 diagonal blocks.  The factored form of A is then used to
  solve the system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D, as

          determined by SSPTRF.  If IPIV(k) > 0, then rows and columns

          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
          then rows and columns k-1 and -IPIV(k) were interchanged and

          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
          diagonal block.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is

                exactly singular, so the solution could not be
                computed.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. External Functions ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        ipiv -
        b -
        ldb -
        info -

      • sspsvx

        public abstract void sspsvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] afp,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
  A = L*D*L**T to compute the solution to a real system of linear
  equations A * X = B, where A is an N-by-N symmetric matrix stored
  in packed format and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A as
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)

     triangular matrices and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

  2. If some D(i,i)=0, so that D is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, AFP and IPIV contain the factored form of
                  A.  AP, AFP and IPIV will not be modified.
          = 'N':  The matrix A will be copied to AFP and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          See below for further details.

  AFP     (input or output) REAL array, dimension
                            (N*(N+1)/2)
          If FACT = 'F', then AFP is an input argument and on entry
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

          If FACT = 'N', then AFP is an output argument and on exit
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
          a packed triangular matrix in the same storage format as A.

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains details of the interchanges and the block structure

          of D, as determined by SSPTRF.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains details of the interchanges and the block structure

          of D, as determined by SSPTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  D(i,i) is exactly zero.  The factorization
                       has been completed but the factor D is exactly
                       singular, so the solution and error bounds could
                       not be computed. RCOND = 0 is returned.
                = N+1: D is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  Further Details
  ===============

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        ap -
        afp -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        iwork -
        info -

      • ssptrd

        public abstract void ssptrd(java.lang.String uplo,
                            int n,
                            float[] ap,
                            float[] d,
                            float[] e,
                            float[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPTRD reduces a real symmetric matrix A stored in packed form to
  symmetric tridiagonal form T by an orthogonal similarity
  transformation: Q**T * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  D       (output) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  overwriting A(i+2:n,i), and tau is stored in TAU(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        d -
        e -
        tau -
        info -

      • ssptrf

        public abstract void ssptrf(java.lang.String uplo,
                            int n,
                            float[] ap,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPTRF computes the factorization of a real symmetric matrix A stored
  in packed format using the Bunch-Kaufman diagonal pivoting method:

     A = U*D*U**T  or  A = L*D*L**T

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with
  1-by-1 and 2-by-2 diagonal blocks.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L, stored as a packed triangular
          matrix overwriting A (see below for further details).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it

               is used to solve a system of equations.

  Further Details
  ===============

  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        info -

      • ssptri

        public abstract void ssptri(java.lang.String uplo,
                            int n,
                            float[] ap,
                            int[] ipiv,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPTRI computes the inverse of a real symmetric indefinite matrix
  A in packed storage using the factorization A = U*D*U**T or
  A = L*D*L**T computed by SSPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by SSPTRF,
          stored as a packed triangular matrix.

          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix, stored as a packed triangular matrix. The j-th column
          of inv(A) is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
          if UPLO = 'L',
             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSPTRF.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

               inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        ap -
        ipiv -
        work -
        info -

      • ssptrs

        public abstract void ssptrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] ap,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSPTRS solves a system of linear equations A*X = B with a real
  symmetric matrix A stored in packed format using the factorization
  A = U*D*U**T or A = L*D*L**T computed by SSPTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by SSPTRF, stored as a
          packed triangular matrix.

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSPTRF.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        ap -
        ipiv -
        b -
        ldb -
        info -

      • sstebz

        public abstract void sstebz(java.lang.String range,
                            java.lang.String order,
                            int n,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            float[] d,
                            float[] e,
                            org.netlib.util.intW m,
                            org.netlib.util.intW nsplit,
                            float[] w,
                            int[] iblock,
                            int[] isplit,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEBZ computes the eigenvalues of a symmetric tridiagonal
  matrix T.  The user may ask for all eigenvalues, all eigenvalues
  in the half-open interval (VL, VU], or the IL-th through IU-th
  eigenvalues.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
  Matrix", Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

  Arguments
  =========

  RANGE   (input) CHARACTER*1
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the

                           entire matrix) will be found.

  ORDER   (input) CHARACTER*1
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and

                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal

          to VL, or greater than VU, will not be returned.  VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute tolerance for the eigenvalues.  An eigenvalue
          (or cluster) is considered to be located if it has been
          determined to lie in an interval whose width is ABSTOL or
          less.  If ABSTOL is less than or equal to zero, then ULP*|T|

          will be used, where |T| means the 1-norm of T.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) REAL array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.


  M       (output) INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

  NSPLIT  (output) INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

  W       (output) REAL array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalues.  (SSTEBZ may use the remaining N-M elements as
          workspace.)

  IBLOCK  (output) INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (SSTEBZ may use the remaining N-M elements as
          workspace.)

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to ISPLIT(1),

          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
          etc., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

  WORK    (workspace) REAL array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the

                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick

                                out eigenvalues IL:IU.  In some cases,

                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  Internal Parameters
  ===================

  RELFAC  REAL, default = 2.0e0
          The relative tolerance.  An interval (a,b] lies within
          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
          where "ulp" is the machine precision (distance from 1 to
          the next larger floating point number.)

  FUDGE   REAL, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

  =====================================================================

     .. Parameters ..
        Parameters:
        range -
        order -
        n -
        vl -
        vu -
        il -
        iu -
        abstol -
        d -
        e -
        m -
        nsplit -
        w -
        iblock -
        isplit -
        work -
        iwork -
        info -

      • sstedc

        public abstract void sstedc(java.lang.String compz,
                            int n,
                            float[] d,
                            float[] e,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEDC computes all eigenvalues and, optionally, eigenvectors of a
  symmetric tridiagonal matrix using the divide and conquer method.
  The eigenvectors of a full or band real symmetric matrix can also be

  found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this
  matrix to tridiagonal form.

  This code makes very mild assumptions about floating point
  arithmetic. It will work on machines with a guard digit in
  add/subtract, or on those binary machines without guard digits
  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
  It could conceivably fail on hexadecimal or decimal machines
  without guard digits, but we know of none.  See SLAED3 for details.

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.
          = 'V':  Compute eigenvectors of original dense symmetric
                  matrix also.  On entry, Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.

  N       (input) INTEGER
          The dimension of the symmetric tridiagonal matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

  Z       (input/output) REAL array, dimension (LDZ,N)
          On entry, if COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If  COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If eigenvectors are desired, then LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 3*N + 2*N*lg N + 3*N**2 ),
                         where lg( N ) = smallest integer k such
                         that 2**k >= N.
          If COMPZ = 'I' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LWORK need
          only be max(1,2*(N-1)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LIWORK must be at least
                         ( 6 + 6*N + 5*N*lg N ).
          If COMPZ = 'I' and N > 1 then LIWORK must be at least
                         ( 3 + 5*N ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LIWORK
          need only be 1.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sstegr

        public abstract void sstegr(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            float[] d,
                            float[] e,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            int[] isuppz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEGR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T. Any such unreduced matrix h
  a well defined set of pairwise different real eigenvalues, the corres
  real eigenvectors are pairwise orthogonal.

  The spectrum may be computed either completely or partially by specif
  either an interval (VL,VU] or a range of indices IL:IU for the desire
  eigenvalues.

  SSTEGR is a compatability wrapper around the improved SSTEMR routine.
  See SSTEMR for further details.

  One important change is that the ABSTOL parameter no longer provides 
  benefit and hence is no longer used.

  Note : SSTEGR and SSTEMR work only on machines which follow
  IEEE-754 floating-point standard in their handling of infinities and

  NaNs.  Normal execution may create these exceptiona values and hence

  may abort due to a floating point exception in environments which
  do not conform to the IEEE-754 standard.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

  E       (input/output) REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          Unused.  Was the absolute error tolerance for the
          eigenvalues/eigenvectors in previous versions.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigen
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

  WORK    (workspace/output) REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in SLARRE,
                if INFO = 2X, internal error in SLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by SLARRE or
                SLARRV, respectively.

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, LBNL/NERSC, USA

  =====================================================================

     .. Local Scalars ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sstein

        public abstract void sstein(int n,
                            float[] d,
                            float[] e,
                            int m,
                            float[] w,
                            int[] iblock,
                            int[] isplit,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEIN computes the eigenvectors of a real symmetric tridiagonal
  matrix T corresponding to specified eigenvalues, using inverse
  iteration.

  The maximum number of iterations allowed for each eigenvector is
  specified by an internal parameter MAXITS (currently set to 5).

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix
          T, in elements 1 to N-1.

  M       (input) INTEGER
          The number of eigenvectors to be found.  0 <= M <= N.

  W       (input) REAL array, dimension (N)
          The first M elements of W contain the eigenvalues for
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block.  ( The output array
          W from SSTEBZ with ORDER = 'B' is expected here. )

  IBLOCK  (input) INTEGER array, dimension (N)
          The submatrix indices associated with the corresponding
          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
          the first submatrix from the top, =2 if W(i) belongs to
          the second submatrix, etc.  ( The output array IBLOCK
          from SSTEBZ is expected here. )

  ISPLIT  (input) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.

          The first submatrix consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.
          ( The output array ISPLIT from SSTEBZ is expected here. )

  Z       (output) REAL array, dimension (LDZ, M)
          The computed eigenvectors.  The eigenvector associated
          with the eigenvalue W(i) is stored in the i-th column of
          Z.  Any vector which fails to converge is set to its current

          iterate after MAXITS iterations.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (5*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  IFAIL   (output) INTEGER array, dimension (M)
          On normal exit, all elements of IFAIL are zero.
          If one or more eigenvectors fail to converge after
          MAXITS iterations, then their indices are stored in
          array IFAIL.

  INFO    (output) INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, then i eigenvectors failed to converge
               in MAXITS iterations.  Their indices are stored in
               array IFAIL.

  Internal Parameters
  ===================

  MAXITS  INTEGER, default = 5
          The maximum number of iterations performed.

  EXTRA   INTEGER, default = 2
          The number of iterations performed after norm growth
          criterion is satisfied, should be at least 1.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        m -
        w -
        iblock -
        isplit -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • sstemr

        public void sstemr(java.lang.String jobz,
                   java.lang.String range,
                   int n,
                   float[] d,
                   float[] e,
                   float vl,
                   float vu,
                   int il,
                   int iu,
                   org.netlib.util.intW m,
                   float[] w,
                   float[] z,
                   int ldz,
                   int nzc,
                   int[] isuppz,
                   org.netlib.util.booleanW tryrac,
                   float[] work,
                   int lwork,
                   int[] iwork,
                   int liwork,
                   org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEMR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T. Any such unreduced matrix h
  a well defined set of pairwise different real eigenvalues, the corres
  real eigenvectors are pairwise orthogonal.

  The spectrum may be computed either completely or partially by specif
  either an interval (VL,VU] or a range of indices IL:IU for the desire
  eigenvalues.

  Depending on the number of desired eigenvalues, these are computed ei
  by bisection or the dqds algorithm. Numerically orthogonal eigenvecto
  computed by the use of various suitable L D L^T factorizations near c
  of close eigenvalues (referred to as RRRs, Relatively Robust
  Representations). An informal sketch of the algorithm follows.

  For each unreduced block (submatrix) of T,
     (a) Compute T - sigma I  = L D L^T, so that L and D
         define all the wanted eigenvalues to high relative accuracy.
         This means that small relative changes in the entries of D and
         cause only small relative changes in the eigenvalues and
         eigenvectors. The standard (unfactored) representation of the

         tridiagonal matrix T does not have this property in general.
     (b) Compute the eigenvalues to suitable accuracy.
         If the eigenvectors are desired, the algorithm attains full
         accuracy of the computed eigenvalues only right before
         the corresponding vectors have to be computed, see steps c) an
     (c) For each cluster of close eigenvalues, select a new
         shift close to the cluster, find a new factorization, and refi
         the shifted eigenvalues to suitable accuracy.
     (d) For each eigenvalue with a large enough relative separation co
         the corresponding eigenvector by forming a rank revealing twis
         factorization. Go back to (c) for any clusters that remain.

  For more details, see:
  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representat
    to compute orthogonal eigenvectors of symmetric tridiagonal matrice
    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors an
    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, V
    2004.  Also LAPACK Working Note 154.
  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
    tridiagonal eigenvalue/eigenvector problem",
    Computer Science Division Technical Report No. UCB/CSD-97-971,
    UC Berkeley, May 1997.

  Notes:
  1.SSTEMR works only on machines which follow IEEE-754
  floating-point standard in their handling of infinities and NaNs.
  This permits the use of efficient inner loops avoiding a check for
  zero divisors.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

  E       (input/output) REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and can be computed with a workspace

          query by setting NZC = -1, see below.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

  NZC     (input) INTEGER
          The number of eigenvectors to be held in the array Z.
          If RANGE = 'A', then NZC >= max(1,N).
          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,
          If RANGE = 'I', then NZC >= IU-IL+1.
          If NZC = -1, then a workspace query is assumed; the
          routine calculates the number of columns of the array Z that

          are needed to hold the eigenvectors.
          This value is returned as the first entry of the Z array, and
          no error message related to NZC is issued by XERBLA.

  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigen
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

  TRYRAC  (input/output) LOGICAL
          If TRYRAC.EQ..TRUE., indicates that the code should check whe
          the tridiagonal matrix defines its eigenvalues to high relati
          accuracy.  If so, the code uses relative-accuracy preserving

          algorithms that might be (a bit) slower depending on the matr
          If the matrix does not define its eigenvalues to high relativ
          accuracy, the code can uses possibly faster algorithms.
          If TRYRAC.EQ..FALSE., the code is not required to guarantee
          relatively accurate eigenvalues and can use the fastest possi
          techniques.
          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
          does not define its eigenvalues to high relative accuracy.

  WORK    (workspace/output) REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in SLARRE,
                if INFO = 2X, internal error in SLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by SLARRE or
                SLARRV, respectively.


  Further Details
  ===============

  Based on contributions by
     Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        m -
        w -
        z -
        ldz -
        nzc -
        isuppz -
        tryrac -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssteqr

        public abstract void ssteqr(java.lang.String compz,
                            int n,
                            float[] d,
                            float[] e,
                            float[] z,
                            int ldz,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
  symmetric tridiagonal matrix using the implicit QL or QR method.
  The eigenvectors of a full or band symmetric matrix can also be found
  if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to

  tridiagonal form.

  Arguments
  =========

  COMPZ   (input) CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvalues and eigenvectors of the original

                  symmetric matrix.  On entry, Z must contain the
                  orthogonal matrix used to reduce the original matrix

                  to tridiagonal form.
          = 'I':  Compute eigenvalues and eigenvectors of the
                  tridiagonal matrix.  Z is initialized to the identity
                  matrix.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  Z       (input/output) REAL array, dimension (LDZ, N)
          On entry, if  COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If COMPZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          eigenvectors are desired, then  LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (max(1,2*N-2))
          If COMPZ = 'N', then WORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm has failed to find all the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero; on exit, D
                and E contain the elements of a symmetric tridiagonal
                matrix which is orthogonally similar to the original
                matrix.

  =====================================================================

     .. Parameters ..
        Parameters:
        compz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • ssterf

        public abstract void ssterf(int n,
                            float[] d,
                            float[] e,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTERF computes all eigenvalues of a symmetric tridiagonal matrix
  using the Pal-Walker-Kahan variant of the QL or QR algorithm.

  Arguments
  =========

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix.

          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm failed to find all of the eigenvalues in

                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        n -
        d -
        e -
        info -

      • sstev

        public abstract void sstev(java.lang.String jobz,
                           int n,
                           float[] d,
                           float[] e,
                           float[] z,
                           int ldz,
                           float[] work,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEV computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric tridiagonal matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (max(1,2*N-2))
          If JOBZ = 'N', WORK is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        info -

      • sstevd

        public abstract void sstevd(java.lang.String jobz,
                            int n,
                            float[] d,
                            float[] e,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEVD computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric tridiagonal matrix. If eigenvectors are desired, it
  uses a divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.

  E       (input/output) REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.

  Z       (output) REAL array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LIWORK must be at least 3+5*N.


          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        n -
        d -
        e -
        z -
        ldz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sstevr

        public abstract void sstevr(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            float[] d,
                            float[] e,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            int[] isuppz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEVR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix T.  Eigenvalues and
  eigenvectors can be selected by specifying either a range of values
  or a range of indices for the desired eigenvalues.

  Whenever possible, SSTEVR calls SSTEMR to compute the
  eigenspectrum using Relatively Robust Representations.  SSTEMR
  computes eigenvalues by the dqds algorithm, while orthogonal
  eigenvectors are computed from various "good" L D L^T representations
  (also known as Relatively Robust Representations). Gram-Schmidt
  orthogonalization is avoided as far as possible. More specifically,
  the various steps of the algorithm are as follows. For the i-th
  unreduced block of T,
     (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
          is a relatively robust representation,
     (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
         relative accuracy by the dqds algorithm,
     (c) If there is a cluster of close eigenvalues, "choose" sigma_i
         close to the cluster, and go to step (a),
     (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
         compute the corresponding eigenvector by forming a
         rank-revealing twisted factorization.
  The desired accuracy of the output can be specified by the input
  parameter ABSTOL.

  For more details, see "A new O(n^2) algorithm for the symmetric
  tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
  Computer Science Division Technical Report No. UCB//CSD-97-971,
  UC Berkeley, May 1997.


  Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
  on machines which conform to the ieee-754 floating point standard.
  SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
  when partial spectrum requests are made.

  Normal execution of SSTEMR may create NaNs and infinities and
  hence may abort due to a floating point exception in environments
  which do not handle NaNs and infinities in the ieee standard default

  manner.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.
 For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
 SSTEIN are called

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, D may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  E       (input/output) REAL array, dimension (max(1,N-1))
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A in elements 1 to N-1 of E.
          On exit, E may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          SLAMCH( 'Safe minimum' ).  Doing so will guarantee that
          eigenvalues are computed to high relative accuracy when
          possible in future releases.  The current code does not
          make any guarantees about high relative accuracy, but
          future releases will. See J. Barlow and J. Demmel,
          "Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices", LAPACK Working Note #7, for a discussion

          of which matrices define their eigenvalues to high relative
          accuracy.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ).
 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal (and
          minimal) LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= 20*N.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal (and
          minimal) LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= 10*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  Internal error

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of
       California at Berkeley, USA
     Jason Riedy, Computer Science Division, University of
       California at Berkeley, USA

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • sstevx

        public abstract void sstevx(java.lang.String jobz,
                            java.lang.String range,
                            int n,
                            float[] d,
                            float[] e,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSTEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric tridiagonal matrix A.  Eigenvalues and
  eigenvectors can be selected by specifying either a range of values
  or a range of indices for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  N       (input) INTEGER
          The order of the matrix.  N >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, D may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  E       (input/output) REAL array, dimension (max(1,N-1))
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A in elements 1 to N-1 of E.
          On exit, E may be multiplied by a constant factor chosen
          to avoid over/underflow in computing the eigenvalues.

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less
          than or equal to zero, then  EPS*|T|  will be used in
          its place, where |T| is the 1-norm of the tridiagonal
          matrix.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge (INFO > 0), then that
          column of Z contains the latest approximation to the
          eigenvector, and the index of the eigenvector is returned
          in IFAIL.  If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace) REAL array, dimension (5*N)

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        range -
        n -
        d -
        e -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        iwork -
        ifail -
        info -

      • ssycon

        public abstract void ssycon(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float anorm,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYCON estimates the reciprocal of the condition number (in the
  1-norm) of a real symmetric matrix A using the factorization
  A = U*D*U**T or A = L*D*L**T computed by SSYTRF.

  An estimate is obtained for norm(inv(A)), and the reciprocal of the
  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by SSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSYTRF.

  ANORM   (input) REAL
          The 1-norm of the original matrix A.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

  WORK    (workspace) REAL array, dimension (2*N)

  IWORK    (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        anorm -
        rcond -
        work -
        iwork -
        info -

      • ssyev

        public abstract void ssyev(java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           float[] a,
                           int lda,
                           float[] w,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYEV computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          orthonormal eigenvectors of the matrix A.
          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

          or the upper triangle (if UPLO='U') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,3*N-1).
          For optimal efficiency, LWORK >= (NB+2)*N,
          where NB is the blocksize for SSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        a -
        lda -
        w -
        work -
        lwork -
        info -

      • ssyevd

        public abstract void ssyevd(java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] w,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYEVD computes all eigenvalues and, optionally, eigenvectors of a
  real symmetric matrix A. If eigenvectors are desired, it uses a
  divide and conquer algorithm.

  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Because of large use of BLAS of level 3, SSYEVD needs N**2 more
  workspace than SSYEVX.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          orthonormal eigenvectors of the matrix A.
          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

          or the upper triangle (if UPLO='U') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) REAL array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK must be at least 1.
          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
          If JOBZ = 'V' and N > 1, LWORK must be at least 
                                                1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If N <= 1,                LIWORK must be at least 1.
          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
                to converge; i off-diagonal elements of an intermediate
                tridiagonal form did not converge to zero;
                if INFO = i and JOBZ = 'V', then the algorithm failed
                to compute an eigenvalue while working on the submatrix
                lying in rows and columns INFO/(N+1) through
                mod(INFO,N+1).

  Further Details
  ===============

  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA
  Modified by Francoise Tisseur, University of Tennessee.

  Modified description of INFO. Sven, 16 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        jobz -
        uplo -
        n -
        a -
        lda -
        w -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssyevr

        public abstract void ssyevr(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            int[] isuppz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYEVR computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of
  indices for the desired eigenvalues.

  SSYEVR first reduces the matrix A to tridiagonal form T with a call
  to SSYTRD.  Then, whenever possible, SSYEVR calls SSTEMR to compute
  the eigenspectrum using Relatively Robust Representations.  SSTEMR
  computes eigenvalues by the dqds algorithm, while orthogonal
  eigenvectors are computed from various "good" L D L^T representations
  (also known as Relatively Robust Representations). Gram-Schmidt
  orthogonalization is avoided as far as possible. More specifically,
  the various steps of the algorithm are as follows.

  For each unreduced block (submatrix) of T,
     (a) Compute T - sigma I  = L D L^T, so that L and D
         define all the wanted eigenvalues to high relative accuracy.
         This means that small relative changes in the entries of D and
         cause only small relative changes in the eigenvalues and
         eigenvectors. The standard (unfactored) representation of the

         tridiagonal matrix T does not have this property in general.
     (b) Compute the eigenvalues to suitable accuracy.
         If the eigenvectors are desired, the algorithm attains full
         accuracy of the computed eigenvalues only right before
         the corresponding vectors have to be computed, see steps c) an
     (c) For each cluster of close eigenvalues, select a new
         shift close to the cluster, find a new factorization, and refi
         the shifted eigenvalues to suitable accuracy.
     (d) For each eigenvalue with a large enough relative separation co
         the corresponding eigenvector by forming a rank revealing twis
         factorization. Go back to (c) for any clusters that remain.

  The desired accuracy of the output can be specified by the input
  parameter ABSTOL.

  For more details, see SSTEMR's documentation and:
  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representat
    to compute orthogonal eigenvectors of symmetric tridiagonal matrice
    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors an
    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, V
    2004.  Also LAPACK Working Note 154.
  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
    tridiagonal eigenvalue/eigenvector problem",
    Computer Science Division Technical Report No. UCB/CSD-97-971,
    UC Berkeley, May 1997.


  Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested
  on machines which conform to the ieee-754 floating point standard.
  SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and
  when partial spectrum requests are made.

  Normal execution of SSTEMR may create NaNs and infinities and
  hence may abort due to a floating point exception in environments
  which do not handle NaNs and infinities in the ieee standard default

  manner.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.
 For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
 SSTEIN are called

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          SLAMCH( 'Safe minimum' ).  Doing so will guarantee that
          eigenvalues are computed to high relative accuracy when
          possible in future releases.  The current code does not
          make any guarantees about high relative accuracy, but
          future releases will. See J. Barlow and J. Demmel,
          "Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices", LAPACK Working Note #7, for a discussion

          of which matrices define their eigenvalues to high relative
          accuracy.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ).
 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,26*N).
          For optimal efficiency, LWORK >= (NB+6)*N,
          where NB is the max of the blocksize for SSYTRD and SORMTR
          returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  Internal error

  Further Details
  ===============

  Based on contributions by
     Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of
       California at Berkeley, USA
     Jason Riedy, Computer Science Division, University of
       California at Berkeley, USA

 =====================================================================


     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        isuppz -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssyevx

        public abstract void ssyevx(java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYEVX computes selected eigenvalues and, optionally, eigenvectors
  of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
  selected by specifying either a range of values or a range of indices
  for the desired eigenvalues.

  Arguments
  =========

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.
          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= 1, when N <= 1;
          otherwise 8*N.
          For optimal efficiency, LWORK >= (NB+3)*N,
          where NB is the max of the blocksize for SSYTRD and SORMTR
          returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.

 =====================================================================


     .. Parameters ..
        Parameters:
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        ifail -
        info -

      • ssygs2

        public abstract void ssygs2(int itype,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYGS2 reduces a real symmetric-definite generalized eigenproblem
  to standard form.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U')*A*inv(U) or inv(L)*A*inv(L')

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U` or L'*A*L.

  B must have been previously factorized as U'*U or L*L' by SPOTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U')*A*inv(U) or inv(L)*A*inv(L');
          = 2 or 3: compute U*A*U' or L'*A*L.

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored, and how B has been factorized.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) REAL array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by SPOTRF.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        info -

      • ssygst

        public abstract void ssygst(int itype,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYGST reduces a real symmetric-definite generalized eigenproblem
  to standard form.

  If ITYPE = 1, the problem is A*x = lambda*B*x,
  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

  B must have been previously factorized as U**T*U or L*L**T by SPOTRF.

  Arguments
  =========

  ITYPE   (input) INTEGER
          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
          = 2 or 3: compute U*A*U**T or L**T*A*L.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**T*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**T.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) REAL array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by SPOTRF.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        info -

      • ssygv

        public abstract void ssygv(int itype,
                           java.lang.String jobz,
                           java.lang.String uplo,
                           int n,
                           float[] a,
                           int lda,
                           float[] b,
                           int ldb,
                           float[] w,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYGV computes all the eigenvalues, and optionally, the eigenvectors

  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
  Here A and B are assumed to be symmetric and B is also
  positive definite.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          matrix Z of eigenvectors.  The eigenvectors are normalized
          as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')

          or the lower triangle (if UPLO='L') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the symmetric positive definite matrix B.
          If UPLO = 'U', the leading N-by-N upper triangular part of B

          contains the upper triangular part of the matrix B.
          If UPLO = 'L', the leading N-by-N lower triangular part of B

          contains the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,3*N-1).
          For optimal efficiency, LWORK >= (NB+2)*N,
          where NB is the blocksize for SSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPOTRF or SSYEV returned an error code:
             <= N:  if INFO = i, SSYEV failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        w -
        work -
        lwork -
        info -

      • ssygvd

        public abstract void ssygvd(int itype,
                            java.lang.String jobz,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] w,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYGVD computes all the eigenvalues, and optionally, the eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
  B are assumed to be symmetric and B is also positive definite.
  If eigenvectors are desired, it uses a divide and conquer algorithm.


  The divide and conquer algorithm makes very mild assumptions about
  floating point arithmetic. It will work on machines with a guard
  digit in add/subtract, or on those binary machines without guard
  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
  Cray-2. It could conceivably fail on hexadecimal or decimal machines

  without guard digits, but we know of none.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
          matrix Z of eigenvectors.  The eigenvectors are normalized
          as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')

          or the lower triangle (if UPLO='L') of A, including the
          diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the symmetric matrix B.  If UPLO = 'U', the
          leading N-by-N upper triangular part of B contains the
          upper triangular part of the matrix B.  If UPLO = 'L',
          the leading N-by-N lower triangular part of B contains
          the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  W       (output) REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If N <= 1,                LIWORK >= 1.
          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPOTRF or SSYEVD returned an error code:
             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
                    failed to converge; i off-diagonal elements of an
                    intermediate tridiagonal form did not converge to
                    zero;
                    if INFO = i and JOBZ = 'V', then the algorithm
                    failed to compute an eigenvalue while working on
                    the submatrix lying in rows and columns INFO/(N+1)

                    through mod(INFO,N+1);
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

  Modified so that no backsubstitution is performed if SSYEVD fails to

  converge (NEIG in old code could be greater than N causing out of
  bounds reference to A - reported by Ralf Meyer).  Also corrected the

  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
  =====================================================================

     .. Parameters ..
        Parameters:
        itype -
        jobz -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        w -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • ssygvx

        public abstract void ssygvx(int itype,
                            java.lang.String jobz,
                            java.lang.String range,
                            java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float vl,
                            float vu,
                            int il,
                            int iu,
                            float abstol,
                            org.netlib.util.intW m,
                            float[] w,
                            float[] z,
                            int ldz,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int[] ifail,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYGVX computes selected eigenvalues, and optionally, eigenvectors
  of a real generalized symmetric-definite eigenproblem, of the form
  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
  and B are assumed to be symmetric and B is also positive definite.
  Eigenvalues and eigenvectors can be selected by specifying either a
  range of values or a range of indices for the desired eigenvalues.

  Arguments
  =========

  ITYPE   (input) INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x

  JOBZ    (input) CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

  RANGE   (input) CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A and B are stored;
          = 'L':  Lower triangle of A and B are stored.

  N       (input) INTEGER
          The order of the matrix pencil (A,B).  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains the
          upper triangular part of the matrix A.  If UPLO = 'L',
          the leading N-by-N lower triangular part of A contains
          the lower triangular part of the matrix A.

          On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is
          destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDA, N)
          On entry, the symmetric matrix B.  If UPLO = 'U', the
          leading N-by-N upper triangular part of B contains the
          upper triangular part of the matrix B.  If UPLO = 'L',
          the leading N-by-N lower triangular part of B contains
          the lower triangular part of the matrix B.

          On exit, if INFO <= N, the part of B containing the matrix is
          overwritten by the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  VL      (input) REAL
  VU      (input) REAL
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

  IL      (input) INTEGER
  IU      (input) INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

  ABSTOL  (input) REAL
          The absolute error tolerance for the eigenvalues.
          An approximate eigenvalue is accepted as converged
          when it is determined to lie in an interval [a,b]
          of width less than or equal to

                  ABSTOL + EPS *   max( |a|,|b| ) ,

          where EPS is the machine precision.  If ABSTOL is less than
          or equal to zero, then  EPS*|T|  will be used in its place,
          where |T| is the 1-norm of the tridiagonal matrix obtained
          by reducing A to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*SLAMCH('S').

  M       (output) INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

  W       (output) REAL array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

  Z       (output) REAL array, dimension (LDZ, max(1,M))
          If JOBZ = 'N', then Z is not referenced.
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.

          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,8*N).
          For optimal efficiency, LWORK >= (NB+3)*N,
          where NB is the blocksize for SSYTRD returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (5*N)

  IFAIL   (output) INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPOTRF or SSYEVX returned an error code:
             <= N:  if INFO = i, SSYEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading

                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

  Further Details
  ===============

  Based on contributions by
     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 =====================================================================


     .. Parameters ..
        Parameters:
        itype -
        jobz -
        range -
        uplo -
        n -
        a -
        lda -
        b -
        ldb -
        vl -
        vu -
        il -
        iu -
        abstol -
        m -
        w -
        z -
        ldz -
        work -
        lwork -
        iwork -
        ifail -
        info -

      • ssyrfs

        public abstract void ssyrfs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYRFS improves the computed solution to a system of linear
  equations when the coefficient matrix is symmetric indefinite, and
  provides error bounds and backward error estimates for the solution.


  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input) REAL array, dimension (LDAF,N)
          The factored form of the matrix A.  AF contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by SSYTRF.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSYTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input/output) REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SSYTRS.
          On exit, the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Internal Parameters
  ===================

  ITMAX is the maximum number of steps of iterative refinement.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • ssysv

        public abstract void ssysv(java.lang.String uplo,
                           int n,
                           int nrhs,
                           float[] a,
                           int lda,
                           int[] ipiv,
                           float[] b,
                           int ldb,
                           float[] work,
                           int lwork,
                           org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYSV computes the solution to a real system of linear equations
     A * X = B,
  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
  matrices.

  The diagonal pivoting method is used to factor A as
     A = U * D * U**T,  if UPLO = 'U', or
     A = L * D * L**T,  if UPLO = 'L',
  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with 
  1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
  used to solve the system of equations A * X = B.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, if INFO = 0, the block diagonal matrix D and the
          multipliers used to obtain the factor U or L from the
          factorization A = U*D*U**T or A = L*D*L**T as computed by
          SSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D, as

          determined by SSYTRF.  If IPIV(k) > 0, then rows and columns

          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
          then rows and columns k-1 and -IPIV(k) were interchanged and

          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
          diagonal block.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >= 1, and for best performance
          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
          SSYTRF.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, so the solution could not be computed.

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        work -
        lwork -
        info -

      • ssysvx

        public abstract void ssysvx(java.lang.String fact,
                            java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] af,
                            int ldaf,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            org.netlib.util.floatW rcond,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYSVX uses the diagonal pivoting factorization to compute the
  solution to a real system of linear equations A * X = B,
  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
  matrices.

  Error bounds on the solution and a condition estimate are also
  provided.

  Description
  ===========

  The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A.
     The form of the factorization is
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)

     triangular matrices, and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

  2. If some D(i,i)=0, so that D is exactly singular, then the routine

     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.  If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  Arguments
  =========

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, AF and IPIV contain the factored form of
                  A.  AF and IPIV will not be modified.
          = 'N':  The matrix A will be copied to AF and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A

          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is

          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  AF      (input or output) REAL array, dimension (LDAF,N)
          If FACT = 'F', then AF is an input argument and on entry
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.

          If FACT = 'N', then AF is an output argument and on exit
          returns the block diagonal matrix D and the multipliers used

          to obtain the factor U or L from the factorization
          A = U*D*U**T or A = L*D*L**T.

  LDAF    (input) INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains details of the interchanges and the block structure

          of D, as determined by SSYTRF.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains details of the interchanges and the block structure

          of D, as determined by SSYTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working

          precision.  This condition is indicated by a return code of
          INFO > 0.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >= max(1,3*N), and for best
          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
          NB is the optimal blocksize for SSYTRF.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, and i is
                <= N:  D(i,i) is exactly zero.  The factorization
                       has been completed but the factor D is exactly
                       singular, so the solution and error bounds could
                       not be computed. RCOND = 0 is returned.
                = N+1: D is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the

                       value of RCOND would suggest.

  =====================================================================

     .. Parameters ..
        Parameters:
        fact -
        uplo -
        n -
        nrhs -
        a -
        lda -
        af -
        ldaf -
        ipiv -
        b -
        ldb -
        x -
        ldx -
        rcond -
        ferr -
        berr -
        work -
        lwork -
        iwork -
        info -

      • ssytd2

        public abstract void ssytd2(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] d,
                            float[] e,
                            float[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
  form T by an orthogonal similarity transformation: Q' * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  D       (output) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),

  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi

  denotes an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        d -
        e -
        tau -
        info -

      • ssytf2

        public abstract void ssytf2(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTF2 computes the factorization of a real symmetric matrix A using

  the Bunch-Kaufman diagonal pivoting method:

     A = U*D*U'  or  A = L*D*L'

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, U' is the transpose of U, and D is symmetric and
  block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  This is the unblocked version of the algorithm, calling Level 2 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details).


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization

               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it

               is used to solve a system of equations.

  Further Details
  ===============

  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l.204 and l.372
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J. Lewis, Boeing Computer Services Company
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        info -

      • ssytrd

        public abstract void ssytrd(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            float[] d,
                            float[] e,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTRD reduces a real symmetric matrix A to real symmetric
  tridiagonal form T by an orthogonal similarity transformation:
  Q**T * A * Q = T.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal

          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with

          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  D       (output) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

  E       (output) REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

  TAU     (output) REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further

          Details).

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= 1.
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-1).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),

  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi

  denotes an element of the vector defining H(i).

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        d -
        e -
        tau -
        work -
        lwork -
        info -

      • ssytrf

        public abstract void ssytrf(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTRF computes the factorization of a real symmetric matrix A using

  the Bunch-Kaufman diagonal pivoting method.  The form of the
  factorization is

     A = U*D*U**T  or  A = L*D*L**T

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with 
  1-by-1 and 2-by-2 diagonal blocks.

  This is the blocked version of the algorithm, calling Level 3 BLAS.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower

          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details).


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is

                exactly singular, and division by zero will occur if it
                is used to solve a system of equations.

  Further Details
  ===============

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to

  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

  =====================================================================

     .. Local Scalars ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        work -
        lwork -
        info -

      • ssytri

        public abstract void ssytri(java.lang.String uplo,
                            int n,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTRI computes the inverse of a real symmetric indefinite matrix
  A using the factorization A = U*D*U**T or A = L*D*L**T computed by
  SSYTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by SSYTRF.

          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix.  If UPLO = 'U', the upper triangular part of the
          inverse is formed and the part of A below the diagonal is not
          referenced; if UPLO = 'L' the lower triangular part of the
          inverse is formed and the part of A above the diagonal is
          not referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSYTRF.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

               inverse could not be computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        a -
        lda -
        ipiv -
        work -
        info -

      • ssytrs

        public abstract void ssytrs(java.lang.String uplo,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            int[] ipiv,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  SSYTRS solves a system of linear equations A*X = B with a real
  symmetric matrix A using the factorization A = U*D*U**T or
  A = L*D*L**T computed by SSYTRF.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by SSYTRF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  IPIV    (input) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by SSYTRF.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        n -
        nrhs -
        a -
        lda -
        ipiv -
        b -
        ldb -
        info -

      • stbcon

        public abstract void stbcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            int kd,
                            float[] ab,
                            int ldab,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STBCON estimates the reciprocal of the condition number of a
  triangular band matrix A, in either the 1-norm or the infinity-norm.


  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        kd -
        ab -
        ldab -
        rcond -
        work -
        iwork -
        info -

      • stbrfs

        public abstract void stbrfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int kd,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STBRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular band
  coefficient matrix.

  The solution matrix X must be computed by STBTRS or some other
  means before entering this routine.  STBRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored

          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) REAL array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • stbtrs

        public abstract void stbtrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int kd,
                            int nrhs,
                            float[] ab,
                            int ldab,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STBTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular band matrix of order N, and B is an
  N-by NRHS matrix.  A check is made to verify that A is nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        kd -
        nrhs -
        ab -
        ldab -
        b -
        ldb -
        info -

      • stgevc

        public abstract void stgevc(java.lang.String side,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            float[] s,
                            int lds,
                            float[] p,
                            int ldp,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..


  Purpose
  =======

  STGEVC computes some or all of the right and/or left eigenvectors of

  a pair of real matrices (S,P), where S is a quasi-triangular matrix
  and P is upper triangular.  Matrix pairs of this type are produced by
  the generalized Schur factorization of a matrix pair (A,B):

     A = Q*S*Z**T,  B = Q*P*Z**T

  as computed by SGGHRD + SHGEQZ.

  The right eigenvector x and the left eigenvector y of (S,P)
  corresponding to an eigenvalue w are defined by:
  
     S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
  
  where y**H denotes the conjugate tranpose of y.
  The eigenvalues are not input to this routine, but are computed
  directly from the diagonal blocks of S and P.
  
  This routine returns the matrices X and/or Y of right and left
  eigenvectors of (S,P), or the products Z*X and/or Q*Y,
  where Z and Q are input matrices.
  If Q and Z are the orthogonal factors from the generalized Schur
  factorization of a matrix pair (A,B), then Z*X and Q*Y
  are the matrices of right and left eigenvectors of (A,B).
 
  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

  HOWMNY  (input) CHARACTER*1
          = 'A': compute all right and/or left eigenvectors;
          = 'B': compute all right and/or left eigenvectors,
                 backtransformed by the matrices in VR and/or VL;
          = 'S': compute selected right and/or left eigenvectors,
                 specified by the logical array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY='S', SELECT specifies the eigenvectors to be
          computed.  If w(j) is a real eigenvalue, the corresponding
          real eigenvector is computed if SELECT(j) is .TRUE..
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector
          is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
          and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
          set to .FALSE..
          Not referenced if HOWMNY = 'A' or 'B'.

  N       (input) INTEGER
          The order of the matrices S and P.  N >= 0.

  S       (input) REAL array, dimension (LDS,N)
          The upper quasi-triangular matrix S from a generalized Schur

          factorization, as computed by SHGEQZ.

  LDS     (input) INTEGER
          The leading dimension of array S.  LDS >= max(1,N).

  P       (input) REAL array, dimension (LDP,N)
          The upper triangular matrix P from a generalized Schur
          factorization, as computed by SHGEQZ.
          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
          of S must be in positive diagonal form.

  LDP     (input) INTEGER
          The leading dimension of array P.  LDP >= max(1,N).

  VL      (input/output) REAL array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of left Schur vectors returned by SHGEQZ).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);

          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by

                      SELECT, stored consecutively in the columns of
                      VL, in the same order as their eigenvalues.

          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part.

          Not referenced if SIDE = 'R'.

  LDVL    (input) INTEGER
          The leading dimension of array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.

  VR      (input/output) REAL array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Z (usually the orthogonal matrix Z
          of right Schur vectors returned by SHGEQZ).

          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
          if HOWMNY = 'B' or 'b', the matrix Z*X;
          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                      specified by SELECT, stored consecutively in the

                      columns of VR, in the same order as their
                      eigenvalues.

          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part.
          
          Not referenced if SIDE = 'L'.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B', LDVR >= N.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
          is set to N.  Each selected real eigenvector occupies one
          column and each selected complex eigenvector occupies two
          columns.

  WORK    (workspace) REAL array, dimension (6*N)

  INFO    (output) INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex

                eigenvalue.

  Further Details
  ===============

  Allocation of workspace:
  ---------- -- ---------

     WORK( j ) = 1-norm of j-th column of A, above the diagonal
     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
     WORK( 2*N+1:3*N ) = real part of eigenvector
     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

  Rowwise vs. columnwise solution methods:
  ------- --  ---------- -------- -------

  Finding a generalized eigenvector consists basically of solving the
  singular triangular system

   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

  Consider finding the i-th right eigenvector (assume all eigenvalues
  are real). The equation to be solved is:
       n                   i
  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
      k=j                 k=j

  where  C = (A - w B)  (The components v(i+1:n) are 0.)

  The "rowwise" method is:

  (1)  v(i) := 1
  for j = i-1,. . .,1:
                          i
      (2) compute  s = - sum C(j,k) v(k)   and
                        k=j+1

      (3) v(j) := s / C(j,j)

  Step 2 is sometimes called the "dot product" step, since it is an
  inner product between the j-th row and the portion of the eigenvector
  that has been computed so far.

  The "columnwise" method consists basically in doing the sums
  for all the rows in parallel.  As each v(j) is computed, the
  contribution of v(j) times the j-th column of C is added to the
  partial sums.  Since FORTRAN arrays are stored columnwise, this has
  the advantage that at each step, the elements of C that are accessed

  are adjacent to one another, whereas with the rowwise method, the
  elements accessed at a step are spaced LDS (and LDP) words apart.

  When finding left eigenvectors, the matrix in question is the
  transpose of the one in storage, so the rowwise method then
  actually accesses columns of A and B at each step, and so is the
  preferred method.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        howmny -
        select -
        n -
        s -
        lds -
        p -
        ldp -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        info -

      • stgex2

        public abstract void stgex2(boolean wantq,
                            boolean wantz,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] q,
                            int ldq,
                            float[] z,
                            int ldz,
                            int j1,
                            int n1,
                            int n2,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
  of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
  (A, B) by an orthogonal equivalence transformation.

  (A, B) must be in generalized real Schur canonical form (as returned

  by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
  diagonal blocks. B is upper triangular.

  Optionally, the matrices Q and Z of generalized Schur vectors are
  updated.

         Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
         Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'


  Arguments
  =========

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A      (input/output) REAL arrays, dimensions (LDA,N)
          On entry, the matrix A in the pair (A, B).
          On exit, the updated matrix A.

  LDA     (input)  INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B      (input/output) REAL arrays, dimensions (LDB,N)
          On entry, the matrix B in the pair (A, B).
          On exit, the updated matrix B.

  LDB     (input)  INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  Q       (input/output) REAL array, dimension (LDZ,N)
          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
          On exit, the updated matrix Q.
          Not referenced if WANTQ = .FALSE..

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) REAL array, dimension (LDZ,N)
          On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
          On exit, the updated matrix Z.
          Not referenced if WANTZ = .FALSE..

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.

  J1      (input) INTEGER
          The index to the first block (A11, B11). 1 <= J1 <= N.

  N1      (input) INTEGER
          The order of the first block (A11, B11). N1 = 0, 1 or 2.

  N2      (input) INTEGER
          The order of the second block (A22, B22). N2 = 0, 1 or 2.

  WORK    (workspace) REAL array, dimension (MAX(1,LWORK)).

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >=  MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )

  INFO    (output) INTEGER
            =0: Successful exit
            >0: If INFO = 1, the transformed matrix (A, B) would be
                too far from generalized Schur form; the blocks are
                not swapped and (A, B) and (Q, Z) are unchanged.
                The problem of swapping is too ill-conditioned.
            <0: If INFO = -16: LWORK is too small. Appropriate value
                for LWORK is returned in WORK(1).

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  In the current code both weak and strong stability tests are
  performed. The user can omit the strong stability test by changing
  the internal logical parameter WANDS to .FALSE.. See ref. [2] for
  details.

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  =====================================================================
  Replaced various illegal calls to SCOPY by calls to SLASET, or by DO

  loops. Sven Hammarling, 1/5/02.

     .. Parameters ..
        Parameters:
        wantq -
        wantz -
        n -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        j1 -
        n1 -
        n2 -
        work -
        lwork -
        info -

      • stgexc

        public abstract void stgexc(boolean wantq,
                            boolean wantz,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] q,
                            int ldq,
                            float[] z,
                            int ldz,
                            org.netlib.util.intW ifst,
                            org.netlib.util.intW ilst,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGEXC reorders the generalized real Schur decomposition of a real
  matrix pair (A,B) using an orthogonal equivalence transformation

                 (A, B) = Q * (A, B) * Z',

  so that the diagonal block of (A, B) with row index IFST is moved
  to row ILST.

  (A, B) must be in generalized real Schur canonical form (as returned

  by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
  diagonal blocks. B is upper triangular.

  Optionally, the matrices Q and Z of generalized Schur vectors are
  updated.

         Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
         Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'


  Arguments
  =========

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the matrix A in generalized real Schur canonical
          form.
          On exit, the updated matrix A, again in generalized
          real Schur canonical form.

  LDA     (input)  INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the matrix B in generalized real Schur canonical
          form (A,B).
          On exit, the updated matrix B, again in generalized
          real Schur canonical form (A,B).

  LDB     (input)  INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  Q       (input/output) REAL array, dimension (LDZ,N)
          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
          On exit, the updated matrix Q.
          If WANTQ = .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) REAL array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
          On exit, the updated matrix Z.
          If WANTZ = .FALSE., Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.

  IFST    (input/output) INTEGER
  ILST    (input/output) INTEGER
          Specify the reordering of the diagonal blocks of (A, B).
          The block with row index IFST is moved to row ILST, by a
          sequence of swapping between adjacent blocks.
          On exit, if IFST pointed on entry to the second row of
          a 2-by-2 block, it is changed to point to the first row;
          ILST always points to the first row of the block in its
          final position (which may differ from its input value by
          +1 or -1). 1 <= IFST, ILST <= N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
           =0:  successful exit.
           <0:  if INFO = -i, the i-th argument had an illegal value.
           =1:  The transformed matrix pair (A, B) would be too far
                from generalized Schur form; the problem is ill-
                conditioned. (A, B) may have been partially reordered,

                and ILST points to the first row of the current
                position of the block being moved.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  =====================================================================

     .. Parameters ..
        Parameters:
        wantq -
        wantz -
        n -
        a -
        lda -
        b -
        ldb -
        q -
        ldq -
        z -
        ldz -
        ifst -
        ilst -
        work -
        lwork -
        info -

      • stgsen

        public abstract void stgsen(int ijob,
                            boolean wantq,
                            boolean wantz,
                            boolean[] select,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] alphar,
                            float[] alphai,
                            float[] beta,
                            float[] q,
                            int ldq,
                            float[] z,
                            int ldz,
                            org.netlib.util.intW m,
                            org.netlib.util.floatW pl,
                            org.netlib.util.floatW pr,
                            float[] dif,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGSEN reorders the generalized real Schur decomposition of a real
  matrix pair (A, B) (in terms of an orthonormal equivalence trans-
  formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
  appears in the leading diagonal blocks of the upper quasi-triangular

  matrix A and the upper triangular B. The leading columns of Q and
  Z form orthonormal bases of the corresponding left and right eigen-
  spaces (deflating subspaces). (A, B) must be in generalized real
  Schur canonical form (as returned by SGGES), i.e. A is block upper
  triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
  triangular.

  STGSEN also computes the generalized eigenvalues

              w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

  of the reordered matrix pair (A, B).

  Optionally, STGSEN computes the estimates of reciprocal condition
  numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
  (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
  between the matrix pairs (A11, B11) and (A22,B22) that correspond to

  the selected cluster and the eigenvalues outside the cluster, resp.,

  and norms of "projections" onto left and right eigenspaces w.r.t.
  the selected cluster in the (1,1)-block.

  Arguments
  =========

  IJOB    (input) INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

  WANTQ   (input) LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

  WANTZ   (input) LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

  SELECT  (input) LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster.
          To select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.

  N       (input) INTEGER
          The order of the matrices A and B. N >= 0.

  A       (input/output) REAL array, dimension(LDA,N)
          On entry, the upper quasi-triangular matrix A, with (A, B) in
          generalized real Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input/output) REAL array, dimension(LDB,N)
          On entry, the upper triangular matrix B, with (A, B) in
          generalized real Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
  ALPHAI  (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real generalized Schur form of (A,B) were further reduced
          to triangular form using complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= 1;
          and if WANTQ = .TRUE., LDQ >= N.

  Z       (input/output) REAL array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.

  LDZ     (input) INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.

  M       (output) INTEGER
          The dimension of the specified pair of left and right eigen-

          spaces (deflating subspaces). 0 <= M <= N.

  PL      (output) REAL
  PR      (output) REAL
          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal of the norm of "projections" onto left and right
          eigenspaces with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3, PL and PR are not referenced.

  DIF     (output) REAL array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.

          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >=  4*N+16.
          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
          IF IJOB = 0, IWORK is not referenced.  Otherwise,
          on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.

  Further Details
  ===============

  STGSEN first collects the selected eigenvalues by computing
  orthogonal U and W that move them to the top left corner of (A, B).
  In other words, the selected eigenvalues are the eigenvalues of
  (A11, B11) in:

                U'*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

  where N = n1+n2 and U' means the transpose of U. The first n1 columns
  of U and W span the specified pair of left and right eigenspaces
  (deflating subspaces) of (A, B).

  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
  reordered generalized real Schur form of (C, D) is given by

           (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',

  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.

  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.

  The Difu and Difl are defined as:

       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix

       Zu = [ kron(In2, A11)  -kron(A22', In1) ]
            [ kron(In2, B11)  -kron(B22', In1) ].

  Here, Inx is the identity matrix of size nx and A22' is the
  transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.

  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is

       EPS * norm((A, B)) / DIF(2),

  where EPS is the machine precision.

  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.

  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where

       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2

  and (L, R) is the solution to the generalized Sylvester equation

       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12

  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / PL.

  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is

   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).


  An approximate bound on x can be computed from DIF(1:2), PL and PR.

  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as

   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))

   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))


  See LAPACK User's Guide section 4.11 or the following references
  for more information.

  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see SLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
  (IJOB = 2 will be used)). See STGSYL for more details.

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  References
  ==========

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
      1996.

  =====================================================================

     .. Parameters ..
        Parameters:
        ijob -
        wantq -
        wantz -
        select -
        n -
        a -
        lda -
        b -
        ldb -
        alphar -
        alphai -
        beta -
        q -
        ldq -
        z -
        ldz -
        m -
        pl -
        pr -
        dif -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • stgsja

        public abstract void stgsja(java.lang.String jobu,
                            java.lang.String jobv,
                            java.lang.String jobq,
                            int m,
                            int p,
                            int n,
                            int k,
                            int l,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float tola,
                            float tolb,
                            float[] alpha,
                            float[] beta,
                            float[] u,
                            int ldu,
                            float[] v,
                            int ldv,
                            float[] q,
                            int ldq,
                            float[] work,
                            org.netlib.util.intW ncycle,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGSJA computes the generalized singular value decomposition (GSVD)
  of two real upper triangular (or trapezoidal) matrices A and B.

  On entry, it is assumed that matrices A and B have the following
  forms, which may be obtained by the preprocessing subroutine SGGSVP
  from a general M-by-N matrix A and P-by-N matrix B:

               N-K-L  K    L
     A =    K ( 0    A12  A13 ) if M-K-L >= 0;
            L ( 0     0   A23 )
        M-K-L ( 0     0    0  )

             N-K-L  K    L
     A =  K ( 0    A12  A13 ) if M-K-L < 0;
        M-K ( 0     0   A23 )

             N-K-L  K    L
     B =  L ( 0     0   B13 )
        P-L ( 0     0    0  )

  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
  otherwise A23 is (M-K)-by-L upper trapezoidal.

  On exit,

              U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),

  where U, V and Q are orthogonal matrices, Z' denotes the transpose
  of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
  ``diagonal'' matrices, which are of the following structures:

  If M-K-L >= 0,

                      K  L
         D1 =     K ( I  0 )
                  L ( 0  C )
              M-K-L ( 0  0 )

                    K  L
         D2 = L   ( 0  S )
              P-L ( 0  0 )

                 N-K-L  K    L
    ( 0 R ) = K (  0   R11  R12 ) K
              L (  0    0   R22 ) L

  where

    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
    S = diag( BETA(K+1),  ... , BETA(K+L) ),
    C**2 + S**2 = I.

    R is stored in A(1:K+L,N-K-L+1:N) on exit.

  If M-K-L < 0,

                 K M-K K+L-M
      D1 =   K ( I  0    0   )
           M-K ( 0  C    0   )

                   K M-K K+L-M
      D2 =   M-K ( 0  S    0   )
           K+L-M ( 0  0    I   )
             P-L ( 0  0    0   )

                 N-K-L  K   M-K  K+L-M
 ( 0 R ) =    K ( 0    R11  R12  R13  )
            M-K ( 0     0   R22  R23  )
          K+L-M ( 0     0    0   R33  )

  where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.

  R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored

      (  0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.

  The computation of the orthogonal transformation matrices U, V or Q
  is optional.  These matrices may either be formed explicitly, or they
  may be postmultiplied into input matrices U1, V1, or Q1.

  Arguments
  =========

  JOBU    (input) CHARACTER*1
          = 'U':  U must contain an orthogonal matrix U1 on entry, and

                  the product U1*U is returned;
          = 'I':  U is initialized to the unit matrix, and the
                  orthogonal matrix U is returned;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  V must contain an orthogonal matrix V1 on entry, and

                  the product V1*V is returned;
          = 'I':  V is initialized to the unit matrix, and the
                  orthogonal matrix V is returned;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and

                  the product Q1*Q is returned;
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal matrix Q is returned;
          = 'N':  Q is not computed.

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix B.  P >= 0.

  N       (input) INTEGER
          The number of columns of the matrices A and B.  N >= 0.

  K       (input) INTEGER
  L       (input) INTEGER
          K and L specify the subblocks in the input matrices A and B:

          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
          of A and B, whose GSVD is going to be computed by STGSJA.
          See Further details.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
          matrix R or part of R.  See Purpose for details.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input/output) REAL array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
          a part of R.  See Purpose for details.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

  TOLA    (input) REAL
  TOLB    (input) REAL
          TOLA and TOLB are the convergence criteria for the Jacobi-
          Kogbetliantz iteration procedure. Generally, they are the
          same as used in the preprocessing step, say
              TOLA = max(M,N)*norm(A)*MACHEPS,
              TOLB = max(P,N)*norm(B)*MACHEPS.

  ALPHA   (output) REAL array, dimension (N)
  BETA    (output) REAL array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = diag(C),
            BETA(K+1:K+L)  = diag(S),
          or if M-K-L < 0,
            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
          Furthermore, if K+L < N,
            ALPHA(K+L+1:N) = 0 and
            BETA(K+L+1:N)  = 0.

  U       (input/output) REAL array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually

          the orthogonal matrix returned by SGGSVP).
          On exit,
          if JOBU = 'I', U contains the orthogonal matrix U;
          if JOBU = 'U', U contains the product U1*U.
          If JOBU = 'N', U is not referenced.

  LDU     (input) INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

  V       (input/output) REAL array, dimension (LDV,P)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually

          the orthogonal matrix returned by SGGSVP).
          On exit,
          if JOBV = 'I', V contains the orthogonal matrix V;
          if JOBV = 'V', V contains the product V1*V.
          If JOBV = 'N', V is not referenced.

  LDV     (input) INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually

          the orthogonal matrix returned by SGGSVP).
          On exit,
          if JOBQ = 'I', Q contains the orthogonal matrix Q;
          if JOBQ = 'Q', Q contains the product Q1*Q.
          If JOBQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

  WORK    (workspace) REAL array, dimension (2*N)

  NCYCLE  (output) INTEGER
          The number of cycles required for convergence.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the procedure does not converge after MAXIT cycles.

  Internal Parameters
  ===================

  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure

          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1.

  Further Details
  ===============

  STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
  matrix B13 to the form:

           U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

  where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
  of Z.  C1 and S1 are diagonal matrices satisfying

                C1**2 + S1**2 = I,

  and R1 is an L-by-L nonsingular upper triangular matrix.

  =====================================================================

     .. Parameters ..
        Parameters:
        jobu -
        jobv -
        jobq -
        m -
        p -
        n -
        k -
        l -
        a -
        lda -
        b -
        ldb -
        tola -
        tolb -
        alpha -
        beta -
        u -
        ldu -
        v -
        ldv -
        q -
        ldq -
        work -
        ncycle -
        info -

      • stgsna

        public abstract void stgsna(java.lang.String job,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            float[] s,
                            float[] dif,
                            int mm,
                            org.netlib.util.intW m,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGSNA estimates reciprocal condition numbers for specified
  eigenvalues and/or eigenvectors of a matrix pair (A, B) in
  generalized real Schur canonical form (or of any matrix pair
  (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where
  Z' denotes the transpose of Z.

  (A, B) must be in generalized real Schur form (as returned by SGGES),
  i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
  blocks. B is upper triangular.


  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (DIF):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (DIF);
          = 'B': for both eigenvalues and eigenvectors (S and DIF).

  HOWMNY  (input) CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the eigenpair corresponding to a real eigenvalue w(j),
          SELECT(j) must be set to .TRUE.. To select condition numbers

          corresponding to a complex conjugate pair of eigenvalues w(j)
          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be

          set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

  N       (input) INTEGER
          The order of the square matrix pair (A, B). N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The upper quasi-triangular matrix A in the pair (A,B).

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

  B       (input) REAL array, dimension (LDB,N)
          The upper triangular matrix B in the pair (A,B).

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  VL      (input) REAL array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns of VL, as returned by STGEVC.
          If JOB = 'V', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL. LDVL >= 1.
          If JOB = 'E' or 'B', LDVL >= N.

  VR      (input) REAL array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns ov VR, as returned by STGEVC.
          If JOB = 'V', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR. LDVR >= 1.
          If JOB = 'E' or 'B', LDVR >= N.

  S       (output) REAL array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the

          selected eigenvalues, stored in consecutive elements of the
          array. For a complex conjugate pair of eigenvalues two
          consecutive elements of S are set to the same value. Thus
          S(j), DIF(j), and the j-th columns of VL and VR all
          correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.

  DIF     (output) REAL array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array. For a complex eigenvector two
          consecutive elements of DIF are set to the same value. If
          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
          is set to 0; this can only occur when the true value would be
          very small anyway.
          If JOB = 'E', DIF is not referenced.

  MM      (input) INTEGER
          The number of elements in the arrays S and DIF. MM >= M.

  M       (output) INTEGER
          The number of elements of the arrays S and DIF used to store

          the specified condition numbers; for each selected real
          eigenvalue one element is used, and for each selected complex
          conjugate pair of eigenvalues, two elements are used.
          If HOWMNY = 'A', M is set to N.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (N + 6)
          If JOB = 'E', IWORK is not referenced.

  INFO    (output) INTEGER
          =0: Successful exit
          <0: If INFO = -i, the i-th argument had an illegal value


  Further Details
  ===============

  The reciprocal of the condition number of a generalized eigenvalue
  w = (a, b) is defined as

       S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))

  where u and v are the left and right eigenvectors of (A, B)
  corresponding to w; |z| denotes the absolute value of the complex
  number, and norm(u) denotes the 2-norm of the vector u.
  The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)
  of the matrix pair (A, B). If both a and b equal zero, then (A B) is

  singular and S(I) = -1 is returned.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(A, B) / S(I)

  where EPS is the machine precision.

  The reciprocal of the condition number DIF(i) of right eigenvector u

  and left eigenvector v corresponding to the generalized eigenvalue w

  is defined as follows:

  a) If the i-th eigenvalue w = (a,b) is real

     Suppose U and V are orthogonal transformations such that

                U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                                        ( 0  S22 ),( 0 T22 )  n-1
                                          1  n-1     1 n-1

     Then the reciprocal condition number DIF(i) is

                Difl((a, b), (S22, T22)) = sigma-min( Zl ),

     where sigma-min(Zl) denotes the smallest singular value of the
     2(n-1)-by-2(n-1) matrix

         Zl = [ kron(a, In-1)  -kron(1, S22) ]
              [ kron(b, In-1)  -kron(1, T22) ] .

     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
     Kronecker product between the matrices X and Y.

     Note that if the default method for computing DIF(i) is wanted
     (see SLATDF), then the parameter DIFDRI (see below) should be
     changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
     See STGSYL for more details.

  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

     Suppose U and V are orthogonal transformations such that

                U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                                       ( 0    S22 ),( 0    T22) n-2
                                         2    n-2     2    n-2

     and (S11, T11) corresponds to the complex conjugate eigenvalue
     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
     that

         U1'*S11*V1 = ( s11 s12 )   and U1'*T11*V1 = ( t11 t12 )
                      (  0  s22 )                    (  0  t22 )

     where the generalized eigenvalues w = s11/t11 and
     conjg(w) = s22/t22.

     Then the reciprocal condition number DIF(i) is bounded by

         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
     Z1 is the complex 2-by-2 matrix

              Z1 =  [ s11  -s22 ]
                    [ t11  -t22 ],

     This is done by computing (using real arithmetic) the
     roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
     where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
     the determinant of X.

     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an

     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

              Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]
                   [ kron(T11', In-2)  -kron(I2, T22) ]

     Note that if the default method for computing DIF is wanted (see
     SLATDF), then the parameter DIFDRI (see below) should be changed
     from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
     for more details.

  For each eigenvalue/vector specified by SELECT, DIF stores a
  Frobenius norm-based estimate of Difl.

  An approximate error bound for the i-th computed eigenvector VL(i) or
  VR(i) is given by

             EPS * norm(A, B) / DIF(i).

  See ref. [2-3] for more details and further references.

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  References
  ==========

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
      No 1, 1996.

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        howmny -
        select -
        n -
        a -
        lda -
        b -
        ldb -
        vl -
        ldvl -
        vr -
        ldvr -
        s -
        dif -
        mm -
        m -
        work -
        lwork -
        iwork -
        info -

      • stgsy2

        public abstract void stgsy2(java.lang.String trans,
                            int ijob,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] c,
                            int Ldc,
                            float[] d,
                            int ldd,
                            float[] e,
                            int lde,
                            float[] f,
                            int ldf,
                            org.netlib.util.floatW scale,
                            org.netlib.util.floatW rdsum,
                            org.netlib.util.floatW rdscal,
                            int[] iwork,
                            org.netlib.util.intW pq,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGSY2 solves the generalized Sylvester equation:

              A * R - L * B = scale * C                (1)
              D * R - L * E = scale * F,

  using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,

  (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
  N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
  must be in generalized Schur canonical form, i.e. A, B are upper
  quasi triangular and D, E are upper triangular. The solution (R, L)
  overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
  chosen to avoid overflow.

  In matrix notation solving equation (1) corresponds to solve
  Z*x = scale*b, where Z is defined as

         Z = [ kron(In, A)  -kron(B', Im) ]             (2)
             [ kron(In, D)  -kron(E', Im) ],

  Ik is the identity matrix of size k and X' is the transpose of X.
  kron(X, Y) is the Kronecker product between the matrices X and Y.
  In the process of solving (1), we solve a number of such systems
  where Dim(In), Dim(In) = 1 or 2.

  If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
  which is equivalent to solve for R and L in

              A' * R  + D' * L   = scale *  C           (3)
              R  * B' + L  * E'  = scale * -F

  This case is used to compute an estimate of Dif[(A, D), (B, E)] =
  sigma_min(Z) using reverse communicaton with SLACON.

  STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL
  of an upper bound on the separation between to matrix pairs. Then
  the input (A, D), (B, E) are sub-pencils of the matrix pair in
  STGSYL. See STGSYL for details.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N', solve the generalized Sylvester equation (1).
          = 'T': solve the 'transposed' system (3).

  IJOB    (input) INTEGER
          Specifies what kind of functionality to be performed.
          = 0: solve (1) only.
          = 1: A contribution from this subsystem to a Frobenius
               norm-based estimate of the separation between two matrix
               pairs is computed. (look ahead strategy is used).
          = 2: A contribution from this subsystem to a Frobenius
               norm-based estimate of the separation between two matrix
               pairs is computed. (SGECON on sub-systems is used.)
          Not referenced if TRANS = 'T'.

  M       (input) INTEGER
          On entry, M specifies the order of A and D, and the row
          dimension of C, F, R and L.

  N       (input) INTEGER
          On entry, N specifies the order of B and E, and the column
          dimension of C, F, R and L.

  A       (input) REAL array, dimension (LDA, M)
          On entry, A contains an upper quasi triangular matrix.

  LDA     (input) INTEGER
          The leading dimension of the matrix A. LDA >= max(1, M).

  B       (input) REAL array, dimension (LDB, N)
          On entry, B contains an upper quasi triangular matrix.

  LDB     (input) INTEGER
          The leading dimension of the matrix B. LDB >= max(1, N).

  C       (input/output) REAL array, dimension (LDC, N)
          On entry, C contains the right-hand-side of the first matrix

          equation in (1).
          On exit, if IJOB = 0, C has been overwritten by the
          solution R.

  LDC     (input) INTEGER
          The leading dimension of the matrix C. LDC >= max(1, M).

  D       (input) REAL array, dimension (LDD, M)
          On entry, D contains an upper triangular matrix.

  LDD     (input) INTEGER
          The leading dimension of the matrix D. LDD >= max(1, M).

  E       (input) REAL array, dimension (LDE, N)
          On entry, E contains an upper triangular matrix.

  LDE     (input) INTEGER
          The leading dimension of the matrix E. LDE >= max(1, N).

  F       (input/output) REAL array, dimension (LDF, N)
          On entry, F contains the right-hand-side of the second matrix
          equation in (1).
          On exit, if IJOB = 0, F has been overwritten by the
          solution L.

  LDF     (input) INTEGER
          The leading dimension of the matrix F. LDF >= max(1, M).

  SCALE   (output) REAL
          On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
          R and L (C and F on entry) will hold the solutions to a
          slightly perturbed system but the input matrices A, B, D and

          E have not been changed. If SCALE = 0, R and L will hold the

          solutions to the homogeneous system with C = F = 0. Normally,
          SCALE = 1.

  RDSUM   (input/output) REAL
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by STGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

  RDSCAL  (input/output) REAL
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when STGSY2 is called by
                STGSYL.

  IWORK   (workspace) INTEGER array, dimension (M+N+2)

  PQ      (output) INTEGER
          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
          8-by-8) solved by this routine.

  INFO    (output) INTEGER
          On exit, if INFO is set to
            =0: Successful exit
            <0: If INFO = -i, the i-th argument had an illegal value.
            >0: The matrix pairs (A, D) and (B, E) have common or very

                close eigenvalues.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  =====================================================================
  Replaced various illegal calls to SCOPY by calls to SLASET.
  Sven Hammarling, 27/5/02.

     .. Parameters ..
        Parameters:
        trans -
        ijob -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        d -
        ldd -
        e -
        lde -
        f -
        ldf -
        scale -
        rdsum -
        rdscal -
        iwork -
        pq -
        info -

      • stgsyl

        public abstract void stgsyl(java.lang.String trans,
                            int ijob,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] c,
                            int Ldc,
                            float[] d,
                            int ldd,
                            float[] e,
                            int lde,
                            float[] f,
                            int ldf,
                            org.netlib.util.floatW scale,
                            org.netlib.util.floatW dif,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STGSYL solves the generalized Sylvester equation:

              A * R - L * B = scale * C                 (1)
              D * R - L * E = scale * F

  where R and L are unknown m-by-n matrices, (A, D), (B, E) and
  (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
  respectively, with real entries. (A, D) and (B, E) must be in
  generalized (real) Schur canonical form, i.e. A, B are upper quasi
  triangular and D, E are upper triangular.

  The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
  scaling factor chosen to avoid overflow.

  In matrix notation (1) is equivalent to solve  Zx = scale b, where
  Z is defined as

             Z = [ kron(In, A)  -kron(B', Im) ]         (2)
                 [ kron(In, D)  -kron(E', Im) ].

  Here Ik is the identity matrix of size k and X' is the transpose of
  X. kron(X, Y) is the Kronecker product between the matrices X and Y.


  If TRANS = 'T', STGSYL solves the transposed system Z'*y = scale*b,
  which is equivalent to solve for R and L in

              A' * R  + D' * L   = scale *  C           (3)
              R  * B' + L  * E'  = scale * (-F)

  This case (TRANS = 'T') is used to compute an one-norm-based estimate
  of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
  and (B,E), using SLACON.

  If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate
  of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the

  reciprocal of the smallest singular value of Z. See [1-2] for more
  information.

  This is a level 3 BLAS algorithm.

  Arguments
  =========

  TRANS   (input) CHARACTER*1
          = 'N', solve the generalized Sylvester equation (1).
          = 'T', solve the 'transposed' system (3).

  IJOB    (input) INTEGER
          Specifies what kind of functionality to be performed.
           =0: solve (1) only.
           =1: The functionality of 0 and 3.
           =2: The functionality of 0 and 4.
           =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
               (look ahead strategy IJOB  = 1 is used).
           =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
               ( SGECON on sub-systems is used ).
          Not referenced if TRANS = 'T'.

  M       (input) INTEGER
          The order of the matrices A and D, and the row dimension of
          the matrices C, F, R and L.

  N       (input) INTEGER
          The order of the matrices B and E, and the column dimension
          of the matrices C, F, R and L.

  A       (input) REAL array, dimension (LDA, M)
          The upper quasi triangular matrix A.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1, M).

  B       (input) REAL array, dimension (LDB, N)
          The upper quasi triangular matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1, N).

  C       (input/output) REAL array, dimension (LDC, N)
          On entry, C contains the right-hand-side of the first matrix

          equation in (1) or (3).
          On exit, if IJOB = 0, 1 or 2, C has been overwritten by
          the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,

          the solution achieved during the computation of the
          Dif-estimate.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1, M).

  D       (input) REAL array, dimension (LDD, M)
          The upper triangular matrix D.

  LDD     (input) INTEGER
          The leading dimension of the array D. LDD >= max(1, M).

  E       (input) REAL array, dimension (LDE, N)
          The upper triangular matrix E.

  LDE     (input) INTEGER
          The leading dimension of the array E. LDE >= max(1, N).

  F       (input/output) REAL array, dimension (LDF, N)
          On entry, F contains the right-hand-side of the second matrix
          equation in (1) or (3).
          On exit, if IJOB = 0, 1 or 2, F has been overwritten by
          the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,

          the solution achieved during the computation of the
          Dif-estimate.

  LDF     (input) INTEGER
          The leading dimension of the array F. LDF >= max(1, M).

  DIF     (output) REAL
          On exit DIF is the reciprocal of a lower bound of the
          reciprocal of the Dif-function, i.e. DIF is an upper bound of
          Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
          IF IJOB = 0 or TRANS = 'T', DIF is not touched.

  SCALE   (output) REAL
          On exit SCALE is the scaling factor in (1) or (3).
          If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,

          to a slightly perturbed system but the input matrices A, B, D
          and E have not been changed. If SCALE = 0, C and F hold the
          solutions R and L, respectively, to the homogeneous system
          with C = F = 0. Normally, SCALE = 1.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK. LWORK > = 1.
          If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (M+N+6)

  INFO    (output) INTEGER
            =0: successful exit
            <0: If INFO = -i, the i-th argument had an illegal value.
            >0: (A, D) and (B, E) have common or close eigenvalues.

  Further Details
  ===============

  Based on contributions by
     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
     Umea University, S-901 87 Umea, Sweden.

  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,

      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
      No 1, 1996.

  [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
      Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
      Appl., 15(4):1045-1060, 1994

  [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
      Condition Estimators for Solving the Generalized Sylvester
      Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
      July 1989, pp 745-751.

  =====================================================================
  Replaced various illegal calls to SCOPY by calls to SLASET.
  Sven Hammarling, 1/5/02.

     .. Parameters ..
        Parameters:
        trans -
        ijob -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        d -
        ldd -
        e -
        lde -
        f -
        ldf -
        scale -
        dif -
        work -
        lwork -
        iwork -
        info -

      • stpcon

        public abstract void stpcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            float[] ap,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STPCON estimates the reciprocal of the condition number of a packed
  triangular matrix A, in either the 1-norm or the infinity-norm.

  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        ap -
        rcond -
        work -
        iwork -
        info -

      • stprfs

        public abstract void stprfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STPRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular packed
  coefficient matrix.

  The solution matrix X must be computed by STPTRS or some other
  means before entering this routine.  STPRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          If DIAG = 'U', the diagonal elements of A are not referenced

          and are assumed to be 1.

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) REAL array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        ap -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • stptri

        public abstract void stptri(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            float[] ap,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STPTRI computes the inverse of a real upper or lower triangular
  matrix A stored in packed format.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangular matrix A, stored
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.
          On exit, the (triangular) inverse of the original matrix, in

          the same packed storage format.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, A(i,i) is exactly zero.  The triangular
                matrix is singular and its inverse can not be computed.

  Further Details
  ===============

  A triangular matrix A can be transferred to packed storage using one

  of the following program segments:

  UPLO = 'U':                      UPLO = 'L':

        JC = 1                           JC = 1
        DO 2 J = 1, N                    DO 2 J = 1, N
           DO 1 I = 1, J                    DO 1 I = J, N
              AP(JC+I-1) = A(I,J)              AP(JC+I-J) = A(I,J)
      1    CONTINUE                    1    CONTINUE
           JC = JC + J                      JC = JC + N - J + 1
      2 CONTINUE                       2 CONTINUE

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        ap -
        info -

      • stptrs

        public abstract void stptrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            float[] ap,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STPTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular matrix of order N stored in packed format,
  and B is an N-by-NRHS matrix.  A check is made to verify that A is
  nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in

          a linear array.  The j-th column of A is stored in the array

          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        ap -
        b -
        ldb -
        info -

      • strcon

        public abstract void strcon(java.lang.String norm,
                            java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.floatW rcond,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRCON estimates the reciprocal of the condition number of a
  triangular matrix A, in either the 1-norm or the infinity-norm.

  The norm of A is computed and an estimate is obtained for
  norm(inv(A)), then the reciprocal of the condition number is
  computed as
     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  Arguments
  =========

  NORM    (input) CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  RCOND   (output) REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        norm -
        uplo -
        diag -
        n -
        a -
        lda -
        rcond -
        work -
        iwork -
        info -

      • strevc

        public abstract void strevc(java.lang.String side,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            float[] t,
                            int ldt,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            int mm,
                            org.netlib.util.intW m,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STREVC computes some or all of the right and/or left eigenvectors of

  a real upper quasi-triangular matrix T.
  Matrices of this type are produced by the Schur factorization of
  a real general matrix:  A = Q*T*Q**T, as computed by SHSEQR.
  
  The right eigenvector x and the left eigenvector y of T corresponding
  to an eigenvalue w are defined by:
  
     T*x = w*x,     (y**H)*T = w*(y**H)
  
  where y**H denotes the conjugate transpose of y.
  The eigenvalues are not input to this routine, but are read directly

  from the diagonal blocks of T.
  
  This routine returns the matrices X and/or Y of right and left
  eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
  input matrix.  If Q is the orthogonal factor that reduces a matrix
  A to Schur form T, then Q*X and Q*Y are the matrices of right and
  left eigenvectors of A.

  Arguments
  =========

  SIDE    (input) CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.

  HOWMNY  (input) CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed by the matrices in VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.

  SELECT  (input/output) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          If w(j) is a real eigenvalue, the corresponding real
          eigenvector is computed if SELECT(j) is .TRUE..
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector is

          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to

          .FALSE..
          Not referenced if HOWMNY = 'A' or 'B'.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input) REAL array, dimension (LDT,N)
          The upper quasi-triangular matrix T in Schur canonical form.


  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  VL      (input/output) REAL array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by SHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns

                           of VL, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part.
          Not referenced if SIDE = 'R'.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.

  VR      (input/output) REAL array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by SHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns

                           of VR, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part.
          Not referenced if SIDE = 'L'.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B', LDVR >= N.

  MM      (input) INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

  M       (output) INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.
          If HOWMNY = 'A' or 'B', M is set to N.
          Each selected real eigenvector occupies one column and each
          selected complex eigenvector occupies two columns.

  WORK    (workspace) REAL array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.

  =====================================================================

     .. Parameters ..
        Parameters:
        side -
        howmny -
        select -
        n -
        t -
        ldt -
        vl -
        ldvl -
        vr -
        ldvr -
        mm -
        m -
        work -
        info -

      • strexc

        public abstract void strexc(java.lang.String compq,
                            int n,
                            float[] t,
                            int ldt,
                            float[] q,
                            int ldq,
                            org.netlib.util.intW ifst,
                            org.netlib.util.intW ilst,
                            float[] work,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STREXC reorders the real Schur factorization of a real matrix
  A = Q*T*Q**T, so that the diagonal block of T with row index IFST is

  moved to row ILST.

  The real Schur form T is reordered by an orthogonal similarity
  transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
  is updated by postmultiplying it with Z.

  T must be in Schur canonical form (as returned by SHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  COMPQ   (input) CHARACTER*1
          = 'V':  update the matrix Q of Schur vectors;
          = 'N':  do not update Q.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) REAL array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          Schur canonical form.
          On exit, the reordered upper quasi-triangular matrix, again
          in Schur canonical form.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          orthogonal transformation matrix Z which reorders T.
          If COMPQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= max(1,N).

  IFST    (input/output) INTEGER
  ILST    (input/output) INTEGER
          Specify the reordering of the diagonal blocks of T.
          The block with row index IFST is moved to row ILST, by a
          sequence of transpositions between adjacent blocks.
          On exit, if IFST pointed on entry to the second row of a
          2-by-2 block, it is changed to point to the first row; ILST
          always points to the first row of the block in its final
          position (which may differ from its input value by +1 or -1).
          1 <= IFST <= N; 1 <= ILST <= N.

  WORK    (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          = 1:  two adjacent blocks were too close to swap (the problem
                is very ill-conditioned); T may have been partially
                reordered, and ILST points to the first row of the
                current position of the block being moved.

  =====================================================================

     .. Parameters ..
        Parameters:
        compq -
        n -
        t -
        ldt -
        q -
        ldq -
        ifst -
        ilst -
        work -
        info -

      • strrfs

        public abstract void strrfs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] x,
                            int ldx,
                            float[] ferr,
                            float[] berr,
                            float[] work,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRRFS provides error bounds and backward error estimates for the
  solution to a system of linear equations with a triangular
  coefficient matrix.

  The solution matrix X must be computed by STRTRS or some other
  means before entering this routine.  STRRFS does not do iterative
  refinement because doing so cannot improve the backward error.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (input) REAL array, dimension (LDX,NRHS)
          The solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)

          is an estimated upper bound for the magnitude of the largest

          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

  WORK    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        x -
        ldx -
        ferr -
        berr -
        work -
        iwork -
        info -

      • strsen

        public abstract void strsen(java.lang.String job,
                            java.lang.String compq,
                            boolean[] select,
                            int n,
                            float[] t,
                            int ldt,
                            float[] q,
                            int ldq,
                            float[] wr,
                            float[] wi,
                            org.netlib.util.intW m,
                            org.netlib.util.floatW s,
                            org.netlib.util.floatW sep,
                            float[] work,
                            int lwork,
                            int[] iwork,
                            int liwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRSEN reorders the real Schur factorization of a real matrix
  A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
  the leading diagonal blocks of the upper quasi-triangular matrix T,
  and the leading columns of Q form an orthonormal basis of the
  corresponding right invariant subspace.

  Optionally the routine computes the reciprocal condition numbers of
  the cluster of eigenvalues and/or the invariant subspace.

  T must be in Schur canonical form (as returned by SHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elemnts equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (S) or the invariant subspace (SEP):
          = 'N': none;
          = 'E': for eigenvalues only (S);
          = 'V': for invariant subspace only (SEP);
          = 'B': for both eigenvalues and invariant subspace (S and
                 SEP).

  COMPQ   (input) CHARACTER*1
          = 'V': update the matrix Q of Schur vectors;
          = 'N': do not update Q.

  SELECT  (input) LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To

          select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) REAL array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, T is overwritten by the reordered matrix T, again in
          Schur canonical form, with the selected eigenvalues in the
          leading diagonal blocks.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  Q       (input/output) REAL array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          orthogonal transformation matrix which reorders T; the
          leading M columns of Q form an orthonormal basis for the
          specified invariant subspace.
          If COMPQ = 'N', Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

  WR      (output) REAL array, dimension (N)
  WI      (output) REAL array, dimension (N)
          The real and imaginary parts, respectively, of the reordered

          eigenvalues of T. The eigenvalues are stored in the same
          order as on the diagonal of T, with WR(i) = T(i,i) and, if
          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
          sufficiently ill-conditioned, then its value may differ
          significantly from its value before reordering.

  M       (output) INTEGER
          The dimension of the specified invariant subspace.
          0 < = M <= N.

  S       (output) REAL
          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
          condition number for the selected cluster of eigenvalues.
          S cannot underestimate the true reciprocal condition number
          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
          If JOB = 'N' or 'V', S is not referenced.

  SEP     (output) REAL
          If JOB = 'V' or 'B', SEP is the estimated reciprocal
          condition number of the specified invariant subspace. If
          M = 0 or N, SEP = norm(T).
          If JOB = 'N' or 'E', SEP is not referenced.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.
          If JOB = 'N', LWORK >= max(1,N);
          if JOB = 'E', LWORK >= max(1,M*(N-M));
          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

  LIWORK  (input) INTEGER
          The dimension of the array IWORK.
          If JOB = 'N' or 'E', LIWORK >= 1;
          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,

          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1: reordering of T failed because some eigenvalues are too

               close to separate (the problem is very ill-conditioned);
               T may have been partially reordered, and WR and WI
               contain the eigenvalues in the same order as in T; S and
               SEP (if requested) are set to zero.

  Further Details
  ===============

  STRSEN first collects the selected eigenvalues by computing an
  orthogonal transformation Z to move them to the top left corner of T.
  In other words, the selected eigenvalues are the eigenvalues of T11
  in:

                Z'*T*Z = ( T11 T12 ) n1
                         (  0  T22 ) n2
                            n1  n2

  where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
  of Z span the specified invariant subspace of T.

  If T has been obtained from the real Schur factorization of a matrix

  A = Q*T*Q', then the reordered real Schur factorization of A is given
  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
  the corresponding invariant subspace of A.

  The reciprocal condition number of the average of the eigenvalues of

  T11 may be returned in S. S lies between 0 (very badly conditioned)
  and 1 (very well conditioned). It is computed as follows. First we
  compute R so that

                         P = ( I  R ) n1
                             ( 0  0 ) n2
                               n1 n2

  is the projector on the invariant subspace associated with T11.
  R is the solution of the Sylvester equation:

                        T11*R - R*T22 = T12.

  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
  the two-norm of M. Then S is computed as the lower bound

                      (1 + F-norm(R)**2)**(-1/2)

  on the reciprocal of 2-norm(P), the true reciprocal condition number.
  S cannot underestimate 1 / 2-norm(P) by more than a factor of
  sqrt(N).

  An approximate error bound for the computed average of the
  eigenvalues of T11 is

                         EPS * norm(T) / S

  where EPS is the machine precision.

  The reciprocal condition number of the right invariant subspace
  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.

  SEP is defined as the separation of T11 and T22:

                     sep( T11, T22 ) = sigma-min( C )

  where sigma-min(C) is the smallest singular value of the
  n1*n2-by-n1*n2 matrix

     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
  product. We estimate sigma-min(C) by the reciprocal of an estimate of
  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

  When SEP is small, small changes in T can cause large changes in
  the invariant subspace. An approximate bound on the maximum angular
  error in the computed right invariant subspace is

                      EPS * norm(T) / SEP

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        compq -
        select -
        n -
        t -
        ldt -
        q -
        ldq -
        wr -
        wi -
        m -
        s -
        sep -
        work -
        lwork -
        iwork -
        liwork -
        info -

      • strsna

        public abstract void strsna(java.lang.String job,
                            java.lang.String howmny,
                            boolean[] select,
                            int n,
                            float[] t,
                            int ldt,
                            float[] vl,
                            int ldvl,
                            float[] vr,
                            int ldvr,
                            float[] s,
                            float[] sep,
                            int mm,
                            org.netlib.util.intW m,
                            float[] work,
                            int ldwork,
                            int[] iwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRSNA estimates reciprocal condition numbers for specified
  eigenvalues and/or right eigenvectors of a real upper
  quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
  orthogonal).

  T must be in Schur canonical form (as returned by SHSEQR), that is,
  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
  2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  JOB     (input) CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (SEP):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (SEP);
          = 'B': for both eigenvalues and eigenvectors (S and SEP).

  HOWMNY  (input) CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the eigenpair corresponding to a real eigenvalue w(j),
          SELECT(j) must be set to .TRUE.. To select condition numbers

          corresponding to a complex conjugate pair of eigenvalues w(j)
          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be

          set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input) REAL array, dimension (LDT,N)
          The upper quasi-triangular matrix T, in Schur canonical form.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  VL      (input) REAL array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
          (or of any Q*T*Q**T with Q orthogonal), corresponding to the

          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VL, as returned by
          SHSEIN or STREVC.
          If JOB = 'V', VL is not referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.
          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.

  VR      (input) REAL array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of T

          (or of any Q*T*Q**T with Q orthogonal), corresponding to the

          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VR, as returned by
          SHSEIN or STREVC.
          If JOB = 'V', VR is not referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.
          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

  S       (output) REAL array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the

          selected eigenvalues, stored in consecutive elements of the
          array. For a complex conjugate pair of eigenvalues two
          consecutive elements of S are set to the same value. Thus
          S(j), SEP(j), and the j-th columns of VL and VR all
          correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.

  SEP     (output) REAL array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array. For a complex eigenvector two
          consecutive elements of SEP are set to the same value. If
          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
          is set to 0; this can only occur when the true value would be
          very small anyway.
          If JOB = 'E', SEP is not referenced.

  MM      (input) INTEGER
          The number of elements in the arrays S (if JOB = 'E' or 'B')

           and/or SEP (if JOB = 'V' or 'B'). MM >= M.

  M       (output) INTEGER
          The number of elements of the arrays S and/or SEP actually
          used to store the estimated condition numbers.
          If HOWMNY = 'A', M is set to N.

  WORK    (workspace) REAL array, dimension (LDWORK,N+6)
          If JOB = 'E', WORK is not referenced.

  LDWORK  (input) INTEGER
          The leading dimension of the array WORK.
          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

  IWORK   (workspace) INTEGER array, dimension (2*(N-1))
          If JOB = 'E', IWORK is not referenced.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The reciprocal of the condition number of an eigenvalue lambda is
  defined as

          S(lambda) = |v'*u| / (norm(u)*norm(v))

  where u and v are the right and left eigenvectors of T corresponding

  to lambda; v' denotes the conjugate-transpose of v, and norm(u)
  denotes the Euclidean norm. These reciprocal condition numbers always
  lie between zero (very badly conditioned) and one (very well
  conditioned). If n = 1, S(lambda) is defined to be 1.

  An approximate error bound for a computed eigenvalue W(i) is given by

                      EPS * norm(T) / S(i)

  where EPS is the machine precision.

  The reciprocal of the condition number of the right eigenvector u
  corresponding to lambda is defined as follows. Suppose

              T = ( lambda  c  )
                  (   0    T22 )

  Then the reciprocal condition number is

          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

  where sigma-min denotes the smallest singular value. We approximate
  the smallest singular value by the reciprocal of an estimate of the
  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
  defined to be abs(T(1,1)).

  An approximate error bound for a computed right eigenvector VR(i)
  is given by

                      EPS * norm(T) / SEP(i)

  =====================================================================

     .. Parameters ..
        Parameters:
        job -
        howmny -
        select -
        n -
        t -
        ldt -
        vl -
        ldvl -
        vr -
        ldvr -
        s -
        sep -
        mm -
        m -
        work -
        ldwork -
        iwork -
        info -

      • strsyl

        public abstract void strsyl(java.lang.String trana,
                            java.lang.String tranb,
                            int isgn,
                            int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            float[] c,
                            int Ldc,
                            org.netlib.util.floatW scale,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRSYL solves the real Sylvester matrix equation:

     op(A)*X + X*op(B) = scale*C or
     op(A)*X - X*op(B) = scale*C,

  where op(A) = A or A**T, and  A and B are both upper quasi-
  triangular. A is M-by-M and B is N-by-N; the right hand side C and
  the solution X are M-by-N; and scale is an output scale factor, set
  <= 1 to avoid overflow in X.

  A and B must be in Schur canonical form (as returned by SHSEQR), that
  is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
  each 2-by-2 diagonal block has its diagonal elements equal and its
  off-diagonal elements of opposite sign.

  Arguments
  =========

  TRANA   (input) CHARACTER*1
          Specifies the option op(A):
          = 'N': op(A) = A    (No transpose)
          = 'T': op(A) = A**T (Transpose)
          = 'C': op(A) = A**H (Conjugate transpose = Transpose)

  TRANB   (input) CHARACTER*1
          Specifies the option op(B):
          = 'N': op(B) = B    (No transpose)
          = 'T': op(B) = B**T (Transpose)
          = 'C': op(B) = B**H (Conjugate transpose = Transpose)

  ISGN    (input) INTEGER
          Specifies the sign in the equation:
          = +1: solve op(A)*X + X*op(B) = scale*C
          = -1: solve op(A)*X - X*op(B) = scale*C

  M       (input) INTEGER
          The order of the matrix A, and the number of rows in the
          matrices X and C. M >= 0.

  N       (input) INTEGER
          The order of the matrix B, and the number of columns in the
          matrices X and C. N >= 0.

  A       (input) REAL array, dimension (LDA,M)
          The upper quasi-triangular matrix A, in Schur canonical form.

  LDA     (input) INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

  B       (input) REAL array, dimension (LDB,N)
          The upper quasi-triangular matrix B, in Schur canonical form.

  LDB     (input) INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

  C       (input/output) REAL array, dimension (LDC,N)
          On entry, the M-by-N right hand side matrix C.
          On exit, C is overwritten by the solution matrix X.

  LDC     (input) INTEGER
          The leading dimension of the array C. LDC >= max(1,M)

  SCALE   (output) REAL
          The scale factor, scale, set <= 1 to avoid overflow in X.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1: A and B have common or very close eigenvalues; perturbed
               values were used to solve the equation (but the matrices
               A and B are unchanged).

  =====================================================================

     .. Parameters ..
        Parameters:
        trana -
        tranb -
        isgn -
        m -
        n -
        a -
        lda -
        b -
        ldb -
        c -
        Ldc -
        scale -
        info -

      • strti2

        public abstract void strti2(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRTI2 computes the inverse of a real upper or lower triangular
  matrix.

  This is the Level 2 BLAS version of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.

          = 'U':  Upper triangular
          = 'L':  Lower triangular

  DIAG    (input) CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading n by n upper triangular part of the array A contains

          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of the array A contains

          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.

          On exit, the (triangular) inverse of the original matrix, in

          the same storage format.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        a -
        lda -
        info -

      • strtri

        public abstract void strtri(java.lang.String uplo,
                            java.lang.String diag,
                            int n,
                            float[] a,
                            int lda,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRTRI computes the inverse of a real upper or lower triangular
  matrix A.

  This is the Level 3 BLAS version of the algorithm.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains

          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains

          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.
          On exit, the (triangular) inverse of the original matrix, in

          the same storage format.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can not be computed.


  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        diag -
        n -
        a -
        lda -
        info -

      • strtrs

        public abstract void strtrs(java.lang.String uplo,
                            java.lang.String trans,
                            java.lang.String diag,
                            int n,
                            int nrhs,
                            float[] a,
                            int lda,
                            float[] b,
                            int ldb,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STRTRS solves a triangular system of the form

     A * X = B  or  A**T * X = B,

  where A is a triangular matrix of order N, and B is an N-by-NRHS
  matrix.  A check is made to verify that A is nonsingular.

  Arguments
  =========

  UPLO    (input) CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

  DIAG    (input) CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

  A       (input) REAL array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of

          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular

          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the i-th diagonal element of A is zero,
               indicating that the matrix is singular and the solutions
               X have not been computed.

  =====================================================================

     .. Parameters ..
        Parameters:
        uplo -
        trans -
        diag -
        n -
        nrhs -
        a -
        lda -
        b -
        ldb -
        info -

      • stzrqf

        public abstract void stzrqf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  This routine is deprecated and has been replaced by routine STZRZF.

  STZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
  to upper triangular form by means of orthogonal transformations.

  The upper trapezoidal matrix A is factored as

     A = ( R  0 ) * Z,

  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
  triangular matrix.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= M.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements M+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (M)
          The scalar factors of the elementary reflectors.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an ( n - m ) element vector.
  tau and z( k ) are chosen to annihilate the elements of the kth row
  of X.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A, such that the elements of z( k ) are
  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        info -

      • stzrzf

        public abstract void stzrzf(int m,
                            int n,
                            float[] a,
                            int lda,
                            float[] tau,
                            float[] work,
                            int lwork,
                            org.netlib.util.intW info)
        
     ..

  Purpose
  =======

  STZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
  to upper triangular form by means of orthogonal transformations.

  The upper trapezoidal matrix A is factored as

     A = ( R  0 ) * Z,

  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
  triangular matrix.

  Arguments
  =========

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= M.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements M+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.


  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  TAU     (output) REAL array, dimension (M)
          The scalar factors of the elementary reflectors.

  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  Further Details
  ===============

  Based on contributions by
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA


  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into

  the ( m - k + 1 )th row of A, is given in the form

     Z( k ) = ( I     0   ),
              ( 0  T( k ) )

  where

     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )

  tau is a scalar and z( k ) is an ( n - m ) element vector.
  tau and z( k ) are chosen to annihilate the elements of the kth row
  of X.

  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A, such that the elements of z( k ) are
  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in

  the upper triangular part of A.

  Z is given by

     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

  =====================================================================

     .. Parameters ..
        Parameters:
        m -
        n -
        a -
        lda -
        tau -
        work -
        lwork -
        info -

      • dlamch

        public abstract double dlamch(java.lang.String cmach)
        
     ..

  Purpose
  =======

  DLAMCH determines double precision machine parameters.

  Arguments
  =========

  CMACH   (input) CHARACTER*1
          Specifies the value to be returned by DLAMCH:
          = 'E' or 'e',   DLAMCH := eps
          = 'S' or 's ,   DLAMCH := sfmin
          = 'B' or 'b',   DLAMCH := base
          = 'P' or 'p',   DLAMCH := eps*base
          = 'N' or 'n',   DLAMCH := t
          = 'R' or 'r',   DLAMCH := rnd
          = 'M' or 'm',   DLAMCH := emin
          = 'U' or 'u',   DLAMCH := rmin
          = 'L' or 'l',   DLAMCH := emax
          = 'O' or 'o',   DLAMCH := rmax

          where

          eps   = relative machine precision
          sfmin = safe minimum, such that 1/sfmin does not overflow
          base  = base of the machine
          prec  = eps*base
          t     = number of (base) digits in the mantissa
          rnd   = 1.0 when rounding occurs in addition, 0.0 otherwise
          emin  = minimum exponent before (gradual) underflow
          rmin  = underflow threshold - base**(emin-1)
          emax  = largest exponent before overflow
          rmax  = overflow threshold  - (base**emax)*(1-eps)

 =====================================================================


     .. Parameters ..
        Parameters:
        cmach -

      • dlamc1

        public abstract void dlamc1(org.netlib.util.intW beta,
                            org.netlib.util.intW t,
                            org.netlib.util.booleanW rnd,
                            org.netlib.util.booleanW ieee1)
        
     ..

  Purpose
  =======

  DLAMC1 determines the machine parameters given by BETA, T, RND, and
  IEEE1.

  Arguments
  =========

  BETA    (output) INTEGER
          The base of the machine.

  T       (output) INTEGER
          The number of ( BETA ) digits in the mantissa.

  RND     (output) LOGICAL
          Specifies whether proper rounding  ( RND = .TRUE. )  or
          chopping  ( RND = .FALSE. )  occurs in addition. This may not
          be a reliable guide to the way in which the machine performs

          its arithmetic.

  IEEE1   (output) LOGICAL
          Specifies whether rounding appears to be done in the IEEE
          'round to nearest' style.

  Further Details
  ===============

  The routine is based on the routine  ENVRON  by Malcolm and
  incorporates suggestions by Gentleman and Marovich. See

     Malcolm M. A. (1972) Algorithms to reveal properties of
        floating-point arithmetic. Comms. of the ACM, 15, 949-951.

     Gentleman W. M. and Marovich S. B. (1974) More on algorithms
        that reveal properties of floating point arithmetic units.
        Comms. of the ACM, 17, 276-277.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        beta -
        t -
        rnd -
        ieee1 -

      • dlamc2

        public abstract void dlamc2(org.netlib.util.intW beta,
                            org.netlib.util.intW t,
                            org.netlib.util.booleanW rnd,
                            org.netlib.util.doubleW eps,
                            org.netlib.util.intW emin,
                            org.netlib.util.doubleW rmin,
                            org.netlib.util.intW emax,
                            org.netlib.util.doubleW rmax)
        
     ..

  Purpose
  =======

  DLAMC2 determines the machine parameters specified in its argument
  list.

  Arguments
  =========

  BETA    (output) INTEGER
          The base of the machine.

  T       (output) INTEGER
          The number of ( BETA ) digits in the mantissa.

  RND     (output) LOGICAL
          Specifies whether proper rounding  ( RND = .TRUE. )  or
          chopping  ( RND = .FALSE. )  occurs in addition. This may not
          be a reliable guide to the way in which the machine performs

          its arithmetic.

  EPS     (output) DOUBLE PRECISION
          The smallest positive number such that

             fl( 1.0 - EPS ) .LT. 1.0,

          where fl denotes the computed value.

  EMIN    (output) INTEGER
          The minimum exponent before (gradual) underflow occurs.

  RMIN    (output) DOUBLE PRECISION
          The smallest normalized number for the machine, given by
          BASE**( EMIN - 1 ), where  BASE  is the floating point value

          of BETA.

  EMAX    (output) INTEGER
          The maximum exponent before overflow occurs.

  RMAX    (output) DOUBLE PRECISION
          The largest positive number for the machine, given by
          BASE**EMAX * ( 1 - EPS ), where  BASE  is the floating point

          value of BETA.

  Further Details
  ===============

  The computation of  EPS  is based on a routine PARANOIA by
  W. Kahan of the University of California at Berkeley.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        beta -
        t -
        rnd -
        eps -
        emin -
        rmin -
        emax -
        rmax -

      • dlamc3

        public double dlamc3(double a,
                     double b)
        
     ..

  Purpose
  =======

  DLAMC3  is intended to force  A  and  B  to be stored prior to doing

  the addition of  A  and  B ,  for use in situations where optimizers

  might hold one of these in a register.

  Arguments
  =========

  A       (input) DOUBLE PRECISION
  B       (input) DOUBLE PRECISION
          The values A and B.

 =====================================================================


     .. Executable Statements ..
        Parameters:
        a -
        b -

      • dlamc4

        public abstract void dlamc4(org.netlib.util.intW emin,
                            double start,
                            int base)
        
     ..

  Purpose
  =======

  DLAMC4 is a service routine for DLAMC2.

  Arguments
  =========

  EMIN    (output) INTEGER 
          The minimum exponent before (gradual) underflow, computed by

          setting A = START and dividing by BASE until the previous A
          can not be recovered.

  START   (input) DOUBLE PRECISION
          The starting point for determining EMIN.

  BASE    (input) INTEGER
          The base of the machine.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        emin -
        start -
        base -

      • dlamc5

        public abstract void dlamc5(int beta,
                            int p,
                            int emin,
                            boolean ieee,
                            org.netlib.util.intW emax,
                            org.netlib.util.doubleW rmax)
        
     ..

  Purpose
  =======

  DLAMC5 attempts to compute RMAX, the largest machine floating-point
  number, without overflow.  It assumes that EMAX + abs(EMIN) sum
  approximately to a power of 2.  It will fail on machines where this
  assumption does not hold, for example, the Cyber 205 (EMIN = -28625,

  EMAX = 28718).  It will also fail if the value supplied for EMIN is
  too large (i.e. too close to zero), probably with overflow.

  Arguments
  =========

  BETA    (input) INTEGER
          The base of floating-point arithmetic.

  P       (input) INTEGER
          The number of base BETA digits in the mantissa of a
          floating-point value.

  EMIN    (input) INTEGER
          The minimum exponent before (gradual) underflow.

  IEEE    (input) LOGICAL
          A logical flag specifying whether or not the arithmetic
          system is thought to comply with the IEEE standard.

  EMAX    (output) INTEGER
          The largest exponent before overflow

  RMAX    (output) DOUBLE PRECISION
          The largest machine floating-point number.

 =====================================================================


     .. Parameters ..
        Parameters:
        beta -
        p -
        emin -
        ieee -
        emax -
        rmax -

      • dsecnd

        public double dsecnd()
        

  -- LAPACK auxiliary routine (version 3.1.1) --
     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
     February 2007

  Purpose
  =======

  DSECND returns the user time for a process in seconds.
  This version gets the time from the EXTERNAL system function ETIME.

 =====================================================================


     .. Local Scalars ..

      • lsame

        public boolean lsame(java.lang.String ca,
                     java.lang.String cb)
        
     ..

  Purpose
  =======

  LSAME returns .TRUE. if CA is the same letter as CB regardless of
  case.

  Arguments
  =========

  CA      (input) CHARACTER*1
  CB      (input) CHARACTER*1
          CA and CB specify the single characters to be compared.

 =====================================================================


     .. Intrinsic Functions ..
        Parameters:
        ca -
        cb -

      • second

        public float second()
        

  -- LAPACK auxiliary routine (version 3.1.1) --
     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
     February 2007

  Purpose
  =======

  SECOND returns the user time for a process in seconds.
  This version gets the time from the EXTERNAL system function ETIME.

 =====================================================================


     .. Local Scalars ..

      • slamch

        public float slamch(java.lang.String cmach)
        
     ..

  Purpose
  =======

  SLAMCH determines single precision machine parameters.

  Arguments
  =========

  CMACH   (input) CHARACTER*1
          Specifies the value to be returned by SLAMCH:
          = 'E' or 'e',   SLAMCH := eps
          = 'S' or 's ,   SLAMCH := sfmin
          = 'B' or 'b',   SLAMCH := base
          = 'P' or 'p',   SLAMCH := eps*base
          = 'N' or 'n',   SLAMCH := t
          = 'R' or 'r',   SLAMCH := rnd
          = 'M' or 'm',   SLAMCH := emin
          = 'U' or 'u',   SLAMCH := rmin
          = 'L' or 'l',   SLAMCH := emax
          = 'O' or 'o',   SLAMCH := rmax

          where

          eps   = relative machine precision
          sfmin = safe minimum, such that 1/sfmin does not overflow
          base  = base of the machine
          prec  = eps*base
          t     = number of (base) digits in the mantissa
          rnd   = 1.0 when rounding occurs in addition, 0.0 otherwise
          emin  = minimum exponent before (gradual) underflow
          rmin  = underflow threshold - base**(emin-1)
          emax  = largest exponent before overflow
          rmax  = overflow threshold  - (base**emax)*(1-eps)

 =====================================================================


     .. Parameters ..
        Parameters:
        cmach -

      • slamc1

        public abstract void slamc1(org.netlib.util.intW beta,
                            org.netlib.util.intW t,
                            org.netlib.util.booleanW rnd,
                            org.netlib.util.booleanW ieee1)
        
     ..

  Purpose
  =======

  SLAMC1 determines the machine parameters given by BETA, T, RND, and
  IEEE1.

  Arguments
  =========

  BETA    (output) INTEGER
          The base of the machine.

  T       (output) INTEGER
          The number of ( BETA ) digits in the mantissa.

  RND     (output) LOGICAL
          Specifies whether proper rounding  ( RND = .TRUE. )  or
          chopping  ( RND = .FALSE. )  occurs in addition. This may not
          be a reliable guide to the way in which the machine performs

          its arithmetic.

  IEEE1   (output) LOGICAL
          Specifies whether rounding appears to be done in the IEEE
          'round to nearest' style.

  Further Details
  ===============

  The routine is based on the routine  ENVRON  by Malcolm and
  incorporates suggestions by Gentleman and Marovich. See

     Malcolm M. A. (1972) Algorithms to reveal properties of
        floating-point arithmetic. Comms. of the ACM, 15, 949-951.

     Gentleman W. M. and Marovich S. B. (1974) More on algorithms
        that reveal properties of floating point arithmetic units.
        Comms. of the ACM, 17, 276-277.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        beta -
        t -
        rnd -
        ieee1 -

      • slamc2

        public abstract void slamc2(org.netlib.util.intW beta,
                            org.netlib.util.intW t,
                            org.netlib.util.booleanW rnd,
                            org.netlib.util.floatW eps,
                            org.netlib.util.intW emin,
                            org.netlib.util.floatW rmin,
                            org.netlib.util.intW emax,
                            org.netlib.util.floatW rmax)
        
     ..

  Purpose
  =======

  SLAMC2 determines the machine parameters specified in its argument
  list.

  Arguments
  =========

  BETA    (output) INTEGER
          The base of the machine.

  T       (output) INTEGER
          The number of ( BETA ) digits in the mantissa.

  RND     (output) LOGICAL
          Specifies whether proper rounding  ( RND = .TRUE. )  or
          chopping  ( RND = .FALSE. )  occurs in addition. This may not
          be a reliable guide to the way in which the machine performs

          its arithmetic.

  EPS     (output) REAL
          The smallest positive number such that

             fl( 1.0 - EPS ) .LT. 1.0,

          where fl denotes the computed value.

  EMIN    (output) INTEGER
          The minimum exponent before (gradual) underflow occurs.

  RMIN    (output) REAL
          The smallest normalized number for the machine, given by
          BASE**( EMIN - 1 ), where  BASE  is the floating point value

          of BETA.

  EMAX    (output) INTEGER
          The maximum exponent before overflow occurs.

  RMAX    (output) REAL
          The largest positive number for the machine, given by
          BASE**EMAX * ( 1 - EPS ), where  BASE  is the floating point

          value of BETA.

  Further Details
  ===============

  The computation of  EPS  is based on a routine PARANOIA by
  W. Kahan of the University of California at Berkeley.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        beta -
        t -
        rnd -
        eps -
        emin -
        rmin -
        emax -
        rmax -

      • slamc3

        public float slamc3(float a,
                    float b)
        
     ..

  Purpose
  =======

  SLAMC3  is intended to force  A  and  B  to be stored prior to doing

  the addition of  A  and  B ,  for use in situations where optimizers

  might hold one of these in a register.

  Arguments
  =========

  A       (input) REAL
  B       (input) REAL
          The values A and B.

 =====================================================================


     .. Executable Statements ..
        Parameters:
        a -
        b -

      • slamc4

        public abstract void slamc4(org.netlib.util.intW emin,
                            float start,
                            int base)
        
     ..

  Purpose
  =======

  SLAMC4 is a service routine for SLAMC2.

  Arguments
  =========

  EMIN    (output) INTEGER 
          The minimum exponent before (gradual) underflow, computed by

          setting A = START and dividing by BASE until the previous A
          can not be recovered.

  START   (input) REAL
          The starting point for determining EMIN.

  BASE    (input) INTEGER
          The base of the machine.

 =====================================================================


     .. Local Scalars ..
        Parameters:
        emin -
        start -
        base -

      • slamc5

        public abstract void slamc5(int beta,
                            int p,
                            int emin,
                            boolean ieee,
                            org.netlib.util.intW emax,
                            org.netlib.util.floatW rmax)
        
     ..

  Purpose
  =======

  SLAMC5 attempts to compute RMAX, the largest machine floating-point
  number, without overflow.  It assumes that EMAX + abs(EMIN) sum
  approximately to a power of 2.  It will fail on machines where this
  assumption does not hold, for example, the Cyber 205 (EMIN = -28625,

  EMAX = 28718).  It will also fail if the value supplied for EMIN is
  too large (i.e. too close to zero), probably with overflow.

  Arguments
  =========

  BETA    (input) INTEGER
          The base of floating-point arithmetic.

  P       (input) INTEGER
          The number of base BETA digits in the mantissa of a
          floating-point value.

  EMIN    (input) INTEGER
          The minimum exponent before (gradual) underflow.

  IEEE    (input) LOGICAL
          A logical flag specifying whether or not the arithmetic
          system is thought to comply with the IEEE standard.

  EMAX    (output) INTEGER
          The largest exponent before overflow

  RMAX    (output) REAL
          The largest machine floating-point number.

 =====================================================================


     .. Parameters ..
        Parameters:
        beta -
        p -
        emin -
        ieee -
        emax -
        rmax -
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