jvx.numeric

Class PnMatrix

java.lang.Object

jvx.numeric.PnMatrix

public class PnMatrix
extends java.lang.Object

Numerical routines for bigger matrix linear algebra problems. Methods are taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.

See Also:

PdMatrix

Author:

Konrad Polthier

Version:

21.02.05, 1.30 revised (ep) New method tridag.
11.01.05, 1.20 revised (ep) New methods for solving symmetric pentadiagonal systems.
20.07.99, 1.00 revised (kp) All array indices start at 0.
20.07.99, 1.00 created (kp)

Constructor Summary

Constructors

Constructor and Description
PnMatrix()

Method Summary

Modifier and Type	Method and Description
static boolean	cyclic(double[] a, double[] b, double[] c, double[] r, double[] x, double[] bb, double[] u, double[] z, double[] tmp, int n, double eps) Solves the equation Au=r, where A is a cyclic tridiagonal matrix with diagonal b, lower offdiagonal a and upper offdiagonal c.
static double	determinant(double[][] aIn, int n) Compute determinant of a given NxN matrix A.
static boolean	invert(double[][] y, double[][] aIn, int n) Invert a given NxN matrix A and return its invers.
static void	lubksb(double[][] a, int n, int[] indx, double[] b) Solves the set of N linear equations AX=B.
static double	ludcmp(double[][] a, int n, int[] indx) Given an NxN matrix A this routine replaces it by the LU decomposition of a rowwise permutation of itself.
static void	main(java.lang.String[] args) Test routine for matrix inversion.
static boolean	penpes(int n, double[] a, double[] b, double[] c, double[] f, double[] d, double[] e, double[] h, double eps) Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix with additional perodicity coefficients (1,n-1),(1,n) and (2,n).
static boolean	pentas(int n, double[] a, double[] b, double[] c, double[] f, double eps) Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix.
static boolean	tridag(double[] a, double[] b, double[] c, double[] r, double[] u, double[] tmp, int n, double eps) Solves the equation Au=r, where A is a tridiagonal matrix with diagonal b, lower offdiagonal a and upper offdiagonal c.

Methods inherited from class java.lang.Object

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

Constructor Detail

PnMatrix

public PnMatrix()

Method Detail

main

public static void main(java.lang.String[] args)

Test routine for matrix inversion.

invert

public static boolean invert(double[][] y,
                             double[][] aIn,
                             int n)

Invert a given NxN matrix A and return its invers. Apply LU decomposition. A and N are input. Y is output containing the invers of matrix A. Input matrix is not modified since copied to a temporary matrix. All array indices start with 0 rather than the standard fortran 1.

Returns:

false if A was singular and therefore LU failed.

See Also:

ludcmp(double[][], int, int[]), lubksb(double[][], int, int[], double[])

Author:

Konrad Polthier

Version:

08.09.99, 1.20 revised (kp) Keep input matrix untouched.
07.09.99, 1.10 revised (kp) Return false instead of throwing ArithmeticException.
20.07.99, 1.00 created (kp)

determinant

public static double determinant(double[][] aIn,
                                 int n)

Compute determinant of a given NxN matrix A. Apply LU decomposition. A and N are input. Y is output containing the invers of matrix A. Input matrix is not modified since copied to a temporary matrix.

All array indices start with 0 rather than the standard fortran 1.

Returns:

determinant, or 0. if matrix is singular.

See Also:

ludcmp(double[][], int, int[]), lubksb(double[][], int, int[], double[])

Author:

Konrad Polthier

Version:

15.04.06, 1.20 revised (kp) Singular matrix just returns 0, but issues no warning.
08.09.99, 1.10 revised (kp) Keep input matrix untouched.
20.07.99, 1.00 created (kp)

ludcmp

public static double ludcmp(double[][] a,
                            int n,
                            int[] indx)
                     throws java.lang.ArithmeticException

Given an NxN matrix A this routine replaces it by the LU decomposition of a rowwise permutation of itself. A and N are input. A is output, arranged as in equation (2.3.14) described in Numerical Recipes; INDX is an output vector which record the row permutation effected by the partial pivoting; the return value is +/-1 depending on whether the number of row interchanges was even (1.) or odd (-1.). This routine is used in combination with LUBKSB to solve linear equations or invert a matrix.

This version is taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.

Throws:

java.lang.ArithmeticException - if A was singular.

See Also:

lubksb(double[][], int, int[], double[])

Author:

Numerical Recipes, changed to Java and adapted by Konrad Polthier

Version:

20.07.99, 1.00 revised (kp) All array indices start at 0.
20.07.99, 1.00 created (kp)

lubksb

public static void lubksb(double[][] a,
                          int n,
                          int[] indx,
                          double[] b)

Solves the set of N linear equations AX=B. Here A is input, not as the matrix A but rather as its LU decomposition, determined by the routine LUDCMP. INDX is input as the permutation vector returned by LUDCMP. B is input as the right-hand side vector B, and returns with the solution vector X. A, N, NP and INDX are not modified by the routine and can be left in place for successive calls with different right-hand sides B. This routine takes into account the possibility that B will begin with many zero elements, so it is efficient for the use in matrix inversion.

This version is taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.

See Also:

ludcmp(double[][], int, int[])

Author:

Numerical Recipes, changed to Java and adapted by Konrad Polthier

Version:

20.07.99, 1.00 revised (kp) All array indices start at 0.
20.07.99, 1.00 created (kp)

pentas

public static final boolean pentas(int n,
                                   double[] a,
                                   double[] b,
                                   double[] c,
                                   double[] f,
                                   double eps)

Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix. The arrays a, b, c and f are modified by the routine.

Parameters:

n - dimension of matrix; must be greater or equal to 3

a - diagonal of length n

b - 1st side diagonal of length n-1 (but array must be of length n)

c - 2nd side diagonal of length n-2 (but array must be of length n)

f - input: right hand side; output: solution

eps - precision; recommended is 10^(-t+2) where t is machine precision

Returns:

true at success; false if matrix is (numerically) singular

Author:

From 'Eindimensionale Spline-Interpolations-Algorithmen, H. Späth', ported to Java by Eike Preuss

penpes

public static final boolean penpes(int n,
                                   double[] a,
                                   double[] b,
                                   double[] c,
                                   double[] f,
                                   double[] d,
                                   double[] e,
                                   double[] h,
                                   double eps)

Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix with additional perodicity coefficients (1,n-1),(1,n) and (2,n). The arrays a, b, c and f are modified by the routine.

Parameters:

n - dimension of matrix; must be greater or equal to 5

a - diagonal of length n

b - 1st side diagonal of length n-1; and b[n-1] must contain matrix element (1,n)

c - 2nd side diagonal of length n-2; and b[n-2],b[n-1] must contain matrix elements (1,n-1),(2,n)

f - input: right hand side; output: solution

d - helper array of length n

e - helper array of length n

h - helper array of length n

eps - precision; recommended is 10^(-t+2) where t is machine precision

Returns:

true at success; false if matrix is (numerically) singular

Author:

From 'Eindimensionale Spline-Interpolations-Algorithmen, H. Späth', ported to Java by Eike Preuss

tridag

public static boolean tridag(double[] a,
                             double[] b,
                             double[] c,
                             double[] r,
                             double[] u,
                             double[] tmp,
                             int n,
                             double eps)

Solves the equation Au=r, where A is a tridiagonal matrix with diagonal b, lower offdiagonal a and upper offdiagonal c.

Input vectors a, b, c and r are not modified. The method only succeeds if no pivoting is required!

Parameters:

a - lower offdiagonal; only the entries 1,...,n-1 are used

b - diagonal

c - upper offdiagonal; only the entries 0,...,n-2 are used

r - right hand side

u - output: solution

tmp - temporary vector of size n; may be null

n - dimension of matrix/vectors

eps - precision

Returns:

true at success

Version:

21.02.05, 1.00 created (ep) Created and adapted from numerical recipes (2.4).

cyclic

public static boolean cyclic(double[] a,
                             double[] b,
                             double[] c,
                             double[] r,
                             double[] x,
                             double[] bb,
                             double[] u,
                             double[] z,
                             double[] tmp,
                             int n,
                             double eps)

Solves the equation Au=r, where A is a cyclic tridiagonal matrix with diagonal b, lower offdiagonal a and upper offdiagonal c.

Input vectors a, b, c and r are not modified. The method only succeeds if no pivoting is required!

Parameters:

a - lower offdiagonal; contains the element in upper right corner as a[0]

b - diagonal

c - upper offdiagonal; contains the element in lower left corner as c[n-1]

r - right hand side

x - output: solution

bb - temporary vector of size n; may be null

u - temporary vector of size n; may be null

z - temporary vector of size n; may be null

tmp - temporary vector of size n; may be null

n - dimension of matrix/vectors

eps - precision

Returns:

true at success

Version:

22.02.05, 1.00 created (ep) Created and adapted from numerical recipes (2.7).