/*
Copyright (C) 2008 Richard Gomes
This source code is release under the BSD License.
This file is part of JQuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://jquantlib.org/
JQuantLib is free software: you can redistribute it and/or modify it
under the terms of the JQuantLib license. You should have received a
copy of the license along with this program; if not, please email
. The license is also available online at
.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
JQuantLib is based on QuantLib. http://quantlib.org/
When applicable, the original copyright notice follows this notice.
*/
package org.jquantlib.math.matrixutilities;
import org.jquantlib.QL;
import org.jquantlib.lang.annotation.QualityAssurance;
import org.jquantlib.lang.annotation.QualityAssurance.Quality;
import org.jquantlib.lang.annotation.QualityAssurance.Version;
/**
* Eigenvalues and eigenvectors of a real matrix.
*
* If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. I.e. A =
* V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the identity matrix.
*
* If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex
* eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense
* that A*V = V*D, i.e. A.times(V) equals V.times(D). The matrix V may be badly conditioned, or even singular, so the validity of
* the equation A = V*D*inverse(V) depends upon V.cond().
*
* @note This class is adapted from JAMA
* @see JAMA
*
* @author Richard Gomes
*/
@QualityAssurance(quality = Quality.Q1_TRANSLATION, version = Version.OTHER, reviewers = { "Richard Gomes" })
public class EigenvalueDecomposition {
//
// private fields
//
/** Row and column dimension (square matrix). */
private final int n;
/** Symmetry flag. */
private boolean issymmetric;
/** Arrays for internal storage of eigenvalues. */
private final double[] d, e;
/** Array for internal storage of eigenvectors. */
private final Matrix V;
/** Array for internal storage of nonsymmetric Hessenberg form. */
private Matrix H;
/** Working storage for nonsymmetric algorithm. */
private double[] ort;
//
// private transient fields
//
private transient double cdivr, cdivi;
//
// public constructors
//
/**
* Check for symmetry, then construct the eigenvalue decomposition
*
* @param A is a square matrix
* @return Structure to access D and V.
*/
public EigenvalueDecomposition(final Matrix A) {
QL.require(A.rows() == A.cols(), Matrix.MATRIX_MUST_BE_SQUARE); // QA:[RG]::verified
this.n = A.cols();
V = new Matrix(n, n);
d = new double[n];
e = new double[n];
issymmetric = true;
for (int j = 0; (j < n) & issymmetric; j++) {
for (int i = 0; (i < n) & issymmetric; i++) {
issymmetric = (A.$[A.addr.op(i, j)] == A.$[A.addr.op(j, i)]);
}
}
if (issymmetric) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
V.$[V.addr.op(i, j)] = A.$[A.addr.op(i, j)];
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
H = new Matrix(n, n);
ort = new double[n];
for (int j = 0; j < n; j++) {
for (int i = 0; i < n; i++) {
H.$[H.addr.op(i, j)] = A.$[A.addr.op(i, j)];
}
}
// Reduce to Hessenberg form.
orthes();
// Reduce Hessenberg to real Schur form.
hqr2();
}
}
//
// public methods
//
/**
* Return the eigenvector matrix
*
* @return V
*/
public Matrix getV() {
return V.clone();
}
/**
* Return the real parts of the eigenvalues
*
* @return real(diag(D))
*/
public double[] getRealEigenvalues() {
return d;
}
/**
* Return the imaginary parts of the eigenvalues
*
* @return imag(diag(D))
*/
public double[] getImagEigenvalues() {
return e;
}
/**
* Return the block diagonal eigenvalue matrix
*
* @return D
*/
public Matrix getD() {
final Matrix D = new Matrix(n, n);
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
D.$[D.addr.op(i, j)] = 0.0;
}
D.$[D.addr.op(i, i)] = d[i];
if (e[i] > 0) {
D.$[D.addr.op(i, i + 1)] = e[i];
} else if (e[i] < 0) {
D.$[D.addr.op(i, i - 1)] = e[i];
}
}
return D;
}
//
// private methods
//
/**
* Symmetric Householder reduction to tridiagonal form.
*/
private void tred2() {
// This is derived from the Algol procedures tred2 by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int j = 0; j < n; j++) {
d[j] = V.$[V.addr.op(n - 1, j)];
}
// Householder reduction to tridiagonal form.
for (int i = n - 1; i > 0; i--) {
// Scale to avoid under/overflow.
double scale = 0.0;
double h = 0.0;
for (int k = 0; k < i; k++) {
scale = scale + Math.abs(d[k]);
}
if (scale == 0.0) {
e[i] = d[i - 1];
for (int j = 0; j < i; j++) {
d[j] = V.$[V.addr.op(i - 1, j)];
V.$[V.addr.op(i, j)] = 0.0;
V.$[V.addr.op(j, i)] = 0.0;
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
d[k] /= scale;
h += d[k] * d[k];
}
double f = d[i - 1];
double g = Math.sqrt(h);
if (f > 0) {
g = -g;
}
e[i] = scale * g;
h = h - f * g;
d[i - 1] = f - g;
for (int j = 0; j < i; j++) {
e[j] = 0.0;
}
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) {
f = d[j];
V.$[V.addr.op(j, i)] = f;
g = e[j] + V.$[V.addr.op(j, j)] * f;
for (int k = j + 1; k <= i - 1; k++) {
g += V.$[V.addr.op(k, j)] * d[k];
e[k] += V.$[V.addr.op(k, j)] * f;
}
e[j] = g;
}
f = 0.0;
for (int j = 0; j < i; j++) {
e[j] /= h;
f += e[j] * d[j];
}
final double hh = f / (h + h);
for (int j = 0; j < i; j++) {
e[j] -= hh * d[j];
}
for (int j = 0; j < i; j++) {
f = d[j];
g = e[j];
for (int k = j; k <= i - 1; k++) {
V.$[V.addr.op(k, j)] -= (f * e[k] + g * d[k]);
}
d[j] = V.$[V.addr.op(i - 1, j)];
V.$[V.addr.op(i, j)] = 0.0;
}
}
d[i] = h;
}
// Accumulate transformations.
for (int i = 0; i < n - 1; i++) {
V.$[V.addr.op(n - 1, i)] = V.$[V.addr.op(i, i)];
V.$[V.addr.op(i, i)] = 1.0;
final double h = d[i + 1];
if (h != 0.0) {
for (int k = 0; k <= i; k++) {
d[k] = V.$[V.addr.op(k, i + 1)] / h;
}
for (int j = 0; j <= i; j++) {
double g = 0.0;
for (int k = 0; k <= i; k++) {
g += V.$[V.addr.op(k, i + 1)] * V.$[V.addr.op(k, j)];
}
for (int k = 0; k <= i; k++) {
V.$[V.addr.op(k, j)] -= g * d[k];
}
}
}
for (int k = 0; k <= i; k++) {
V.$[V.addr.op(k, i + 1)] = 0.0;
}
}
for (int j = 0; j < n; j++) {
d[j] = V.$[V.addr.op(n - 1, j)];
V.$[V.addr.op(n - 1, j)] = 0.0;
}
V.$[V.addr.op(n - 1, n - 1)] = 1.0;
e[0] = 0.0;
}
/**
* Symmetric tridiagonal QL algorithm.
*/
private void tql2() {
// This is derived from the Algol procedures tql2, by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int i = 1; i < n; i++) {
e[i - 1] = e[i];
}
e[n - 1] = 0.0;
double f = 0.0;
double tst1 = 0.0;
final double eps = Math.pow(2.0, -52.0);
for (int l = 0; l < n; l++) {
// Find small subdiagonal element
tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
int m = l;
while (m < n) {
if (Math.abs(e[m]) <= eps * tst1) {
break;
}
m++;
}
// If m == l, d[l] is an eigenvalue,
// otherwise, iterate.
if (m > l) {
int iter = 0;
do {
iter = iter + 1; // (Could check iteration count here.)
// Compute implicit shift
double g = d[l];
double p = (d[l + 1] - g) / (2.0 * e[l]);
double r = Math.hypot(p, 1.0);
if (p < 0) {
r = -r;
}
d[l] = e[l] / (p + r);
d[l + 1] = e[l] * (p + r);
final double dl1 = d[l + 1];
double h = g - d[l];
for (int i = l + 2; i < n; i++) {
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
final double el1 = e[l + 1];
double s = 0.0;
double s2 = 0.0;
for (int i = m - 1; i >= l; i--) {
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = Math.hypot(p, e[i]);
e[i + 1] = s * r;
s = e[i] / r;
c = p / r;
p = c * d[i] - s * g;
d[i + 1] = h + s * (c * g + s * d[i]);
// Accumulate transformation.
for (int k = 0; k < n; k++) {
h = V.$[V.addr.op(k, i + 1)];
V.$[V.addr.op(k, i + 1)] = s * V.$[V.addr.op(k, i)] + c * h;
V.$[V.addr.op(k, i)] = c * V.$[V.addr.op(k, i)] - s * h;
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
// Check for convergence.
} while (Math.abs(e[l]) > eps * tst1);
}
d[l] = d[l] + f;
e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n - 1; i++) {
int k = i;
double p = d[i];
for (int j = i + 1; j < n; j++) {
if (d[j] < p) {
k = j;
p = d[j];
}
}
if (k != i) {
d[k] = d[i];
d[i] = p;
for (int j = 0; j < n; j++) {
p = V.$[V.addr.op(j, i)];
V.$[V.addr.op(j, i)] = V.$[V.addr.op(j, k)];
V.$[V.addr.op(j, k)] = p;
}
}
}
}
/**
* Nonsymmetric reduction to Hessenberg form.
*/
private void orthes() {
// This is derived from the Algol procedures orthes and ortran,
// by Martin and Wilkinson, Handbook for Auto. Comp.,
// Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutines in EISPACK.
final int low = 0;
final int high = n - 1;
for (int m = low + 1; m <= high - 1; m++) {
// Scale column.
double scale = 0.0;
for (int i = m; i <= high; i++) {
scale = scale + Math.abs(H.$[H.addr.op(i, m - 1)]);
}
if (scale != 0.0) {
// Compute Householder transformation.
double h = 0.0;
for (int i = high; i >= m; i--) {
ort[i] = H.$[H.addr.op(i, m - 1)] / scale;
h += ort[i] * ort[i];
}
double g = Math.sqrt(h);
if (ort[m] > 0) {
g = -g;
}
h = h - ort[m] * g;
ort[m] = ort[m] - g;
// Apply Householder similarity transformation
// H = (I-u*u'/h)*H*(I-u*u')/h)
for (int j = m; j < n; j++) {
double f = 0.0;
for (int i = high; i >= m; i--) {
f += ort[i] * H.$[H.addr.op(i, j)];
}
f = f / h;
for (int i = m; i <= high; i++) {
H.$[H.addr.op(i, j)] -= f * ort[i];
}
}
for (int i = 0; i <= high; i++) {
double f = 0.0;
for (int j = high; j >= m; j--) {
f += ort[j] * H.$[H.addr.op(i, j)];
}
f = f / h;
for (int j = m; j <= high; j++) {
H.$[H.addr.op(i, j)] -= f * ort[j];
}
}
ort[m] = scale * ort[m];
H.$[H.addr.op(m, m - 1)] = scale * g;
}
}
// Accumulate transformations (Algol's ortran).
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
V.$[V.addr.op(i, j)] = (i == j ? 1.0 : 0.0);
}
}
for (int m = high - 1; m >= low + 1; m--) {
if (H.$[H.addr.op(m, m - 1)] != 0.0) {
for (int i = m + 1; i <= high; i++) {
ort[i] = H.$[H.addr.op(i, m - 1)];
}
for (int j = m; j <= high; j++) {
double g = 0.0;
for (int i = m; i <= high; i++) {
g += ort[i] * V.$[V.addr.op(i, j)];
}
// Double division avoids possible underflow
g = (g / ort[m]) / H.$[H.addr.op(m, m - 1)];
for (int i = m; i <= high; i++) {
V.$[V.addr.op(i, j)] += g * ort[i];
}
}
}
}
}
/**
* Complex scalar division.
*
* @param xr
* @param xi
* @param yr
* @param yi
*/
private void cdiv(final double xr, final double xi, final double yr, final double yi) {
double r, d;
if (Math.abs(yr) > Math.abs(yi)) {
r = yi / yr;
d = yr + r * yi;
cdivr = (xr + r * xi) / d;
cdivi = (xi - r * xr) / d;
} else {
r = yr / yi;
d = yi + r * yr;
cdivr = (r * xr + xi) / d;
cdivi = (r * xi - xr) / d;
}
}
// Nonsymmetric reduction from Hessenberg to real Schur form.
private void hqr2() {
// This is derived from the Algol procedure hqr2,
// by Martin and Wilkinson, Handbook for Auto. Comp.,
// Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
// Initialize
final int nn = this.n;
int n = nn - 1;
final int low = 0;
final int high = nn - 1;
final double eps = Math.pow(2.0, -52.0);
double exshift = 0.0;
double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
// Store roots isolated by balanc and compute matrix norm
double norm = 0.0;
for (int i = 0; i < nn; i++) {
if (i < low | i > high) {
d[i] = H.$[H.addr.op(i, i)];
e[i] = 0.0;
}
for (int j = Math.max(i - 1, 0); j < nn; j++) {
norm = norm + Math.abs(H.$[H.addr.op(i, j)]);
}
}
// Outer loop over eigenvalue index
int iter = 0;
while (n >= low) {
// Look for single small sub-diagonal element
int l = n;
while (l > low) {
s = Math.abs(H.$[H.addr.op(l - 1, l - 1)]) + Math.abs(H.$[H.addr.op(l, l)]);
if (s == 0.0) {
s = norm;
}
if (Math.abs(H.$[H.addr.op(l, l - 1)]) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n) {
H.$[H.addr.op(n, n)] = H.$[H.addr.op(n, n)] + exshift;
d[n] = H.$[H.addr.op(n, n)];
e[n] = 0.0;
n--;
iter = 0;
// Two roots found
} else if (l == n - 1) {
w = H.$[H.addr.op(n, n - 1)] * H.$[H.addr.op(n - 1, n)];
p = (H.$[H.addr.op(n - 1, n - 1)] - H.$[H.addr.op(n, n)]) / 2.0;
q = p * p + w;
z = Math.sqrt(Math.abs(q));
H.$[H.addr.op(n, n)] = H.$[H.addr.op(n, n)] + exshift;
H.$[H.addr.op(n - 1, n - 1)] = H.$[H.addr.op(n - 1, n - 1)] + exshift;
x = H.$[H.addr.op(n, n)];
// Real pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
d[n - 1] = x + z;
d[n] = d[n - 1];
if (z != 0.0) {
d[n] = x - w / z;
}
e[n - 1] = 0.0;
e[n] = 0.0;
x = H.$[H.addr.op(n, n - 1)];
s = Math.abs(x) + Math.abs(z);
p = x / s;
q = z / s;
r = Math.sqrt(p * p + q * q);
p = p / r;
q = q / r;
// Row modification
for (int j = n - 1; j < nn; j++) {
z = H.$[H.addr.op(n - 1, j)];
H.$[H.addr.op(n - 1, j)] = q * z + p * H.$[H.addr.op(n, j)];
H.$[H.addr.op(n, j)] = q * H.$[H.addr.op(n, j)] - p * z;
}
// Column modification
for (int i = 0; i <= n; i++) {
z = H.$[H.addr.op(i, n - 1)];
H.$[H.addr.op(i, n - 1)] = q * z + p * H.$[H.addr.op(i, n)];
H.$[H.addr.op(i, n)] = q * H.$[H.addr.op(i, n)] - p * z;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = V.$[V.addr.op(i, n - 1)];
V.$[V.addr.op(i, n - 1)] = q * z + p * V.$[V.addr.op(i, n)];
V.$[V.addr.op(i, n)] = q * V.$[V.addr.op(i, n)] - p * z;
}
// Complex pair
} else {
d[n - 1] = x + p;
d[n] = x + p;
e[n - 1] = z;
e[n] = -z;
}
n = n - 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = H.$[H.addr.op(n, n)];
y = 0.0;
w = 0.0;
if (l < n) {
y = H.$[H.addr.op(n - 1, n - 1)];
w = H.$[H.addr.op(n, n - 1)] * H.$[H.addr.op(n - 1, n)];
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n; i++) {
H.$[H.addr.op(i, i)] -= x;
}
s = Math.abs(H.$[H.addr.op(n, n - 1)]) + Math.abs(H.$[H.addr.op(n - 1, n - 2)]);
x = y = 0.75 * s;
w = -0.4375 * s * s;
}
// MATLAB's new ad hoc shift
if (iter == 30) {
s = (y - x) / 2.0;
s = s * s + w;
if (s > 0) {
s = Math.sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n; i++) {
H.$[H.addr.op(i, i)] -= s;
}
exshift += s;
x = y = w = 0.964;
}
}
iter = iter + 1; // (Could check iteration count here.)
// Look for two consecutive small sub-diagonal elements
int m = n - 2;
while (m >= l) {
z = H.$[H.addr.op(m, m)];
r = x - z;
s = y - z;
p = (r * s - w) / H.$[H.addr.op(m + 1, m)] + H.$[H.addr.op(m, m + 1)];
q = H.$[H.addr.op(m + 1, m + 1)] - z - r - s;
r = H.$[H.addr.op(m + 2, m + 1)];
s = Math.abs(p) + Math.abs(q) + Math.abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m == l) {
break;
}
if (Math.abs(H.$[H.addr.op(m, m - 1)]) * (Math.abs(q) + Math.abs(r)) < eps
* (Math.abs(p) * (Math.abs(H.$[H.addr.op(m - 1, m - 1)]) + Math.abs(z) + Math.abs(H.$[H.addr.op(
m + 1, m + 1)])))) {
break;
}
m--;
}
for (int i = m + 2; i <= n; i++) {
H.$[H.addr.op(i, i - 2)] = 0.0;
if (i > m + 2) {
H.$[H.addr.op(i, i - 3)] = 0.0;
}
}
// Double QR step involving rows l:n and columns m:n
for (int k = m; k <= n - 1; k++) {
final boolean notlast = (k != n - 1);
if (k != m) {
p = H.$[H.addr.op(k, k - 1)];
q = H.$[H.addr.op(k + 1, k - 1)];
r = (notlast ? H.$[H.addr.op(k + 2, k - 1)] : 0.0);
x = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x != 0.0) {
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0.0) {
break;
}
s = Math.sqrt(p * p + q * q + r * r);
if (p < 0) {
s = -s;
}
if (s != 0) {
if (k != m) {
H.$[H.addr.op(k, k - 1)] = -s * x;
} else if (l != m) {
H.$[H.addr.op(k, k - 1)] = -H.$[H.addr.op(k, k - 1)];
}
p = p + s;
x = p / s;
y = q / s;
z = r / s;
q = q / p;
r = r / p;
// Row modification
for (int j = k; j < nn; j++) {
p = H.$[H.addr.op(k, j)] + q * H.$[H.addr.op(k + 1, j)];
if (notlast) {
p = p + r * H.$[H.addr.op(k + 2, j)];
H.$[H.addr.op(k + 2, j)] = H.$[H.addr.op(k + 2, j)] - p * z;
}
H.$[H.addr.op(k, j)] = H.$[H.addr.op(k, j)] - p * x;
H.$[H.addr.op(k + 1, j)] = H.$[H.addr.op(k + 1, j)] - p * y;
}
// Column modification
for (int i = 0; i <= Math.min(n, k + 3); i++) {
p = x * H.$[H.addr.op(i, k)] + y * H.$[H.addr.op(i, k + 1)];
if (notlast) {
p = p + z * H.$[H.addr.op(i, k + 2)];
H.$[H.addr.op(i, k + 2)] = H.$[H.addr.op(i, k + 2)] - p * r;
}
H.$[H.addr.op(i, k)] = H.$[H.addr.op(i, k)] - p;
H.$[H.addr.op(i, k + 1)] = H.$[H.addr.op(i, k + 1)] - p * q;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * V.$[V.addr.op(i, k)] + y * V.$[V.addr.op(i, k + 1)];
if (notlast) {
p = p + z * V.$[V.addr.op(i, k + 2)];
V.$[V.addr.op(i, k + 2)] = V.$[V.addr.op(i, k + 2)] - p * r;
}
V.$[V.addr.op(i, k)] = V.$[V.addr.op(i, k)] - p;
V.$[V.addr.op(i, k + 1)] = V.$[V.addr.op(i, k + 1)] - p * q;
}
} // (s != 0)
} // k loop
} // check convergence
} // while (n >= low)
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0)
return;
for (n = nn - 1; n >= 0; n--) {
p = d[n];
q = e[n];
// Real vector
if (q == 0) {
int l = n;
H.$[H.addr.op(n, n)] = 1.0;
for (int i = n - 1; i >= 0; i--) {
w = H.$[H.addr.op(i, i)] - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r = r + H.$[H.addr.op(i, j)] * H.$[H.addr.op(j, n)];
}
if (e[i] < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e[i] == 0.0) {
if (w != 0.0) {
H.$[H.addr.op(i, n)] = -r / w;
} else {
H.$[H.addr.op(i, n)] = -r / (eps * norm);
}
// Solve real equations
} else {
x = H.$[H.addr.op(i, i + 1)];
y = H.$[H.addr.op(i + 1, i)];
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
t = (x * s - z * r) / q;
H.$[H.addr.op(i, n)] = t;
if (Math.abs(x) > Math.abs(z)) {
H.$[H.addr.op(i + 1, n)] = (-r - w * t) / x;
} else {
H.$[H.addr.op(i + 1, n)] = (-s - y * t) / z;
}
}
// Overflow control
t = Math.abs(H.$[H.addr.op(i, n)]);
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.$[H.addr.op(j, n)] = H.$[H.addr.op(j, n)] / t;
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n - 1;
// Last vector component imaginary so matrix is triangular
if (Math.abs(H.$[H.addr.op(n, n - 1)]) > Math.abs(H.$[H.addr.op(n - 1, n)])) {
H.$[H.addr.op(n - 1, n - 1)] = q / H.$[H.addr.op(n, n - 1)];
H.$[H.addr.op(n - 1, n)] = -(H.$[H.addr.op(n, n)] - p) / H.$[H.addr.op(n, n - 1)];
} else {
cdiv(0.0, -H.$[H.addr.op(n - 1, n)], H.$[H.addr.op(n - 1, n - 1)] - p, q);
H.$[H.addr.op(n - 1, n - 1)] = cdivr;
H.$[H.addr.op(n - 1, n)] = cdivi;
}
H.$[H.addr.op(n, n - 1)] = 0.0;
H.$[H.addr.op(n, n)] = 1.0;
for (int i = n - 2; i >= 0; i--) {
double ra, sa, vr, vi;
ra = 0.0;
sa = 0.0;
for (int j = l; j <= n; j++) {
ra = ra + H.$[H.addr.op(i, j)] * H.$[H.addr.op(j, n - 1)];
sa = sa + H.$[H.addr.op(i, j)] * H.$[H.addr.op(j, n)];
}
w = H.$[H.addr.op(i, i)] - p;
if (e[i] < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e[i] == 0) {
cdiv(-ra, -sa, w, q);
H.$[H.addr.op(i, n - 1)] = cdivr;
H.$[H.addr.op(i, n)] = cdivi;
} else {
// Solve complex equations
x = H.$[H.addr.op(i, i + 1)];
y = H.$[H.addr.op(i + 1, i)];
vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
vi = (d[i] - p) * 2.0 * q;
if (vr == 0.0 & vi == 0.0) {
vr = eps * norm * (Math.abs(w) + Math.abs(q) + Math.abs(x) + Math.abs(y) + Math.abs(z));
}
cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
H.$[H.addr.op(i, n - 1)] = cdivr;
H.$[H.addr.op(i, n)] = cdivi;
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
H.$[H.addr.op(i + 1, n - 1)] = (-ra - w * H.$[H.addr.op(i, n - 1)] + q
* H.$[H.addr.op(i, n)])
/ x;
H.$[H.addr.op(i + 1, n)] = (-sa - w * H.$[H.addr.op(i, n)] - q * H.$[H.addr.op(i, n - 1)])
/ x;
} else {
cdiv(-r - y * H.$[H.addr.op(i, n - 1)], -s - y * H.$[H.addr.op(i, n)], z, q);
H.$[H.addr.op(i + 1, n - 1)] = cdivr;
H.$[H.addr.op(i + 1, n)] = cdivi;
}
}
// Overflow control
t = Math.max(Math.abs(H.$[H.addr.op(i, n - 1)]), Math.abs(H.$[H.addr.op(i, n)]));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.$[H.addr.op(j, n - 1)] = H.$[H.addr.op(j, n - 1)] / t;
H.$[H.addr.op(j, n)] = H.$[H.addr.op(j, n)] / t;
}
}
}
}
}
}
// Vectors of isolated roots
for (int i = 0; i < nn; i++) {
if (i < low | i > high) {
for (int j = i; j < nn; j++) {
V.$[V.addr.op(i, j)] = H.$[H.addr.op(i, j)];
}
}
}
// Back transformation to get eigenvectors of original matrix
for (int j = nn - 1; j >= low; j--) {
for (int i = low; i <= high; i++) {
z = 0.0;
for (int k = low; k <= Math.min(j, high); k++) {
z = z + V.$[V.addr.op(i, k)] * H.$[H.addr.op(k, j)];
}
V.$[V.addr.op(i, j)] = z;
}
}
}
}
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