/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/**
* Adapted from the public domain Jama code.
*/
package org.apache.mahout.math.solver;
import org.apache.mahout.math.DenseMatrix;
import org.apache.mahout.math.DenseVector;
import org.apache.mahout.math.Matrix;
import org.apache.mahout.math.Vector;
import org.apache.mahout.math.function.Functions;
/**
* Eigenvalues and eigenvectors of a real matrix.
* <p/>
* 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.
* <p/>
* 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().
*/
public class EigenDecomposition {
/** Row and column dimension (square matrix). */
private final int n;
/** Arrays for internal storage of eigenvalues. */
private final Vector d;
private final Vector e;
/** Array for internal storage of eigenvectors. */
private final Matrix v;
public EigenDecomposition(Matrix x) {
this(x, isSymmetric(x));
}
public EigenDecomposition(Matrix x, boolean isSymmetric) {
n = x.columnSize();
d = new DenseVector(n);
e = new DenseVector(n);
v = new DenseMatrix(n, n);
if (isSymmetric) {
v.assign(x);
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
// Reduce to Hessenberg form.
// Reduce Hessenberg to real Schur form.
hqr2(orthes(x));
}
}
/**
* Return the eigenvector matrix
*
* @return V
*/
public Matrix getV() {
return v.like().assign(v);
}
/**
* Return the real parts of the eigenvalues
*/
public Vector getRealEigenvalues() {
return d;
}
/**
* Return the imaginary parts of the eigenvalues
*/
public Vector getImagEigenvalues() {
return e;
}
/**
* Return the block diagonal eigenvalue matrix
*
* @return D
*/
public Matrix getD() {
Matrix x = new DenseMatrix(n, n);
x.assign(0);
x.viewDiagonal().assign(d);
for (int i = 0; i < n; i++) {
double v = e.getQuick(i);
if (v > 0) {
x.setQuick(i, i + 1, v);
} else if (v < 0) {
x.setQuick(i, i - 1, v);
}
}
return x;
}
// 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.
d.assign(v.viewColumn(n - 1));
// Householder reduction to tridiagonal form.
for (int i = n - 1; i > 0; i--) {
// Scale to avoid under/overflow.
double scale = d.viewPart(0, i).norm(1);
double h = 0.0;
if (scale == 0.0) {
e.setQuick(i, d.getQuick(i - 1));
for (int j = 0; j < i; j++) {
d.setQuick(j, v.getQuick(i - 1, j));
v.setQuick(i, j, 0.0);
v.setQuick(j, i, 0.0);
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
d.setQuick(k, d.getQuick(k) / scale);
h += d.getQuick(k) * d.getQuick(k);
}
double f = d.getQuick(i - 1);
double g = Math.sqrt(h);
if (f > 0) {
g = -g;
}
e.setQuick(i, scale * g);
h -= f * g;
d.setQuick(i - 1, f - g);
for (int j = 0; j < i; j++) {
e.setQuick(j, 0.0);
}
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) {
f = d.getQuick(j);
v.setQuick(j, i, f);
g = e.getQuick(j) + v.getQuick(j, j) * f;
for (int k = j + 1; k <= i - 1; k++) {
g += v.getQuick(k, j) * d.getQuick(k);
e.setQuick(k, e.getQuick(k) + v.getQuick(k, j) * f);
}
e.setQuick(j, g);
}
f = 0.0;
for (int j = 0; j < i; j++) {
e.setQuick(j, e.getQuick(j) / h);
f += e.getQuick(j) * d.getQuick(j);
}
double hh = f / (h + h);
for (int j = 0; j < i; j++) {
e.setQuick(j, e.getQuick(j) - hh * d.getQuick(j));
}
for (int j = 0; j < i; j++) {
f = d.getQuick(j);
g = e.getQuick(j);
for (int k = j; k <= i - 1; k++) {
v.setQuick(k, j, v.getQuick(k, j) - (f * e.getQuick(k) + g * d.getQuick(k)));
}
d.setQuick(j, v.getQuick(i - 1, j));
v.setQuick(i, j, 0.0);
}
}
d.setQuick(i, h);
}
// Accumulate transformations.
for (int i = 0; i < n - 1; i++) {
v.setQuick(n - 1, i, v.getQuick(i, i));
v.setQuick(i, i, 1.0);
double h = d.getQuick(i + 1);
if (h != 0.0) {
for (int k = 0; k <= i; k++) {
d.setQuick(k, v.getQuick(k, i + 1) / h);
}
for (int j = 0; j <= i; j++) {
double g = 0.0;
for (int k = 0; k <= i; k++) {
g += v.getQuick(k, i + 1) * v.getQuick(k, j);
}
for (int k = 0; k <= i; k++) {
v.setQuick(k, j, v.getQuick(k, j) - g * d.getQuick(k));
}
}
}
for (int k = 0; k <= i; k++) {
v.setQuick(k, i + 1, 0.0);
}
}
d.assign(v.viewRow(n - 1));
v.viewRow(n - 1).assign(0);
v.setQuick(n - 1, n - 1, 1.0);
e.setQuick(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.
e.viewPart(0, n - 1).assign(e.viewPart(1, n - 1));
e.setQuick(n - 1, 0.0);
double f = 0.0;
double tst1 = 0.0;
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.getQuick(l)) + Math.abs(e.getQuick(l)));
int m = l;
while (m < n) {
if (Math.abs(e.getQuick(m)) <= eps * tst1) {
break;
}
m++;
}
// If m == l, d.getQuick(l) is an eigenvalue,
// otherwise, iterate.
if (m > l) {
do {
// Compute implicit shift
double g = d.getQuick(l);
double p = (d.getQuick(l + 1) - g) / (2.0 * e.getQuick(l));
double r = Math.hypot(p, 1.0);
if (p < 0) {
r = -r;
}
d.setQuick(l, e.getQuick(l) / (p + r));
d.setQuick(l + 1, e.getQuick(l) * (p + r));
double dl1 = d.getQuick(l + 1);
double h = g - d.getQuick(l);
for (int i = l + 2; i < n; i++) {
d.setQuick(i, d.getQuick(i) - h);
}
f += h;
// Implicit QL transformation.
p = d.getQuick(m);
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e.getQuick(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.getQuick(i);
h = c * p;
r = Math.hypot(p, e.getQuick(i));
e.setQuick(i + 1, s * r);
s = e.getQuick(i) / r;
c = p / r;
p = c * d.getQuick(i) - s * g;
d.setQuick(i + 1, h + s * (c * g + s * d.getQuick(i)));
// Accumulate transformation.
for (int k = 0; k < n; k++) {
h = v.getQuick(k, i + 1);
v.setQuick(k, i + 1, s * v.getQuick(k, i) + c * h);
v.setQuick(k, i, c * v.getQuick(k, i) - s * h);
}
}
p = -s * s2 * c3 * el1 * e.getQuick(l) / dl1;
e.setQuick(l, s * p);
d.setQuick(l, c * p);
// Check for convergence.
} while (Math.abs(e.getQuick(l)) > eps * tst1);
}
d.setQuick(l, d.getQuick(l) + f);
e.setQuick(l, 0.0);
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n - 1; i++) {
int k = i;
double p = d.getQuick(i);
for (int j = i + 1; j < n; j++) {
if (d.getQuick(j) < p) {
k = j;
p = d.getQuick(j);
}
}
if (k != i) {
d.setQuick(k, d.getQuick(i));
d.setQuick(i, p);
for (int j = 0; j < n; j++) {
p = v.getQuick(j, i);
v.setQuick(j, i, v.getQuick(j, k));
v.setQuick(j, k, p);
}
}
}
}
// Nonsymmetric reduction to Hessenberg form.
private Matrix orthes(Matrix x) {
// Working storage for nonsymmetric algorithm.
Vector ort = new DenseVector(n);
Matrix hessenBerg = new DenseMatrix(n, n).assign(x);
// 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.
int low = 0;
int high = n - 1;
for (int m = low + 1; m <= high - 1; m++) {
// Scale column.
Vector hColumn = hessenBerg.viewColumn(m - 1).viewPart(m, high - m + 1);
double scale = hColumn.norm(1);
if (scale != 0.0) {
// Compute Householder transformation.
ort.viewPart(m, high - m + 1).assign(hColumn, Functions.plusMult(1 / scale));
double h = ort.viewPart(m, high - m + 1).getLengthSquared();
double g = Math.sqrt(h);
if (ort.getQuick(m) > 0) {
g = -g;
}
h -= ort.getQuick(m) * g;
ort.setQuick(m, ort.getQuick(m) - g);
// Apply Householder similarity transformation
// H = (I-u*u'/h)*H*(I-u*u')/h)
Vector ortPiece = ort.viewPart(m, high - m + 1);
for (int j = m; j < n; j++) {
double f = ortPiece.dot(hessenBerg.viewColumn(j).viewPart(m, high - m + 1)) / h;
hessenBerg.viewColumn(j).viewPart(m, high - m + 1).assign(ortPiece, Functions.plusMult(-f));
}
for (int i = 0; i <= high; i++) {
double f = ortPiece.dot(hessenBerg.viewRow(i).viewPart(m, high - m + 1)) / h;
hessenBerg.viewRow(i).viewPart(m, high - m + 1).assign(ortPiece, Functions.plusMult(-f));
}
ort.setQuick(m, scale * ort.getQuick(m));
hessenBerg.setQuick(m, m - 1, scale * g);
}
}
// Accumulate transformations (Algol's ortran).
v.assign(0);
v.viewDiagonal().assign(1);
for (int m = high - 1; m >= low + 1; m--) {
if (hessenBerg.getQuick(m, m - 1) != 0.0) {
ort.viewPart(m + 1, high - m).assign(hessenBerg.viewColumn(m - 1).viewPart(m + 1, high - m));
for (int j = m; j <= high; j++) {
double g = ort.viewPart(m, high - m + 1).dot(v.viewColumn(j).viewPart(m, high - m + 1));
// Double division avoids possible underflow
g = g / ort.getQuick(m) / hessenBerg.getQuick(m, m - 1);
v.viewColumn(j).viewPart(m, high - m + 1).assign(ort.viewPart(m, high - m + 1), Functions.plusMult(g));
}
}
}
return hessenBerg;
}
// Complex scalar division.
private double cdivr;
private double cdivi;
private void cdiv(double xr, double xi, double yr, double yi) {
double r;
double 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(Matrix h) {
// 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
int nn = this.n;
int n = nn - 1;
int low = 0;
int high = nn - 1;
double eps = Math.pow(2.0, -52.0);
double exshift = 0.0;
double p = 0;
double q = 0;
double r = 0;
double s = 0;
double z = 0;
double w;
double x;
double y;
// Store roots isolated by balanc and compute matrix norm
double norm = h.aggregate(Functions.PLUS, Functions.ABS);
// 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.getQuick(l - 1, l - 1)) + Math.abs(h.getQuick(l, l));
if (s == 0.0) {
s = norm;
}
if (Math.abs(h.getQuick(l, l - 1)) < eps * s) {
break;
}
l--;
}
// Check for convergence
if (l == n) {
// One root found
h.setQuick(n, n, h.getQuick(n, n) + exshift);
d.setQuick(n, h.getQuick(n, n));
e.setQuick(n, 0.0);
n--;
iter = 0;
} else if (l == n - 1) {
// Two roots found
w = h.getQuick(n, n - 1) * h.getQuick(n - 1, n);
p = (h.getQuick(n - 1, n - 1) - h.getQuick(n, n)) / 2.0;
q = p * p + w;
z = Math.sqrt(Math.abs(q));
h.setQuick(n, n, h.getQuick(n, n) + exshift);
h.setQuick(n - 1, n - 1, h.getQuick(n - 1, n - 1) + exshift);
x = h.getQuick(n, n);
// Real pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
d.setQuick(n - 1, x + z);
d.setQuick(n, d.getQuick(n - 1));
if (z != 0.0) {
d.setQuick(n, x - w / z);
}
e.setQuick(n - 1, 0.0);
e.setQuick(n, 0.0);
x = h.getQuick(n, n - 1);
s = Math.abs(x) + Math.abs(z);
p = x / s;
q = z / s;
r = Math.sqrt(p * p + q * q);
p /= r;
q /= r;
// Row modification
for (int j = n - 1; j < nn; j++) {
z = h.getQuick(n - 1, j);
h.setQuick(n - 1, j, q * z + p * h.getQuick(n, j));
h.setQuick(n, j, q * h.getQuick(n, j) - p * z);
}
// Column modification
for (int i = 0; i <= n; i++) {
z = h.getQuick(i, n - 1);
h.setQuick(i, n - 1, q * z + p * h.getQuick(i, n));
h.setQuick(i, n, q * h.getQuick(i, n) - p * z);
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = v.getQuick(i, n - 1);
v.setQuick(i, n - 1, q * z + p * v.getQuick(i, n));
v.setQuick(i, n, q * v.getQuick(i, n) - p * z);
}
// Complex pair
} else {
d.setQuick(n - 1, x + p);
d.setQuick(n, x + p);
e.setQuick(n - 1, z);
e.setQuick(n, -z);
}
n -= 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = h.getQuick(n, n);
y = 0.0;
w = 0.0;
if (l < n) {
y = h.getQuick(n - 1, n - 1);
w = h.getQuick(n, n - 1) * h.getQuick(n - 1, n);
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n; i++) {
h.setQuick(i, i, x);
}
s = Math.abs(h.getQuick(n, n - 1)) + Math.abs(h.getQuick(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.setQuick(i, i, h.getQuick(i, i) - s);
}
exshift += s;
x = y = w = 0.964;
}
}
iter++; // (Could check iteration count here.)
// Look for two consecutive small sub-diagonal elements
int m = n - 2;
while (m >= l) {
z = h.getQuick(m, m);
r = x - z;
s = y - z;
p = (r * s - w) / h.getQuick(m + 1, m) + h.getQuick(m, m + 1);
q = h.getQuick(m + 1, m + 1) - z - r - s;
r = h.getQuick(m + 2, m + 1);
s = Math.abs(p) + Math.abs(q) + Math.abs(r);
p /= s;
q /= s;
r /= s;
if (m == l) {
break;
}
double hmag = Math.abs(h.getQuick(m - 1, m - 1)) + Math.abs(h.getQuick(m + 1, m + 1));
double threshold = eps * Math.abs(p) * (Math.abs(z) + hmag);
if (Math.abs(h.getQuick(m, m - 1)) * (Math.abs(q) + Math.abs(r)) < threshold) {
break;
}
m--;
}
for (int i = m + 2; i <= n; i++) {
h.setQuick(i, i - 2, 0.0);
if (i > m + 2) {
h.setQuick(i, i - 3, 0.0);
}
}
// Double QR step involving rows l:n and columns m:n
for (int k = m; k <= n - 1; k++) {
boolean notlast = k != n - 1;
if (k != m) {
p = h.getQuick(k, k - 1);
q = h.getQuick(k + 1, k - 1);
r = notlast ? h.getQuick(k + 2, k - 1) : 0.0;
x = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x != 0.0) {
p /= x;
q /= x;
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.setQuick(k, k - 1, -s * x);
} else if (l != m) {
h.setQuick(k, k - 1, -h.getQuick(k, k - 1));
}
p += s;
x = p / s;
y = q / s;
z = r / s;
q /= p;
r /= p;
// Row modification
for (int j = k; j < nn; j++) {
p = h.getQuick(k, j) + q * h.getQuick(k + 1, j);
if (notlast) {
p += r * h.getQuick(k + 2, j);
h.setQuick(k + 2, j, h.getQuick(k + 2, j) - p * z);
}
h.setQuick(k, j, h.getQuick(k, j) - p * x);
h.setQuick(k + 1, j, h.getQuick(k + 1, j) - p * y);
}
// Column modification
for (int i = 0; i <= Math.min(n, k + 3); i++) {
p = x * h.getQuick(i, k) + y * h.getQuick(i, k + 1);
if (notlast) {
p += z * h.getQuick(i, k + 2);
h.setQuick(i, k + 2, h.getQuick(i, k + 2) - p * r);
}
h.setQuick(i, k, h.getQuick(i, k) - p);
h.setQuick(i, k + 1, h.getQuick(i, k + 1) - p * q);
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * v.getQuick(i, k) + y * v.getQuick(i, k + 1);
if (notlast) {
p += z * v.getQuick(i, k + 2);
v.setQuick(i, k + 2, v.getQuick(i, k + 2) - p * r);
}
v.setQuick(i, k, v.getQuick(i, k) - p);
v.setQuick(i, k + 1, v.getQuick(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.getQuick(n);
q = e.getQuick(n);
// Real vector
double t;
if (q == 0) {
int l = n;
h.setQuick(n, n, 1.0);
for (int i = n - 1; i >= 0; i--) {
w = h.getQuick(i, i) - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r += h.getQuick(i, j) * h.getQuick(j, n);
}
if (e.getQuick(i) < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e.getQuick(i) == 0.0) {
if (w == 0.0) {
h.setQuick(i, n, -r / (eps * norm));
} else {
h.setQuick(i, n, -r / w);
}
// Solve real equations
} else {
x = h.getQuick(i, i + 1);
y = h.getQuick(i + 1, i);
q = (d.getQuick(i) - p) * (d.getQuick(i) - p) + e.getQuick(i) * e.getQuick(i);
t = (x * s - z * r) / q;
h.setQuick(i, n, t);
if (Math.abs(x) > Math.abs(z)) {
h.setQuick(i + 1, n, (-r - w * t) / x);
} else {
h.setQuick(i + 1, n, (-s - y * t) / z);
}
}
// Overflow control
t = Math.abs(h.getQuick(i, n));
if (eps * t * t > 1) {
for (int j = i; j <= n; j++) {
h.setQuick(j, n, h.getQuick(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.getQuick(n, n - 1)) > Math.abs(h.getQuick(n - 1, n))) {
h.setQuick(n - 1, n - 1, q / h.getQuick(n, n - 1));
h.setQuick(n - 1, n, -(h.getQuick(n, n) - p) / h.getQuick(n, n - 1));
} else {
cdiv(0.0, -h.getQuick(n - 1, n), h.getQuick(n - 1, n - 1) - p, q);
h.setQuick(n - 1, n - 1, cdivr);
h.setQuick(n - 1, n, cdivi);
}
h.setQuick(n, n - 1, 0.0);
h.setQuick(n, n, 1.0);
for (int i = n - 2; i >= 0; i--) {
double ra = 0.0;
double sa = 0.0;
for (int j = l; j <= n; j++) {
ra += h.getQuick(i, j) * h.getQuick(j, n - 1);
sa += h.getQuick(i, j) * h.getQuick(j, n);
}
w = h.getQuick(i, i) - p;
if (e.getQuick(i) < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e.getQuick(i) == 0) {
cdiv(-ra, -sa, w, q);
h.setQuick(i, n - 1, cdivr);
h.setQuick(i, n, cdivi);
} else {
// Solve complex equations
x = h.getQuick(i, i + 1);
y = h.getQuick(i + 1, i);
double vr = (d.getQuick(i) - p) * (d.getQuick(i) - p) + e.getQuick(i) * e.getQuick(i) - q * q;
double vi = (d.getQuick(i) - p) * 2.0 * q;
if (vr == 0.0 && vi == 0.0) {
double hmag = Math.abs(x) + Math.abs(y);
vr = eps * norm * (Math.abs(w) + Math.abs(q) + hmag + Math.abs(z));
}
cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
h.setQuick(i, n - 1, cdivr);
h.setQuick(i, n, cdivi);
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
h.setQuick(i + 1, n - 1, (-ra - w * h.getQuick(i, n - 1) + q * h.getQuick(i, n)) / x);
h.setQuick(i + 1, n, (-sa - w * h.getQuick(i, n) - q * h.getQuick(i, n - 1)) / x);
} else {
cdiv(-r - y * h.getQuick(i, n - 1), -s - y * h.getQuick(i, n), z, q);
h.setQuick(i + 1, n - 1, cdivr);
h.setQuick(i + 1, n, cdivi);
}
}
// Overflow control
t = Math.max(Math.abs(h.getQuick(i, n - 1)), Math.abs(h.getQuick(i, n)));
if (eps * t * t > 1) {
for (int j = i; j <= n; j++) {
h.setQuick(j, n - 1, h.getQuick(j, n - 1) / t);
h.setQuick(j, n, h.getQuick(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.setQuick(i, j, h.getQuick(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 += v.getQuick(i, k) * h.getQuick(k, j);
}
v.setQuick(i, j, z);
}
}
}
private static boolean isSymmetric(Matrix a) {
/*
Symmetry flag.
*/
int n = a.columnSize();
boolean isSymmetric = true;
for (int j = 0; (j < n) && isSymmetric; j++) {
for (int i = 0; (i < n) && isSymmetric; i++) {
isSymmetric = a.getQuick(i, j) == a.getQuick(j, i);
}
}
return isSymmetric;
}
}