/*
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* This software is a cooperative product of The MathWorks and the National
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* Institute of Standards and Technology (NIST) which has been released to the
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* public domain. Neither The MathWorks nor NIST assumes any responsibility
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* whatsoever for its use by other parties, and makes no guarantees, expressed
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* or implied, about its quality, reliability, or any other characteristic.
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*/
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/*
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* EigenvalueDecomposition.java
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* Copyright (C) 1999 The Mathworks and NIST
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*
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*/
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package //weka.core.
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matrix;
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import weka.core.RevisionHandler;
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import weka.core.RevisionUtils;
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import java.io.Serializable;
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/**
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* Eigenvalues and eigenvectors of a real matrix.
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* <P>
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* If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal
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* and the eigenvector matrix V is orthogonal. I.e. A =
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* V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the
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* identity matrix.
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* <P>
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* If A is not symmetric, then the eigenvalue matrix D is block diagonal with
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* the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda +
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* i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V
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* represent the eigenvectors in the sense that A*V = V*D, i.e. A.times(V)
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* equals V.times(D). The matrix V may be badly conditioned, or even singular,
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* so the validity of the equation A = V*D*inverse(V) depends upon V.cond().
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* <p/>
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* Adapted from the <a href="http://math.nist.gov/javanumerics/jama/" target="_blank">JAMA</a> package.
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*
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* @author The Mathworks and NIST
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* @author Fracpete (fracpete at waikato dot ac dot nz)
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* @version $Revision: 1.4 $
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*/
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public class EigenvalueDecomposition
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implements Serializable, RevisionHandler {
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/** for serialization */
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private static final long serialVersionUID = 4011654467211422319L;
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/**
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* Row and column dimension (square matrix).
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* @serial matrix dimension.
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*/
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private int n;
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/**
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* Symmetry flag.
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* @serial internal symmetry flag.
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*/
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private boolean issymmetric;
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/**
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* Arrays for internal storage of eigenvalues.
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* @serial internal storage of eigenvalues.
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*/
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private double[] d, e;
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/**
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* Array for internal storage of eigenvectors.
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* @serial internal storage of eigenvectors.
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*/
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private double[][] V;
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/**
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* Array for internal storage of nonsymmetric Hessenberg form.
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* @serial internal storage of nonsymmetric Hessenberg form.
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*/
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private double[][] H;
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/**
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* Working storage for nonsymmetric algorithm.
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* @serial working storage for nonsymmetric algorithm.
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*/
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private double[] ort;
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/**
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* helper variables for the comples scalar division
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* @see #cdiv(double,double,double,double)
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*/
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private transient double cdivr, cdivi;
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/**
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* Symmetric Householder reduction to tridiagonal form.
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* <p/>
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* This is derived from the Algol procedures tred2 by Bowdler, Martin,
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* Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra,
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* and the corresponding Fortran subroutine in EISPACK.
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*/
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private void tred2() {
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for (int j = 0; j < n; j++) {
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d[j] = V[n-1][j];
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}
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// Householder reduction to tridiagonal form.
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for (int i = n-1; i > 0; i--) {
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// Scale to avoid under/overflow.
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double scale = 0.0;
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double h = 0.0;
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for (int k = 0; k < i; k++) {
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scale = scale + Math.abs(d[k]);
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}
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if (scale == 0.0) {
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e[i] = d[i-1];
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for (int j = 0; j < i; j++) {
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d[j] = V[i-1][j];
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V[i][j] = 0.0;
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V[j][i] = 0.0;
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}
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} else {
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// Generate Householder vector.
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for (int k = 0; k < i; k++) {
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d[k] /= scale;
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h += d[k] * d[k];
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}
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double f = d[i-1];
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double g = Math.sqrt(h);
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if (f > 0) {
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g = -g;
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}
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e[i] = scale * g;
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h = h - f * g;
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d[i-1] = f - g;
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for (int j = 0; j < i; j++) {
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e[j] = 0.0;
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}
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// Apply similarity transformation to remaining columns.
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for (int j = 0; j < i; j++) {
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f = d[j];
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V[j][i] = f;
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g = e[j] + V[j][j] * f;
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for (int k = j+1; k <= i-1; k++) {
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g += V[k][j] * d[k];
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e[k] += V[k][j] * f;
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}
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e[j] = g;
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}
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f = 0.0;
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for (int j = 0; j < i; j++) {
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e[j] /= h;
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f += e[j] * d[j];
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}
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double hh = f / (h + h);
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for (int j = 0; j < i; j++) {
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e[j] -= hh * d[j];
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}
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for (int j = 0; j < i; j++) {
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f = d[j];
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g = e[j];
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for (int k = j; k <= i-1; k++) {
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V[k][j] -= (f * e[k] + g * d[k]);
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}
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d[j] = V[i-1][j];
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V[i][j] = 0.0;
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}
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}
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d[i] = h;
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}
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// Accumulate transformations.
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for (int i = 0; i < n-1; i++) {
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V[n-1][i] = V[i][i];
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V[i][i] = 1.0;
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double h = d[i+1];
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if (h != 0.0) {
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for (int k = 0; k <= i; k++) {
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d[k] = V[k][i+1] / h;
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}
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for (int j = 0; j <= i; j++) {
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double g = 0.0;
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for (int k = 0; k <= i; k++) {
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g += V[k][i+1] * V[k][j];
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}
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for (int k = 0; k <= i; k++) {
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V[k][j] -= g * d[k];
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}
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}
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}
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for (int k = 0; k <= i; k++) {
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V[k][i+1] = 0.0;
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}
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}
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for (int j = 0; j < n; j++) {
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d[j] = V[n-1][j];
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V[n-1][j] = 0.0;
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}
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V[n-1][n-1] = 1.0;
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e[0] = 0.0;
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}
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/**
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* Symmetric tridiagonal QL algorithm.
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* <p/>
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* This is derived from the Algol procedures tql2, by Bowdler, Martin,
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* Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra,
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* and the corresponding Fortran subroutine in EISPACK.
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*/
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private void tql2() {
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for (int i = 1; i < n; i++) {
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e[i-1] = e[i];
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}
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e[n-1] = 0.0;
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double f = 0.0;
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double tst1 = 0.0;
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double eps = Math.pow(2.0,-52.0);
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for (int l = 0; l < n; l++) {
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// Find small subdiagonal element
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tst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));
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int m = l;
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while (m < n) {
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if (Math.abs(e[m]) <= eps*tst1) {
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break;
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}
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m++;
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}
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// If m == l, d[l] is an eigenvalue,
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// otherwise, iterate.
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if (m > l) {
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int iter = 0;
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do {
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iter = iter + 1; // (Could check iteration count here.)
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// Compute implicit shift
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double g = d[l];
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double p = (d[l+1] - g) / (2.0 * e[l]);
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double r = Maths.hypot(p,1.0);
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if (p < 0) {
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r = -r;
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}
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d[l] = e[l] / (p + r);
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d[l+1] = e[l] * (p + r);
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double dl1 = d[l+1];
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double h = g - d[l];
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for (int i = l+2; i < n; i++) {
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d[i] -= h;
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}
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f = f + h;
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// Implicit QL transformation.
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p = d[m];
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double c = 1.0;
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double c2 = c;
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double c3 = c;
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double el1 = e[l+1];
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double s = 0.0;
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double s2 = 0.0;
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for (int i = m-1; i >= l; i--) {
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c3 = c2;
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c2 = c;
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s2 = s;
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g = c * e[i];
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h = c * p;
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r = Maths.hypot(p,e[i]);
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e[i+1] = s * r;
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s = e[i] / r;
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c = p / r;
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p = c * d[i] - s * g;
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d[i+1] = h + s * (c * g + s * d[i]);
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// Accumulate transformation.
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for (int k = 0; k < n; k++) {
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h = V[k][i+1];
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V[k][i+1] = s * V[k][i] + c * h;
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V[k][i] = c * V[k][i] - s * h;
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}
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}
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p = -s * s2 * c3 * el1 * e[l] / dl1;
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e[l] = s * p;
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d[l] = c * p;
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// Check for convergence.
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} while (Math.abs(e[l]) > eps*tst1);
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}
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d[l] = d[l] + f;
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e[l] = 0.0;
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}
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// Sort eigenvalues and corresponding vectors.
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for (int i = 0; i < n-1; i++) {
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int k = i;
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double p = d[i];
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for (int j = i+1; j < n; j++) {
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if (d[j] < p) {
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k = j;
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p = d[j];
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}
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}
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if (k != i) {
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d[k] = d[i];
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d[i] = p;
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for (int j = 0; j < n; j++) {
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p = V[j][i];
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V[j][i] = V[j][k];
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V[j][k] = p;
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}
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}
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}
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}
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/**
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* Nonsymmetric reduction to Hessenberg form.
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* <p/>
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* This is derived from the Algol procedures orthes and ortran, by Martin
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* and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the
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* corresponding Fortran subroutines in EISPACK.
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*/
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private void orthes() {
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int low = 0;
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int high = n-1;
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for (int m = low+1; m <= high-1; m++) {
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// Scale column.
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double scale = 0.0;
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for (int i = m; i <= high; i++) {
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scale = scale + Math.abs(H[i][m-1]);
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}
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if (scale != 0.0) {
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// Compute Householder transformation.
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double h = 0.0;
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for (int i = high; i >= m; i--) {
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ort[i] = H[i][m-1]/scale;
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h += ort[i] * ort[i];
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}
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double g = Math.sqrt(h);
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if (ort[m] > 0) {
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g = -g;
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}
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h = h - ort[m] * g;
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ort[m] = ort[m] - g;
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// Apply Householder similarity transformation
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// H = (I-u*u'/h)*H*(I-u*u')/h)
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for (int j = m; j < n; j++) {
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double f = 0.0;
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for (int i = high; i >= m; i--) {
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f += ort[i]*H[i][j];
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}
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f = f/h;
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for (int i = m; i <= high; i++) {
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H[i][j] -= f*ort[i];
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}
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}
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for (int i = 0; i <= high; i++) {
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double f = 0.0;
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for (int j = high; j >= m; j--) {
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f += ort[j]*H[i][j];
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}
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f = f/h;
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for (int j = m; j <= high; j++) {
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H[i][j] -= f*ort[j];
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}
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}
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ort[m] = scale*ort[m];
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H[m][m-1] = scale*g;
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}
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}
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// Accumulate transformations (Algol's ortran).
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for (int i = 0; i < n; i++) {
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for (int j = 0; j < n; j++) {
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V[i][j] = (i == j ? 1.0 : 0.0);
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}
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}
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for (int m = high-1; m >= low+1; m--) {
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if (H[m][m-1] != 0.0) {
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for (int i = m+1; i <= high; i++) {
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ort[i] = H[i][m-1];
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}
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for (int j = m; j <= high; j++) {
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double g = 0.0;
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for (int i = m; i <= high; i++) {
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g += ort[i] * V[i][j];
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}
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// Double division avoids possible underflow
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g = (g / ort[m]) / H[m][m-1];
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for (int i = m; i <= high; i++) {
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V[i][j] += g * ort[i];
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}
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}
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}
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}
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}
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/**
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* Complex scalar division.
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*/
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private void cdiv(double xr, double xi, double yr, double yi) {
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double r,d;
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if (Math.abs(yr) > Math.abs(yi)) {
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r = yi/yr;
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d = yr + r*yi;
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cdivr = (xr + r*xi)/d;
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cdivi = (xi - r*xr)/d;
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} else {
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r = yr/yi;
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d = yi + r*yr;
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cdivr = (r*xr + xi)/d;
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cdivi = (r*xi - xr)/d;
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}
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}
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|
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/**
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* Nonsymmetric reduction from Hessenberg to real Schur form.
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* <p/>
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* This is derived from the Algol procedure hqr2, by Martin and Wilkinson,
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* Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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* Fortran subroutine in EISPACK.
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*/
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private void hqr2() {
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// Initialize
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int nn = this.n;
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int n = nn-1;
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int low = 0;
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int high = nn-1;
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double eps = Math.pow(2.0,-52.0);
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double exshift = 0.0;
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double p=0,q=0,r=0,s=0,z=0,t,w,x,y;
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// Store roots isolated by balanc and compute matrix norm
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double norm = 0.0;
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for (int i = 0; i < nn; i++) {
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if (i < low | i > high) {
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d[i] = H[i][i];
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e[i] = 0.0;
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}
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for (int j = Math.max(i-1,0); j < nn; j++) {
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norm = norm + Math.abs(H[i][j]);
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}
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}
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// Outer loop over eigenvalue index
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int iter = 0;
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while (n >= low) {
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// Look for single small sub-diagonal element
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int l = n;
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while (l > low) {
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s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);
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if (s == 0.0) {
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s = norm;
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}
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if (Math.abs(H[l][l-1]) < eps * s) {
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break;
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}
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l--;
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}
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// Check for convergence
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// One root found
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if (l == n) {
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H[n][n] = H[n][n] + exshift;
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d[n] = H[n][n];
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e[n] = 0.0;
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n--;
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iter = 0;
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// Two roots found
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} else if (l == n-1) {
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w = H[n][n-1] * H[n-1][n];
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p = (H[n-1][n-1] - H[n][n]) / 2.0;
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q = p * p + w;
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z = Math.sqrt(Math.abs(q));
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H[n][n] = H[n][n] + exshift;
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H[n-1][n-1] = H[n-1][n-1] + exshift;
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x = H[n][n];
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// Real pair
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if (q >= 0) {
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if (p >= 0) {
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z = p + z;
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} else {
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z = p - z;
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}
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d[n-1] = x + z;
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d[n] = d[n-1];
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if (z != 0.0) {
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d[n] = x - w / z;
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}
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e[n-1] = 0.0;
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e[n] = 0.0;
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x = H[n][n-1];
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s = Math.abs(x) + Math.abs(z);
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p = x / s;
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q = z / s;
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r = Math.sqrt(p * p+q * q);
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p = p / r;
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q = q / r;
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// Row modification
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for (int j = n-1; j < nn; j++) {
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z = H[n-1][j];
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H[n-1][j] = q * z + p * H[n][j];
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H[n][j] = q * H[n][j] - p * z;
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}
|
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// Column modification
|
|
for (int i = 0; i <= n; i++) {
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z = H[i][n-1];
|
H[i][n-1] = q * z + p * H[i][n];
|
H[i][n] = q * H[i][n] - p * z;
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}
|
|
// Accumulate transformations
|
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for (int i = low; i <= high; i++) {
|
z = V[i][n-1];
|
V[i][n-1] = q * z + p * V[i][n];
|
V[i][n] = q * V[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[n][n];
|
y = 0.0;
|
w = 0.0;
|
if (l < n) {
|
y = H[n-1][n-1];
|
w = H[n][n-1] * H[n-1][n];
|
}
|
|
// Wilkinson's original ad hoc shift
|
|
if (iter == 10) {
|
exshift += x;
|
for (int i = low; i <= n; i++) {
|
H[i][i] -= x;
|
}
|
s = Math.abs(H[n][n-1]) + Math.abs(H[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[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[m][m];
|
r = x - z;
|
s = y - z;
|
p = (r * s - w) / H[m+1][m] + H[m][m+1];
|
q = H[m+1][m+1] - z - r - s;
|
r = H[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[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
|
eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
|
Math.abs(H[m+1][m+1])))) {
|
break;
|
}
|
m--;
|
}
|
|
for (int i = m+2; i <= n; i++) {
|
H[i][i-2] = 0.0;
|
if (i > m+2) {
|
H[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[k][k-1];
|
q = H[k+1][k-1];
|
r = (notlast ? H[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[k][k-1] = -s * x;
|
} else if (l != m) {
|
H[k][k-1] = -H[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[k][j] + q * H[k+1][j];
|
if (notlast) {
|
p = p + r * H[k+2][j];
|
H[k+2][j] = H[k+2][j] - p * z;
|
}
|
H[k][j] = H[k][j] - p * x;
|
H[k+1][j] = H[k+1][j] - p * y;
|
}
|
|
// Column modification
|
|
for (int i = 0; i <= Math.min(n,k+3); i++) {
|
p = x * H[i][k] + y * H[i][k+1];
|
if (notlast) {
|
p = p + z * H[i][k+2];
|
H[i][k+2] = H[i][k+2] - p * r;
|
}
|
H[i][k] = H[i][k] - p;
|
H[i][k+1] = H[i][k+1] - p * q;
|
}
|
|
// Accumulate transformations
|
|
for (int i = low; i <= high; i++) {
|
p = x * V[i][k] + y * V[i][k+1];
|
if (notlast) {
|
p = p + z * V[i][k+2];
|
V[i][k+2] = V[i][k+2] - p * r;
|
}
|
V[i][k] = V[i][k] - p;
|
V[i][k+1] = V[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[n][n] = 1.0;
|
for (int i = n-1; i >= 0; i--) {
|
w = H[i][i] - p;
|
r = 0.0;
|
for (int j = l; j <= n; j++) {
|
r = r + H[i][j] * H[j][n];
|
}
|
if (e[i] < 0.0) {
|
z = w;
|
s = r;
|
} else {
|
l = i;
|
if (e[i] == 0.0) {
|
if (w != 0.0) {
|
H[i][n] = -r / w;
|
} else {
|
H[i][n] = -r / (eps * norm);
|
}
|
|
// Solve real equations
|
|
} else {
|
x = H[i][i+1];
|
y = H[i+1][i];
|
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
|
t = (x * s - z * r) / q;
|
H[i][n] = t;
|
if (Math.abs(x) > Math.abs(z)) {
|
H[i+1][n] = (-r - w * t) / x;
|
} else {
|
H[i+1][n] = (-s - y * t) / z;
|
}
|
}
|
|
// Overflow control
|
|
t = Math.abs(H[i][n]);
|
if ((eps * t) * t > 1) {
|
for (int j = i; j <= n; j++) {
|
H[j][n] = H[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[n][n-1]) > Math.abs(H[n-1][n])) {
|
H[n-1][n-1] = q / H[n][n-1];
|
H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
|
} else {
|
cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
|
H[n-1][n-1] = cdivr;
|
H[n-1][n] = cdivi;
|
}
|
H[n][n-1] = 0.0;
|
H[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[i][j] * H[j][n-1];
|
sa = sa + H[i][j] * H[j][n];
|
}
|
w = H[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[i][n-1] = cdivr;
|
H[i][n] = cdivi;
|
} else {
|
|
// Solve complex equations
|
|
x = H[i][i+1];
|
y = H[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[i][n-1] = cdivr;
|
H[i][n] = cdivi;
|
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
|
H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
|
H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
|
} else {
|
cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
|
H[i+1][n-1] = cdivr;
|
H[i+1][n] = cdivi;
|
}
|
}
|
|
// Overflow control
|
|
t = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));
|
if ((eps * t) * t > 1) {
|
for (int j = i; j <= n; j++) {
|
H[j][n-1] = H[j][n-1] / t;
|
H[j][n] = H[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[i][j] = H[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[i][k] * H[k][j];
|
}
|
V[i][j] = z;
|
}
|
}
|
}
|
|
|
/**
|
* Check for symmetry, then construct the eigenvalue decomposition
|
*
|
* @param Arg Square matrix
|
*/
|
public EigenvalueDecomposition(Matrix Arg) {
|
double[][] A = Arg.getArray();
|
n = Arg.getColumnDimension();
|
V = new double[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[i][j] == A[j][i]);
|
}
|
}
|
|
if (issymmetric) {
|
for (int i = 0; i < n; i++) {
|
for (int j = 0; j < n; j++) {
|
V[i][j] = A[i][j];
|
}
|
}
|
|
// Tridiagonalize.
|
tred2();
|
|
// Diagonalize.
|
tql2();
|
|
} else {
|
H = new double[n][n];
|
ort = new double[n];
|
|
for (int j = 0; j < n; j++) {
|
for (int i = 0; i < n; i++) {
|
H[i][j] = A[i][j];
|
}
|
}
|
|
// Reduce to Hessenberg form.
|
orthes();
|
|
// Reduce Hessenberg to real Schur form.
|
hqr2();
|
}
|
}
|
|
/**
|
* Return the eigenvector matrix
|
* @return V
|
*/
|
public Matrix getV() {
|
return new Matrix(V,n,n);
|
}
|
|
/**
|
* 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() {
|
Matrix X = new Matrix(n,n);
|
double[][] D = X.getArray();
|
for (int i = 0; i < n; i++) {
|
for (int j = 0; j < n; j++) {
|
D[i][j] = 0.0;
|
}
|
D[i][i] = d[i];
|
// NORMAND if (e[i] > 0) {
|
// D[i][i+1] = e[i];
|
// } else if (e[i] < 0) {
|
// D[i][i-1] = e[i];
|
// }
|
}
|
return X;
|
}
|
|
/**
|
* Returns the revision string.
|
*
|
* @return the revision
|
*/
|
public String getRevision() {
|
return RevisionUtils.extract("$Revision: 1.4 $");
|
}
|
}
|
