2010-08-26 19:14:53 +00:00
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<?php
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/**
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2015-05-12 09:22:06 +00:00
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* @package JAMA
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* Class to obtain eigenvalues and eigenvectors of a real matrix.
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* If A is symmetric, then A = V*D*V' where the eigenvalue matrix D
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* is diagonal and the eigenvector matrix V is orthogonal (i.e.
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* A = V.times(D.times(V.transpose())) and V.times(V.transpose())
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* equals the identity matrix).
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* If A is not symmetric, then the eigenvalue matrix D is block diagonal
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* with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
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* lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The
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* columns of V represent the eigenvectors in the sense that A*V = V*D,
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* i.e. A.times(V) equals V.times(D). The matrix V may be badly
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* conditioned, or even singular, so the validity of the equation
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* A = V*D*inverse(V) depends upon V.cond().
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* @author Paul Meagher
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* @license PHP v3.0
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* @version 1.1
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2010-08-26 19:14:53 +00:00
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*/
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2015-05-17 12:16:53 +00:00
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class EigenvalueDecomposition
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{
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2015-05-12 09:22:06 +00:00
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/**
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* Row and column dimension (square matrix).
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* @var int
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*/
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private $n;
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/**
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* Internal symmetry flag.
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* @var int
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*/
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private $issymmetric;
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/**
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* Arrays for internal storage of eigenvalues.
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* @var array
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*/
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private $d = array();
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private $e = array();
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/**
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* Array for internal storage of eigenvectors.
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* @var array
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*/
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private $V = array();
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/**
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* Array for internal storage of nonsymmetric Hessenberg form.
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* @var array
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*/
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private $H = array();
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/**
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* Working storage for nonsymmetric algorithm.
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* @var array
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*/
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private $ort;
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/**
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* Used for complex scalar division.
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* @var float
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*/
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private $cdivr;
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private $cdivi;
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/**
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* Symmetric Householder reduction to tridiagonal form.
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*
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* @access private
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*/
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2015-05-17 13:00:02 +00:00
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private function tred2()
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2015-05-17 12:16:53 +00:00
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{
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2015-05-12 09:22:06 +00:00
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// This is derived from the Algol procedures tred2 by
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// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
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// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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// Fortran subroutine in EISPACK.
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$this->d = $this->V[$this->n-1];
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// Householder reduction to tridiagonal form.
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for ($i = $this->n-1; $i > 0; --$i) {
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$i_ = $i -1;
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// Scale to avoid under/overflow.
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$h = $scale = 0.0;
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$scale += array_sum(array_map(abs, $this->d));
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if ($scale == 0.0) {
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$this->e[$i] = $this->d[$i_];
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$this->d = array_slice($this->V[$i_], 0, $i_);
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for ($j = 0; $j < $i; ++$j) {
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$this->V[$j][$i] = $this->V[$i][$j] = 0.0;
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}
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} else {
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// Generate Householder vector.
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for ($k = 0; $k < $i; ++$k) {
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$this->d[$k] /= $scale;
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$h += pow($this->d[$k], 2);
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}
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$f = $this->d[$i_];
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$g = sqrt($h);
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if ($f > 0) {
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$g = -$g;
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}
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$this->e[$i] = $scale * $g;
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$h = $h - $f * $g;
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$this->d[$i_] = $f - $g;
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for ($j = 0; $j < $i; ++$j) {
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$this->e[$j] = 0.0;
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}
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// Apply similarity transformation to remaining columns.
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for ($j = 0; $j < $i; ++$j) {
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$f = $this->d[$j];
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$this->V[$j][$i] = $f;
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$g = $this->e[$j] + $this->V[$j][$j] * $f;
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for ($k = $j+1; $k <= $i_; ++$k) {
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$g += $this->V[$k][$j] * $this->d[$k];
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$this->e[$k] += $this->V[$k][$j] * $f;
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}
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$this->e[$j] = $g;
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}
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$f = 0.0;
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for ($j = 0; $j < $i; ++$j) {
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$this->e[$j] /= $h;
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$f += $this->e[$j] * $this->d[$j];
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}
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$hh = $f / (2 * $h);
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for ($j=0; $j < $i; ++$j) {
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$this->e[$j] -= $hh * $this->d[$j];
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}
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for ($j = 0; $j < $i; ++$j) {
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$f = $this->d[$j];
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$g = $this->e[$j];
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for ($k = $j; $k <= $i_; ++$k) {
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$this->V[$k][$j] -= ($f * $this->e[$k] + $g * $this->d[$k]);
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}
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$this->d[$j] = $this->V[$i-1][$j];
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$this->V[$i][$j] = 0.0;
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}
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}
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$this->d[$i] = $h;
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}
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// Accumulate transformations.
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for ($i = 0; $i < $this->n-1; ++$i) {
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$this->V[$this->n-1][$i] = $this->V[$i][$i];
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$this->V[$i][$i] = 1.0;
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$h = $this->d[$i+1];
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if ($h != 0.0) {
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for ($k = 0; $k <= $i; ++$k) {
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$this->d[$k] = $this->V[$k][$i+1] / $h;
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}
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for ($j = 0; $j <= $i; ++$j) {
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$g = 0.0;
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for ($k = 0; $k <= $i; ++$k) {
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$g += $this->V[$k][$i+1] * $this->V[$k][$j];
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}
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for ($k = 0; $k <= $i; ++$k) {
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$this->V[$k][$j] -= $g * $this->d[$k];
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}
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}
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}
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for ($k = 0; $k <= $i; ++$k) {
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$this->V[$k][$i+1] = 0.0;
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}
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}
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$this->d = $this->V[$this->n-1];
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$this->V[$this->n-1] = array_fill(0, $j, 0.0);
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$this->V[$this->n-1][$this->n-1] = 1.0;
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$this->e[0] = 0.0;
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}
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/**
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* Symmetric tridiagonal QL algorithm.
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*
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* This is derived from the Algol procedures tql2, by
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* Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
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* Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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* Fortran subroutine in EISPACK.
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*
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* @access private
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*/
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2015-05-17 12:16:53 +00:00
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private function tql2()
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{
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2015-05-12 09:22:06 +00:00
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for ($i = 1; $i < $this->n; ++$i) {
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$this->e[$i-1] = $this->e[$i];
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}
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$this->e[$this->n-1] = 0.0;
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$f = 0.0;
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$tst1 = 0.0;
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2015-05-17 13:00:02 +00:00
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$eps = pow(2.0, -52.0);
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2015-05-12 09:22:06 +00:00
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for ($l = 0; $l < $this->n; ++$l) {
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// Find small subdiagonal element
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$tst1 = max($tst1, abs($this->d[$l]) + abs($this->e[$l]));
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$m = $l;
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while ($m < $this->n) {
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2015-05-17 12:16:53 +00:00
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if (abs($this->e[$m]) <= $eps * $tst1) {
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2015-05-12 09:22:06 +00:00
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break;
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2015-05-17 12:16:53 +00:00
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}
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2015-05-12 09:22:06 +00:00
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++$m;
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}
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// If m == l, $this->d[l] is an eigenvalue,
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// otherwise, iterate.
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if ($m > $l) {
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$iter = 0;
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do {
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// Could check iteration count here.
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$iter += 1;
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// Compute implicit shift
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$g = $this->d[$l];
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$p = ($this->d[$l+1] - $g) / (2.0 * $this->e[$l]);
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$r = hypo($p, 1.0);
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2015-05-17 12:16:53 +00:00
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if ($p < 0) {
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2015-05-12 09:22:06 +00:00
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$r *= -1;
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2015-05-17 12:16:53 +00:00
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}
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2015-05-12 09:22:06 +00:00
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$this->d[$l] = $this->e[$l] / ($p + $r);
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$this->d[$l+1] = $this->e[$l] * ($p + $r);
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$dl1 = $this->d[$l+1];
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$h = $g - $this->d[$l];
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2015-05-17 12:16:53 +00:00
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for ($i = $l + 2; $i < $this->n; ++$i) {
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2015-05-12 09:22:06 +00:00
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$this->d[$i] -= $h;
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2015-05-17 12:16:53 +00:00
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}
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2015-05-12 09:22:06 +00:00
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$f += $h;
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// Implicit QL transformation.
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$p = $this->d[$m];
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$c = 1.0;
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$c2 = $c3 = $c;
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$el1 = $this->e[$l + 1];
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$s = $s2 = 0.0;
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for ($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 * $this->e[$i];
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$h = $c * $p;
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$r = hypo($p, $this->e[$i]);
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$this->e[$i+1] = $s * $r;
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$s = $this->e[$i] / $r;
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$c = $p / $r;
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$p = $c * $this->d[$i] - $s * $g;
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$this->d[$i+1] = $h + $s * ($c * $g + $s * $this->d[$i]);
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// Accumulate transformation.
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for ($k = 0; $k < $this->n; ++$k) {
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$h = $this->V[$k][$i+1];
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$this->V[$k][$i+1] = $s * $this->V[$k][$i] + $c * $h;
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$this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h;
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}
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}
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$p = -$s * $s2 * $c3 * $el1 * $this->e[$l] / $dl1;
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$this->e[$l] = $s * $p;
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$this->d[$l] = $c * $p;
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// Check for convergence.
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} while (abs($this->e[$l]) > $eps * $tst1);
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}
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$this->d[$l] = $this->d[$l] + $f;
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$this->e[$l] = 0.0;
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}
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// Sort eigenvalues and corresponding vectors.
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for ($i = 0; $i < $this->n - 1; ++$i) {
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$k = $i;
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$p = $this->d[$i];
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for ($j = $i+1; $j < $this->n; ++$j) {
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if ($this->d[$j] < $p) {
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$k = $j;
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$p = $this->d[$j];
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}
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}
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if ($k != $i) {
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$this->d[$k] = $this->d[$i];
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$this->d[$i] = $p;
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for ($j = 0; $j < $this->n; ++$j) {
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$p = $this->V[$j][$i];
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$this->V[$j][$i] = $this->V[$j][$k];
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$this->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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*
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* This is derived from the Algol procedures orthes and ortran,
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* by Martin and Wilkinson, Handbook for Auto. Comp.,
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* Vol.ii-Linear Algebra, and the corresponding
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* Fortran subroutines in EISPACK.
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*
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* @access private
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*/
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2015-05-17 12:16:53 +00:00
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private function orthes()
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{
|
2015-05-12 09:22:06 +00:00
|
|
|
$low = 0;
|
|
|
|
|
$high = $this->n-1;
|
|
|
|
|
|
|
|
|
|
for ($m = $low+1; $m <= $high-1; ++$m) {
|
|
|
|
|
// Scale column.
|
|
|
|
|
$scale = 0.0;
|
|
|
|
|
for ($i = $m; $i <= $high; ++$i) {
|
|
|
|
|
$scale = $scale + abs($this->H[$i][$m-1]);
|
|
|
|
|
}
|
|
|
|
|
if ($scale != 0.0) {
|
|
|
|
|
// Compute Householder transformation.
|
|
|
|
|
$h = 0.0;
|
|
|
|
|
for ($i = $high; $i >= $m; --$i) {
|
|
|
|
|
$this->ort[$i] = $this->H[$i][$m-1] / $scale;
|
|
|
|
|
$h += $this->ort[$i] * $this->ort[$i];
|
|
|
|
|
}
|
|
|
|
|
$g = sqrt($h);
|
|
|
|
|
if ($this->ort[$m] > 0) {
|
|
|
|
|
$g *= -1;
|
|
|
|
|
}
|
|
|
|
|
$h -= $this->ort[$m] * $g;
|
|
|
|
|
$this->ort[$m] -= $g;
|
|
|
|
|
// Apply Householder similarity transformation
|
|
|
|
|
// H = (I -u * u' / h) * H * (I -u * u') / h)
|
|
|
|
|
for ($j = $m; $j < $this->n; ++$j) {
|
|
|
|
|
$f = 0.0;
|
|
|
|
|
for ($i = $high; $i >= $m; --$i) {
|
|
|
|
|
$f += $this->ort[$i] * $this->H[$i][$j];
|
|
|
|
|
}
|
|
|
|
|
$f /= $h;
|
|
|
|
|
for ($i = $m; $i <= $high; ++$i) {
|
|
|
|
|
$this->H[$i][$j] -= $f * $this->ort[$i];
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
for ($i = 0; $i <= $high; ++$i) {
|
|
|
|
|
$f = 0.0;
|
|
|
|
|
for ($j = $high; $j >= $m; --$j) {
|
|
|
|
|
$f += $this->ort[$j] * $this->H[$i][$j];
|
|
|
|
|
}
|
|
|
|
|
$f = $f / $h;
|
|
|
|
|
for ($j = $m; $j <= $high; ++$j) {
|
|
|
|
|
$this->H[$i][$j] -= $f * $this->ort[$j];
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
$this->ort[$m] = $scale * $this->ort[$m];
|
|
|
|
|
$this->H[$m][$m-1] = $scale * $g;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// Accumulate transformations (Algol's ortran).
|
|
|
|
|
for ($i = 0; $i < $this->n; ++$i) {
|
|
|
|
|
for ($j = 0; $j < $this->n; ++$j) {
|
|
|
|
|
$this->V[$i][$j] = ($i == $j ? 1.0 : 0.0);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
for ($m = $high-1; $m >= $low+1; --$m) {
|
|
|
|
|
if ($this->H[$m][$m-1] != 0.0) {
|
|
|
|
|
for ($i = $m+1; $i <= $high; ++$i) {
|
|
|
|
|
$this->ort[$i] = $this->H[$i][$m-1];
|
|
|
|
|
}
|
|
|
|
|
for ($j = $m; $j <= $high; ++$j) {
|
|
|
|
|
$g = 0.0;
|
|
|
|
|
for ($i = $m; $i <= $high; ++$i) {
|
|
|
|
|
$g += $this->ort[$i] * $this->V[$i][$j];
|
|
|
|
|
}
|
|
|
|
|
// Double division avoids possible underflow
|
|
|
|
|
$g = ($g / $this->ort[$m]) / $this->H[$m][$m-1];
|
|
|
|
|
for ($i = $m; $i <= $high; ++$i) {
|
|
|
|
|
$this->V[$i][$j] += $g * $this->ort[$i];
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Performs complex division.
|
|
|
|
|
*
|
|
|
|
|
* @access private
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
private function cdiv($xr, $xi, $yr, $yi)
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
if (abs($yr) > abs($yi)) {
|
|
|
|
|
$r = $yi / $yr;
|
|
|
|
|
$d = $yr + $r * $yi;
|
|
|
|
|
$this->cdivr = ($xr + $r * $xi) / $d;
|
|
|
|
|
$this->cdivi = ($xi - $r * $xr) / $d;
|
|
|
|
|
} else {
|
|
|
|
|
$r = $yr / $yi;
|
|
|
|
|
$d = $yi + $r * $yr;
|
|
|
|
|
$this->cdivr = ($r * $xr + $xi) / $d;
|
|
|
|
|
$this->cdivi = ($r * $xi - $xr) / $d;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Nonsymmetric reduction from Hessenberg to real Schur form.
|
|
|
|
|
*
|
|
|
|
|
* Code 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.
|
|
|
|
|
*
|
|
|
|
|
* @access private
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
private function hqr2()
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
// Initialize
|
|
|
|
|
$nn = $this->n;
|
|
|
|
|
$n = $nn - 1;
|
|
|
|
|
$low = 0;
|
|
|
|
|
$high = $nn - 1;
|
|
|
|
|
$eps = pow(2.0, -52.0);
|
|
|
|
|
$exshift = 0.0;
|
|
|
|
|
$p = $q = $r = $s = $z = 0;
|
|
|
|
|
// Store roots isolated by balanc and compute matrix norm
|
|
|
|
|
$norm = 0.0;
|
|
|
|
|
|
|
|
|
|
for ($i = 0; $i < $nn; ++$i) {
|
2015-05-17 12:16:53 +00:00
|
|
|
if (($i < $low) or ($i > $high)) {
|
2015-05-12 09:22:06 +00:00
|
|
|
$this->d[$i] = $this->H[$i][$i];
|
|
|
|
|
$this->e[$i] = 0.0;
|
|
|
|
|
}
|
|
|
|
|
for ($j = max($i-1, 0); $j < $nn; ++$j) {
|
|
|
|
|
$norm = $norm + abs($this->H[$i][$j]);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// Outer loop over eigenvalue index
|
|
|
|
|
$iter = 0;
|
|
|
|
|
while ($n >= $low) {
|
|
|
|
|
// Look for single small sub-diagonal element
|
|
|
|
|
$l = $n;
|
|
|
|
|
while ($l > $low) {
|
|
|
|
|
$s = abs($this->H[$l-1][$l-1]) + abs($this->H[$l][$l]);
|
|
|
|
|
if ($s == 0.0) {
|
|
|
|
|
$s = $norm;
|
|
|
|
|
}
|
|
|
|
|
if (abs($this->H[$l][$l-1]) < $eps * $s) {
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
--$l;
|
|
|
|
|
}
|
|
|
|
|
// Check for convergence
|
|
|
|
|
// One root found
|
|
|
|
|
if ($l == $n) {
|
|
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift;
|
|
|
|
|
$this->d[$n] = $this->H[$n][$n];
|
|
|
|
|
$this->e[$n] = 0.0;
|
|
|
|
|
--$n;
|
|
|
|
|
$iter = 0;
|
|
|
|
|
// Two roots found
|
2015-05-18 15:39:04 +00:00
|
|
|
} elseif ($l == $n-1) {
|
2015-05-12 09:22:06 +00:00
|
|
|
$w = $this->H[$n][$n-1] * $this->H[$n-1][$n];
|
|
|
|
|
$p = ($this->H[$n-1][$n-1] - $this->H[$n][$n]) / 2.0;
|
|
|
|
|
$q = $p * $p + $w;
|
|
|
|
|
$z = sqrt(abs($q));
|
|
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift;
|
|
|
|
|
$this->H[$n-1][$n-1] = $this->H[$n-1][$n-1] + $exshift;
|
|
|
|
|
$x = $this->H[$n][$n];
|
|
|
|
|
// Real pair
|
|
|
|
|
if ($q >= 0) {
|
|
|
|
|
if ($p >= 0) {
|
|
|
|
|
$z = $p + $z;
|
|
|
|
|
} else {
|
|
|
|
|
$z = $p - $z;
|
|
|
|
|
}
|
|
|
|
|
$this->d[$n-1] = $x + $z;
|
|
|
|
|
$this->d[$n] = $this->d[$n-1];
|
|
|
|
|
if ($z != 0.0) {
|
|
|
|
|
$this->d[$n] = $x - $w / $z;
|
|
|
|
|
}
|
|
|
|
|
$this->e[$n-1] = 0.0;
|
|
|
|
|
$this->e[$n] = 0.0;
|
|
|
|
|
$x = $this->H[$n][$n-1];
|
|
|
|
|
$s = abs($x) + abs($z);
|
|
|
|
|
$p = $x / $s;
|
|
|
|
|
$q = $z / $s;
|
|
|
|
|
$r = sqrt($p * $p + $q * $q);
|
|
|
|
|
$p = $p / $r;
|
|
|
|
|
$q = $q / $r;
|
|
|
|
|
// Row modification
|
|
|
|
|
for ($j = $n-1; $j < $nn; ++$j) {
|
|
|
|
|
$z = $this->H[$n-1][$j];
|
|
|
|
|
$this->H[$n-1][$j] = $q * $z + $p * $this->H[$n][$j];
|
|
|
|
|
$this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z;
|
|
|
|
|
}
|
|
|
|
|
// Column modification
|
2015-05-17 12:16:53 +00:00
|
|
|
for ($i = 0; $i <= $n; ++$i) {
|
2015-05-12 09:22:06 +00:00
|
|
|
$z = $this->H[$i][$n-1];
|
|
|
|
|
$this->H[$i][$n-1] = $q * $z + $p * $this->H[$i][$n];
|
|
|
|
|
$this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z;
|
|
|
|
|
}
|
|
|
|
|
// Accumulate transformations
|
|
|
|
|
for ($i = $low; $i <= $high; ++$i) {
|
|
|
|
|
$z = $this->V[$i][$n-1];
|
|
|
|
|
$this->V[$i][$n-1] = $q * $z + $p * $this->V[$i][$n];
|
|
|
|
|
$this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z;
|
|
|
|
|
}
|
|
|
|
|
// Complex pair
|
|
|
|
|
} else {
|
|
|
|
|
$this->d[$n-1] = $x + $p;
|
|
|
|
|
$this->d[$n] = $x + $p;
|
|
|
|
|
$this->e[$n-1] = $z;
|
|
|
|
|
$this->e[$n] = -$z;
|
|
|
|
|
}
|
|
|
|
|
$n = $n - 2;
|
|
|
|
|
$iter = 0;
|
|
|
|
|
// No convergence yet
|
|
|
|
|
} else {
|
|
|
|
|
// Form shift
|
|
|
|
|
$x = $this->H[$n][$n];
|
|
|
|
|
$y = 0.0;
|
|
|
|
|
$w = 0.0;
|
|
|
|
|
if ($l < $n) {
|
|
|
|
|
$y = $this->H[$n-1][$n-1];
|
|
|
|
|
$w = $this->H[$n][$n-1] * $this->H[$n-1][$n];
|
|
|
|
|
}
|
|
|
|
|
// Wilkinson's original ad hoc shift
|
|
|
|
|
if ($iter == 10) {
|
|
|
|
|
$exshift += $x;
|
|
|
|
|
for ($i = $low; $i <= $n; ++$i) {
|
|
|
|
|
$this->H[$i][$i] -= $x;
|
|
|
|
|
}
|
|
|
|
|
$s = abs($this->H[$n][$n-1]) + abs($this->H[$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) {
|
|
|
|
|
$s = sqrt($s);
|
|
|
|
|
if ($y < $x) {
|
|
|
|
|
$s = -$s;
|
|
|
|
|
}
|
|
|
|
|
$s = $x - $w / (($y - $x) / 2.0 + $s);
|
|
|
|
|
for ($i = $low; $i <= $n; ++$i) {
|
|
|
|
|
$this->H[$i][$i] -= $s;
|
|
|
|
|
}
|
|
|
|
|
$exshift += $s;
|
|
|
|
|
$x = $y = $w = 0.964;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// Could check iteration count here.
|
|
|
|
|
$iter = $iter + 1;
|
|
|
|
|
// Look for two consecutive small sub-diagonal elements
|
|
|
|
|
$m = $n - 2;
|
|
|
|
|
while ($m >= $l) {
|
|
|
|
|
$z = $this->H[$m][$m];
|
|
|
|
|
$r = $x - $z;
|
|
|
|
|
$s = $y - $z;
|
|
|
|
|
$p = ($r * $s - $w) / $this->H[$m+1][$m] + $this->H[$m][$m+1];
|
|
|
|
|
$q = $this->H[$m+1][$m+1] - $z - $r - $s;
|
|
|
|
|
$r = $this->H[$m+2][$m+1];
|
|
|
|
|
$s = abs($p) + abs($q) + abs($r);
|
|
|
|
|
$p = $p / $s;
|
|
|
|
|
$q = $q / $s;
|
|
|
|
|
$r = $r / $s;
|
|
|
|
|
if ($m == $l) {
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
if (abs($this->H[$m][$m-1]) * (abs($q) + abs($r)) <
|
|
|
|
|
$eps * (abs($p) * (abs($this->H[$m-1][$m-1]) + abs($z) + abs($this->H[$m+1][$m+1])))) {
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
--$m;
|
|
|
|
|
}
|
|
|
|
|
for ($i = $m + 2; $i <= $n; ++$i) {
|
|
|
|
|
$this->H[$i][$i-2] = 0.0;
|
|
|
|
|
if ($i > $m+2) {
|
|
|
|
|
$this->H[$i][$i-3] = 0.0;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// Double QR step involving rows l:n and columns m:n
|
|
|
|
|
for ($k = $m; $k <= $n-1; ++$k) {
|
|
|
|
|
$notlast = ($k != $n-1);
|
|
|
|
|
if ($k != $m) {
|
|
|
|
|
$p = $this->H[$k][$k-1];
|
|
|
|
|
$q = $this->H[$k+1][$k-1];
|
|
|
|
|
$r = ($notlast ? $this->H[$k+2][$k-1] : 0.0);
|
|
|
|
|
$x = abs($p) + abs($q) + abs($r);
|
|
|
|
|
if ($x != 0.0) {
|
|
|
|
|
$p = $p / $x;
|
|
|
|
|
$q = $q / $x;
|
|
|
|
|
$r = $r / $x;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if ($x == 0.0) {
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
$s = sqrt($p * $p + $q * $q + $r * $r);
|
|
|
|
|
if ($p < 0) {
|
|
|
|
|
$s = -$s;
|
|
|
|
|
}
|
|
|
|
|
if ($s != 0) {
|
|
|
|
|
if ($k != $m) {
|
|
|
|
|
$this->H[$k][$k-1] = -$s * $x;
|
|
|
|
|
} elseif ($l != $m) {
|
|
|
|
|
$this->H[$k][$k-1] = -$this->H[$k][$k-1];
|
|
|
|
|
}
|
|
|
|
|
$p = $p + $s;
|
|
|
|
|
$x = $p / $s;
|
|
|
|
|
$y = $q / $s;
|
|
|
|
|
$z = $r / $s;
|
|
|
|
|
$q = $q / $p;
|
|
|
|
|
$r = $r / $p;
|
|
|
|
|
// Row modification
|
|
|
|
|
for ($j = $k; $j < $nn; ++$j) {
|
|
|
|
|
$p = $this->H[$k][$j] + $q * $this->H[$k+1][$j];
|
|
|
|
|
if ($notlast) {
|
|
|
|
|
$p = $p + $r * $this->H[$k+2][$j];
|
|
|
|
|
$this->H[$k+2][$j] = $this->H[$k+2][$j] - $p * $z;
|
|
|
|
|
}
|
|
|
|
|
$this->H[$k][$j] = $this->H[$k][$j] - $p * $x;
|
|
|
|
|
$this->H[$k+1][$j] = $this->H[$k+1][$j] - $p * $y;
|
|
|
|
|
}
|
|
|
|
|
// Column modification
|
|
|
|
|
for ($i = 0; $i <= min($n, $k+3); ++$i) {
|
|
|
|
|
$p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k+1];
|
|
|
|
|
if ($notlast) {
|
|
|
|
|
$p = $p + $z * $this->H[$i][$k+2];
|
|
|
|
|
$this->H[$i][$k+2] = $this->H[$i][$k+2] - $p * $r;
|
|
|
|
|
}
|
|
|
|
|
$this->H[$i][$k] = $this->H[$i][$k] - $p;
|
|
|
|
|
$this->H[$i][$k+1] = $this->H[$i][$k+1] - $p * $q;
|
|
|
|
|
}
|
|
|
|
|
// Accumulate transformations
|
|
|
|
|
for ($i = $low; $i <= $high; ++$i) {
|
|
|
|
|
$p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k+1];
|
|
|
|
|
if ($notlast) {
|
|
|
|
|
$p = $p + $z * $this->V[$i][$k+2];
|
|
|
|
|
$this->V[$i][$k+2] = $this->V[$i][$k+2] - $p * $r;
|
|
|
|
|
}
|
|
|
|
|
$this->V[$i][$k] = $this->V[$i][$k] - $p;
|
|
|
|
|
$this->V[$i][$k+1] = $this->V[$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 = $this->d[$n];
|
|
|
|
|
$q = $this->e[$n];
|
|
|
|
|
// Real vector
|
|
|
|
|
if ($q == 0) {
|
|
|
|
|
$l = $n;
|
|
|
|
|
$this->H[$n][$n] = 1.0;
|
|
|
|
|
for ($i = $n-1; $i >= 0; --$i) {
|
|
|
|
|
$w = $this->H[$i][$i] - $p;
|
|
|
|
|
$r = 0.0;
|
|
|
|
|
for ($j = $l; $j <= $n; ++$j) {
|
|
|
|
|
$r = $r + $this->H[$i][$j] * $this->H[$j][$n];
|
|
|
|
|
}
|
|
|
|
|
if ($this->e[$i] < 0.0) {
|
|
|
|
|
$z = $w;
|
|
|
|
|
$s = $r;
|
|
|
|
|
} else {
|
|
|
|
|
$l = $i;
|
|
|
|
|
if ($this->e[$i] == 0.0) {
|
|
|
|
|
if ($w != 0.0) {
|
|
|
|
|
$this->H[$i][$n] = -$r / $w;
|
|
|
|
|
} else {
|
|
|
|
|
$this->H[$i][$n] = -$r / ($eps * $norm);
|
|
|
|
|
}
|
|
|
|
|
// Solve real equations
|
|
|
|
|
} else {
|
|
|
|
|
$x = $this->H[$i][$i+1];
|
|
|
|
|
$y = $this->H[$i+1][$i];
|
|
|
|
|
$q = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i];
|
|
|
|
|
$t = ($x * $s - $z * $r) / $q;
|
|
|
|
|
$this->H[$i][$n] = $t;
|
|
|
|
|
if (abs($x) > abs($z)) {
|
|
|
|
|
$this->H[$i+1][$n] = (-$r - $w * $t) / $x;
|
|
|
|
|
} else {
|
|
|
|
|
$this->H[$i+1][$n] = (-$s - $y * $t) / $z;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// Overflow control
|
|
|
|
|
$t = abs($this->H[$i][$n]);
|
|
|
|
|
if (($eps * $t) * $t > 1) {
|
|
|
|
|
for ($j = $i; $j <= $n; ++$j) {
|
|
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// Complex vector
|
2015-05-18 15:39:04 +00:00
|
|
|
} elseif ($q < 0) {
|
2015-05-12 09:22:06 +00:00
|
|
|
$l = $n-1;
|
|
|
|
|
// Last vector component imaginary so matrix is triangular
|
|
|
|
|
if (abs($this->H[$n][$n-1]) > abs($this->H[$n-1][$n])) {
|
|
|
|
|
$this->H[$n-1][$n-1] = $q / $this->H[$n][$n-1];
|
|
|
|
|
$this->H[$n-1][$n] = -($this->H[$n][$n] - $p) / $this->H[$n][$n-1];
|
|
|
|
|
} else {
|
|
|
|
|
$this->cdiv(0.0, -$this->H[$n-1][$n], $this->H[$n-1][$n-1] - $p, $q);
|
|
|
|
|
$this->H[$n-1][$n-1] = $this->cdivr;
|
|
|
|
|
$this->H[$n-1][$n] = $this->cdivi;
|
|
|
|
|
}
|
|
|
|
|
$this->H[$n][$n-1] = 0.0;
|
|
|
|
|
$this->H[$n][$n] = 1.0;
|
|
|
|
|
for ($i = $n-2; $i >= 0; --$i) {
|
|
|
|
|
// double ra,sa,vr,vi;
|
|
|
|
|
$ra = 0.0;
|
|
|
|
|
$sa = 0.0;
|
|
|
|
|
for ($j = $l; $j <= $n; ++$j) {
|
|
|
|
|
$ra = $ra + $this->H[$i][$j] * $this->H[$j][$n-1];
|
|
|
|
|
$sa = $sa + $this->H[$i][$j] * $this->H[$j][$n];
|
|
|
|
|
}
|
|
|
|
|
$w = $this->H[$i][$i] - $p;
|
|
|
|
|
if ($this->e[$i] < 0.0) {
|
|
|
|
|
$z = $w;
|
|
|
|
|
$r = $ra;
|
|
|
|
|
$s = $sa;
|
|
|
|
|
} else {
|
|
|
|
|
$l = $i;
|
|
|
|
|
if ($this->e[$i] == 0) {
|
|
|
|
|
$this->cdiv(-$ra, -$sa, $w, $q);
|
|
|
|
|
$this->H[$i][$n-1] = $this->cdivr;
|
|
|
|
|
$this->H[$i][$n] = $this->cdivi;
|
|
|
|
|
} else {
|
|
|
|
|
// Solve complex equations
|
|
|
|
|
$x = $this->H[$i][$i+1];
|
|
|
|
|
$y = $this->H[$i+1][$i];
|
|
|
|
|
$vr = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i] - $q * $q;
|
|
|
|
|
$vi = ($this->d[$i] - $p) * 2.0 * $q;
|
|
|
|
|
if ($vr == 0.0 & $vi == 0.0) {
|
|
|
|
|
$vr = $eps * $norm * (abs($w) + abs($q) + abs($x) + abs($y) + abs($z));
|
|
|
|
|
}
|
|
|
|
|
$this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi);
|
|
|
|
|
$this->H[$i][$n-1] = $this->cdivr;
|
|
|
|
|
$this->H[$i][$n] = $this->cdivi;
|
|
|
|
|
if (abs($x) > (abs($z) + abs($q))) {
|
|
|
|
|
$this->H[$i+1][$n-1] = (-$ra - $w * $this->H[$i][$n-1] + $q * $this->H[$i][$n]) / $x;
|
|
|
|
|
$this->H[$i+1][$n] = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n-1]) / $x;
|
|
|
|
|
} else {
|
|
|
|
|
$this->cdiv(-$r - $y * $this->H[$i][$n-1], -$s - $y * $this->H[$i][$n], $z, $q);
|
|
|
|
|
$this->H[$i+1][$n-1] = $this->cdivr;
|
|
|
|
|
$this->H[$i+1][$n] = $this->cdivi;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// Overflow control
|
2015-05-17 13:00:02 +00:00
|
|
|
$t = max(abs($this->H[$i][$n-1]), abs($this->H[$i][$n]));
|
2015-05-12 09:22:06 +00:00
|
|
|
if (($eps * $t) * $t > 1) {
|
|
|
|
|
for ($j = $i; $j <= $n; ++$j) {
|
|
|
|
|
$this->H[$j][$n-1] = $this->H[$j][$n-1] / $t;
|
|
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
} // end else
|
|
|
|
|
} // end for
|
|
|
|
|
} // end else for complex case
|
|
|
|
|
} // end for
|
|
|
|
|
|
|
|
|
|
// Vectors of isolated roots
|
|
|
|
|
for ($i = 0; $i < $nn; ++$i) {
|
|
|
|
|
if ($i < $low | $i > $high) {
|
|
|
|
|
for ($j = $i; $j < $nn; ++$j) {
|
|
|
|
|
$this->V[$i][$j] = $this->H[$i][$j];
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// Back transformation to get eigenvectors of original matrix
|
|
|
|
|
for ($j = $nn-1; $j >= $low; --$j) {
|
|
|
|
|
for ($i = $low; $i <= $high; ++$i) {
|
|
|
|
|
$z = 0.0;
|
2015-05-13 10:27:01 +00:00
|
|
|
for ($k = $low; $k <= min($j, $high); ++$k) {
|
2015-05-12 09:22:06 +00:00
|
|
|
$z = $z + $this->V[$i][$k] * $this->H[$k][$j];
|
|
|
|
|
}
|
|
|
|
|
$this->V[$i][$j] = $z;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
} // end hqr2
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Constructor: Check for symmetry, then construct the eigenvalue decomposition
|
|
|
|
|
*
|
|
|
|
|
* @access public
|
|
|
|
|
* @param A Square matrix
|
|
|
|
|
* @return Structure to access D and V.
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
public function __construct($Arg)
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
$this->A = $Arg->getArray();
|
|
|
|
|
$this->n = $Arg->getColumnDimension();
|
|
|
|
|
|
|
|
|
|
$issymmetric = true;
|
|
|
|
|
for ($j = 0; ($j < $this->n) & $issymmetric; ++$j) {
|
|
|
|
|
for ($i = 0; ($i < $this->n) & $issymmetric; ++$i) {
|
|
|
|
|
$issymmetric = ($this->A[$i][$j] == $this->A[$j][$i]);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
if ($issymmetric) {
|
|
|
|
|
$this->V = $this->A;
|
|
|
|
|
// Tridiagonalize.
|
|
|
|
|
$this->tred2();
|
|
|
|
|
// Diagonalize.
|
|
|
|
|
$this->tql2();
|
|
|
|
|
} else {
|
|
|
|
|
$this->H = $this->A;
|
|
|
|
|
$this->ort = array();
|
|
|
|
|
// Reduce to Hessenberg form.
|
|
|
|
|
$this->orthes();
|
|
|
|
|
// Reduce Hessenberg to real Schur form.
|
|
|
|
|
$this->hqr2();
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Return the eigenvector matrix
|
|
|
|
|
*
|
|
|
|
|
* @access public
|
|
|
|
|
* @return V
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
public function getV()
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
return new Matrix($this->V, $this->n, $this->n);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Return the real parts of the eigenvalues
|
|
|
|
|
*
|
|
|
|
|
* @access public
|
|
|
|
|
* @return real(diag(D))
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
public function getRealEigenvalues()
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
return $this->d;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Return the imaginary parts of the eigenvalues
|
|
|
|
|
*
|
|
|
|
|
* @access public
|
|
|
|
|
* @return imag(diag(D))
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
public function getImagEigenvalues()
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
return $this->e;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Return the block diagonal eigenvalue matrix
|
|
|
|
|
*
|
|
|
|
|
* @access public
|
|
|
|
|
* @return D
|
|
|
|
|
*/
|
2015-05-17 12:16:53 +00:00
|
|
|
public function getD()
|
|
|
|
|
{
|
2015-05-12 09:22:06 +00:00
|
|
|
for ($i = 0; $i < $this->n; ++$i) {
|
|
|
|
|
$D[$i] = array_fill(0, $this->n, 0.0);
|
|
|
|
|
$D[$i][$i] = $this->d[$i];
|
|
|
|
|
if ($this->e[$i] == 0) {
|
|
|
|
|
continue;
|
|
|
|
|
}
|
|
|
|
|
$o = ($this->e[$i] > 0) ? $i + 1 : $i - 1;
|
|
|
|
|
$D[$i][$o] = $this->e[$i];
|
|
|
|
|
}
|
|
|
|
|
return new Matrix($D);
|
|
|
|
|
}
|
2015-05-17 12:16:53 +00:00
|
|
|
}
|
