File Source for EigenvalueDecomposition.php

<?php
/**
* @package JAMA
*
* Class to obtain eigenvalues and eigenvectors of a real matrix.
*
* If A is symmetric, then A = V*D*V' where the eigenvalue matrix D
* is diagonal and the eigenvector matrix V is orthogonal (i.e.
* A = V.times(D.times(V.transpose())) and V.times(V.transpose())
* equals the identity matrix).
*
* If A is not symmetric, then the eigenvalue matrix D is block diagonal
* with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
* lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The
* columns of V represent the eigenvectors in the sense that A*V = V*D,
* i.e. A.times(V) equals V.times(D). The matrix V may be badly
* conditioned, or even singular, so the validity of the equation
* A = V*D*inverse(V) depends upon V.cond().
*
* @author Paul Meagher
* @license PHP v3.0
* @version 1.1
*/
class EigenvalueDecomposition {
/**
* Row and column dimension (square matrix).
* @var int
*/
var $n;
/**
* Internal symmetry flag.
* @var int
*/
var $issymmetric;
/**
* Arrays for internal storage of eigenvalues.
* @var array
*/
var $d = array();
var $e = array();
/**
* Array for internal storage of eigenvectors.
* @var array
*/
var $V = array();
/**
* Array for internal storage of nonsymmetric Hessenberg form.
* @var array
*/
var $H = array();
/**
* Working storage for nonsymmetric algorithm.
* @var array
*/
var $ort;
/**
* Used for complex scalar division.
* @var float
*/
var $cdivr;
var $cdivi;
/**
* Symmetric Householder reduction to tridiagonal form.
* @access private
*/
function 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.
$this->d = $this->V[$this->n-1];
// Householder reduction to tridiagonal form.
for ($i = $this->n-1; $i > 0; $i--) {
$i_ = $i -1;
// Scale to avoid under/overflow.
$h = $scale = 0.0;
$scale += array_sum(array_map(abs, $this->d));
if ($scale == 0.0) {
$this->e[$i] = $this->d[$i_];
$this->d = array_slice($this->V[$i_], 0, $i_);
for ($j = 0; $j < $i; $j++) {
$this->V[$j][$i] = $this->V[$i][$j] = 0.0;
}
} else {
// Generate Householder vector.
for ($k = 0; $k < $i; $k++) {
$this->d[$k] /= $scale;
$h += pow($this->d[$k], 2);
}
$f = $this->d[$i_];
$g = sqrt($h);
if ($f > 0)
$g = -$g;
$this->e[$i] = $scale * $g;
$h = $h - $f * $g;
$this->d[$i_] = $f - $g;
for ($j = 0; $j < $i; $j++)
$this->e[$j] = 0.0;
// Apply similarity transformation to remaining columns.
for ($j = 0; $j < $i; $j++) {
$f = $this->d[$j];
$this->V[$j][$i] = $f;
$g = $this->e[$j] + $this->V[$j][$j] * $f;
for ($k = $j+1; $k <= $i_; $k++) {
$g += $this->V[$k][$j] * $this->d[$k];
$this->e[$k] += $this->V[$k][$j] * $f;
}
$this->e[$j] = $g;
}
$f = 0.0;
for ($j = 0; $j < $i; $j++) {
$this->e[$j] /= $h;
$f += $this->e[$j] * $this->d[$j];
}
$hh = $f / (2 * $h);
for ($j=0; $j < $i; $j++)
$this->e[$j] -= $hh * $this->d[$j];
for ($j = 0; $j < $i; $j++) {
$f = $this->d[$j];
$g = $this->e[$j];
for ($k = $j; $k <= $i_; $k++)
$this->V[$k][$j] -= ($f * $this->e[$k] + $g * $this->d[$k]);
$this->d[$j] = $this->V[$i-1][$j];
$this->V[$i][$j] = 0.0;
}
}
$this->d[$i] = $h;
}
// Accumulate transformations.
for ($i = 0; $i < $this->n-1; $i++) {
$this->V[$this->n-1][$i] = $this->V[$i][$i];
$this->V[$i][$i] = 1.0;
$h = $this->d[$i+1];
if ($h != 0.0) {
for ($k = 0; $k <= $i; $k++)
$this->d[$k] = $this->V[$k][$i+1] / $h;
for ($j = 0; $j <= $i; $j++) {
$g = 0.0;
for ($k = 0; $k <= $i; $k++)
$g += $this->V[$k][$i+1] * $this->V[$k][$j];
for ($k = 0; $k <= $i; $k++)
$this->V[$k][$j] -= $g * $this->d[$k];
}
}
for ($k = 0; $k <= $i; $k++)
$this->V[$k][$i+1] = 0.0;
}
$this->d = $this->V[$this->n-1];
$this->V[$this->n-1] = array_fill(0, $j, 0.0);
$this->V[$this->n-1][$this->n-1] = 1.0;
$this->e[0] = 0.0;
}
/**
* Symmetric tridiagonal QL algorithm.
*
* This is derived from the Algol procedures tql2, by
* Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
* Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
* Fortran subroutine in EISPACK.
*
* @access private
*/
function tql2 () {
for ($i = 1; $i < $this->n; $i++)
$this->e[$i-1] = $this->e[$i];
$this->e[$this->n-1] = 0.0;
$f = 0.0;
$tst1 = 0.0;
$eps = pow(2.0,-52.0);
for ($l = 0; $l < $this->n; $l++) {
// Find small subdiagonal element
$tst1 = max($tst1, abs($this->d[$l]) + abs($this->e[$l]));
$m = $l;
while ($m < $this->n) {
if (abs($this->e[$m]) <= $eps * $tst1)
break;
$m++;
}
// If m == l, $this->d[l] is an eigenvalue,
// otherwise, iterate.
if ($m > $l) {
$iter = 0;
do {
// Could check iteration count here.
$iter += 1;
// Compute implicit shift
$g = $this->d[$l];
$p = ($this->d[$l+1] - $g) / (2.0 * $this->e[$l]);
$r = hypo($p, 1.0);
if ($p < 0)
$r *= -1;
$this->d[$l] = $this->e[$l] / ($p + $r);
$this->d[$l+1] = $this->e[$l] * ($p + $r);
$dl1 = $this->d[$l+1];
$h = $g - $this->d[$l];
for ($i = $l + 2; $i < $this->n; $i++)
$this->d[$i] -= $h;
$f += $h;
// Implicit QL transformation.
$p = $this->d[$m];
$c = 1.0;
$c2 = $c3 = $c;
$el1 = $this->e[$l + 1];
$s = $s2 = 0.0;
for ($i = $m-1; $i >= $l; $i--) {
$c3 = $c2;
$c2 = $c;
$s2 = $s;
$g = $c * $this->e[$i];
$h = $c * $p;
$r = hypo($p, $this->e[$i]);
$this->e[$i+1] = $s * $r;
$s = $this->e[$i] / $r;
$c = $p / $r;
$p = $c * $this->d[$i] - $s * $g;
$this->d[$i+1] = $h + $s * ($c * $g + $s * $this->d[$i]);
// Accumulate transformation.
for ($k = 0; $k < $this->n; $k++) {
$h = $this->V[$k][$i+1];
$this->V[$k][$i+1] = $s * $this->V[$k][$i] + $c * $h;
$this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h;
}
}
$p = -$s * $s2 * $c3 * $el1 * $this->e[$l] / $dl1;
$this->e[$l] = $s * $p;
$this->d[$l] = $c * $p;
// Check for convergence.
} while (abs($this->e[$l]) > $eps * $tst1);
}
$this->d[$l] = $this->d[$l] + $f;
$this->e[$l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for ($i = 0; $i < $this->n - 1; $i++) {
$k = $i;
$p = $this->d[$i];
for ($j = $i+1; $j < $this->n; $j++) {
if ($this->d[$j] < $p) {
$k = $j;
$p = $this->d[$j];
}
}
if ($k != $i) {
$this->d[$k] = $this->d[$i];
$this->d[$i] = $p;
for ($j = 0; $j < $this->n; $j++) {
$p = $this->V[$j][$i];
$this->V[$j][$i] = $this->V[$j][$k];
$this->V[$j][$k] = $p;
}
}
}
}
/**
* Nonsymmetric reduction to Hessenberg form.
*
* 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.
*
* @access private
*/
function orthes () {
$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
*/
function cdiv($xr, $xi, $yr, $yi) {
if (abs($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
*/
function hqr2 () {
// 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; $i++) {
if (($i < $low) OR ($i > $high)) {
$this->d[$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--;
$iter = 0;
// Two roots found
} else if ($l == $n-1) {
$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 * $z;
}
// Column modification
for ($i = 0; $i <= n; $i++) {
$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 * $x;
else if ($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] / $t;
}
}
}
// Complex vector
} else if ($q < 0) {
$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
$t = max(abs($this->H[$i][$n-1]),abs($this->H[$i][$n]));
if (($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++) {
$z = 0.0;
for ($k = $low; $k <= min($j,$high); $k++)
$z = $z + $this->V[$i][$k] * $this->H[$k][$j];
$this->V[$i][$j] = $z;
}
}
} // end hqr2
/**
* Constructor: Check for symmetry, then construct the eigenvalue decomposition
* @access public
* @param A Square matrix
* @return Structure to access D and V.
*/
function EigenvalueDecomposition($Arg) {
$this->A = $Arg->getArray();
$this->n = $Arg->getColumnDimension();
$this->V = array();
$this->d = array();
$this->e = array();
$issymmetric = true;
for ($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.
$this->hqr2();
}
}
/**
* Return the eigenvector matrix
* @access public
* @return V
*/
function getV() {
return new Matrix($this->V, $this->n, $this->n);
}
/**
* Return the real parts of the eigenvalues
* @access public
* @return real(diag(D))
*/
function getRealEigenvalues() {
return $this->d;
}
/**
* Return the imaginary parts of the eigenvalues
* @access public
* @return imag(diag(D))
*/
function getImagEigenvalues() {
return $this->e;
}
/**
* Return the block diagonal eigenvalue matrix
* @access public
* @return D
*/
function getD() {
for ($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);
}
}

Source for file EigenvalueDecomposition.php