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| author | Anton Luka Šijanec <anton@sijanec.eu> | 2022-01-11 12:35:47 +0100 |
|---|---|---|
| committer | Anton Luka Šijanec <anton@sijanec.eu> | 2022-01-11 12:35:47 +0100 |
| commit | 19985dbb8c0aa66dc4bf7905abc1148de909097d (patch) | |
| tree | 2cd5a5d20d7e80fc2a51adf60d838d8a2c40999e /admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php | |
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Diffstat (limited to 'admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php')
| -rw-r--r-- | admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php | 862 |
1 files changed, 862 insertions, 0 deletions
diff --git a/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php b/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php new file mode 100644 index 0000000..716af82 --- /dev/null +++ b/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php @@ -0,0 +1,862 @@ +<?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
+ */
+ private $n;
+
+ /**
+ * Internal symmetry flag.
+ * @var int
+ */
+ private $issymmetric;
+
+ /**
+ * Arrays for internal storage of eigenvalues.
+ * @var array
+ */
+ private $d = array();
+ private $e = array();
+
+ /**
+ * Array for internal storage of eigenvectors.
+ * @var array
+ */
+ private $V = array();
+
+ /**
+ * Array for internal storage of nonsymmetric Hessenberg form.
+ * @var array
+ */
+ private $H = array();
+
+ /**
+ * Working storage for nonsymmetric algorithm.
+ * @var array
+ */
+ private $ort;
+
+ /**
+ * Used for complex scalar division.
+ * @var float
+ */
+ private $cdivr;
+ private $cdivi;
+
+
+ /**
+ * Symmetric Householder reduction to tridiagonal form.
+ *
+ * @access private
+ */
+ 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;
+ $this->e[0] = 0.0;
+ }
+
+
+ /**
+ * Symmetric tridiagonal QL algorithm.
+ *
+ * This is derived from the Algol procedures tql2, by
+ * Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
+ * Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
+ * Fortran subroutine in EISPACK.
+ *
+ * @access private
+ */
+ private function tql2() {
+ for ($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
+ */
+ 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
+ */
+ 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
+ */
+ 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 * $x;
+ } elseif ($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] / $t;
+ }
+ }
+ }
+ }
+ // Complex vector
+ } else if ($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] = $z;
+ }
+ }
+ } // end hqr2
+
+
+ /**
+ * Constructor: Check for symmetry, then construct the eigenvalue decomposition
+ *
+ * @access public
+ * @param A Square matrix
+ * @return Structure to access D and V.
+ */
+ public function __construct($Arg) {
+ $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.
+ $this->hqr2();
+ }
+ }
+
+
+ /**
+ * Return the eigenvector matrix
+ *
+ * @access public
+ * @return V
+ */
+ public function getV() {
+ return new Matrix($this->V, $this->n, $this->n);
+ }
+
+
+ /**
+ * Return the real parts of the eigenvalues
+ *
+ * @access public
+ * @return real(diag(D))
+ */
+ public function getRealEigenvalues() {
+ return $this->d;
+ }
+
+
+ /**
+ * Return the imaginary parts of the eigenvalues
+ *
+ * @access public
+ * @return imag(diag(D))
+ */
+ public function getImagEigenvalues() {
+ return $this->e;
+ }
+
+
+ /**
+ * Return the block diagonal eigenvalue matrix
+ *
+ * @access public
+ * @return D
+ */
+ public function getD() {
+ for ($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);
+ }
+
+} // class EigenvalueDecomposition
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