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Diffstat (limited to 'admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php')
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diff --git a/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php b/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php deleted file mode 100644 index 716af82..0000000 --- a/admin/survey/excel/PHPExcel/Shared/JAMA/EigenvalueDecomposition.php +++ /dev/null @@ -1,862 +0,0 @@ -<?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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