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<?php declare(strict_types=1); |
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/** |
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* |
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* Class to obtain eigenvalues and eigenvectors of a real matrix. |
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* |
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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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* |
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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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* |
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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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* |
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* Slightly changed to adapt the original code to PHP-ML library |
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* @date 2017/04/11 |
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* @author Mustafa Karabulut |
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*/ |
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namespace Phpml\Math\LinAlg; |
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use Phpml\Math\Matrix; |
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class EigenvalueDecomposition |
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{ |
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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 = []; |
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private $e = []; |
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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 = []; |
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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 = []; |
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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 $cdivi; |
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/** |
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* Symmetric Householder reduction to tridiagonal form. |
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*/ |
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private function tred2() |
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{ |
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// This is derived from the Algol procedures tred2 by |
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// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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// Fortran subroutine in EISPACK. |
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$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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if ($h === 0) { |
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$h = 1e-20; |
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} |
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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 ($h != 0.0) { |
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View Code Duplication |
for ($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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$g = 0.0; |
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View Code Duplication |
for ($k = 0; $k <= $i; ++$k) { |
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$g += $this->V[$k][$i+1] * $this->V[$k][$j]; |
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} |
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View Code Duplication |
for ($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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$this->e[0] = 0.0; |
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} |
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/** |
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* Symmetric tridiagonal QL algorithm. |
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* |
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* This is derived from the Algol procedures tql2, by |
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* Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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* Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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* Fortran subroutine in EISPACK. |
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*/ |
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private function tql2() |
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{ |
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for ($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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$eps = pow(2.0, -52.0); |
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for ($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 < $this->n) { |
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if (abs($this->e[$m]) <= $eps * $tst1) { |
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break; |
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} |
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++$m; |
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} |
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// If m == l, $this->d[l] is an eigenvalue, |
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// otherwise, iterate. |
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if ($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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$r = hypot($p, 1.0); |
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if ($p < 0) { |
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$r *= -1; |
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} |
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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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for ($i = $l + 2; $i < $this->n; ++$i) { |
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$this->d[$i] -= $h; |
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} |
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$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 = hypot($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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293
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294
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/** |
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295
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* Nonsymmetric reduction to Hessenberg form. |
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296
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* |
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297
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* This is derived from the Algol procedures orthes and ortran, |
|
298
|
|
|
* by Martin and Wilkinson, Handbook for Auto. Comp., |
|
299
|
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* Vol.ii-Linear Algebra, and the corresponding |
|
300
|
|
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* Fortran subroutines in EISPACK. |
|
301
|
|
|
*/ |
|
302
|
|
|
private function orthes() |
|
303
|
|
|
{ |
|
304
|
|
|
$low = 0; |
|
305
|
|
|
$high = $this->n-1; |
|
306
|
|
|
|
|
307
|
|
|
for ($m = $low+1; $m <= $high-1; ++$m) { |
|
308
|
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|
// Scale column. |
|
309
|
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|
$scale = 0.0; |
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310
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View Code Duplication |
for ($i = $m; $i <= $high; ++$i) { |
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|
|
|
|
|
311
|
|
|
$scale = $scale + abs($this->H[$i][$m-1]); |
|
312
|
|
|
} |
|
313
|
|
|
if ($scale != 0.0) { |
|
314
|
|
|
// Compute Householder transformation. |
|
315
|
|
|
$h = 0.0; |
|
316
|
|
|
for ($i = $high; $i >= $m; --$i) { |
|
317
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$this->ort[$i] = $this->H[$i][$m-1] / $scale; |
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318
|
|
|
$h += $this->ort[$i] * $this->ort[$i]; |
|
319
|
|
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} |
|
320
|
|
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$g = sqrt($h); |
|
321
|
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if ($this->ort[$m] > 0) { |
|
322
|
|
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$g *= -1; |
|
323
|
|
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} |
|
324
|
|
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$h -= $this->ort[$m] * $g; |
|
325
|
|
|
$this->ort[$m] -= $g; |
|
326
|
|
|
// Apply Householder similarity transformation |
|
327
|
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// H = (I -u * u' / h) * H * (I -u * u') / h) |
|
328
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|
View Code Duplication |
for ($j = $m; $j < $this->n; ++$j) { |
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|
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|
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329
|
|
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$f = 0.0; |
|
330
|
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for ($i = $high; $i >= $m; --$i) { |
|
331
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|
$f += $this->ort[$i] * $this->H[$i][$j]; |
|
332
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} |
|
333
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|
$f /= $h; |
|
334
|
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|
for ($i = $m; $i <= $high; ++$i) { |
|
335
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|
$this->H[$i][$j] -= $f * $this->ort[$i]; |
|
336
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|
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} |
|
337
|
|
|
} |
|
338
|
|
View Code Duplication |
for ($i = 0; $i <= $high; ++$i) { |
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|
|
|
|
|
339
|
|
|
$f = 0.0; |
|
340
|
|
|
for ($j = $high; $j >= $m; --$j) { |
|
341
|
|
|
$f += $this->ort[$j] * $this->H[$i][$j]; |
|
342
|
|
|
} |
|
343
|
|
|
$f = $f / $h; |
|
344
|
|
|
for ($j = $m; $j <= $high; ++$j) { |
|
345
|
|
|
$this->H[$i][$j] -= $f * $this->ort[$j]; |
|
346
|
|
|
} |
|
347
|
|
|
} |
|
348
|
|
|
$this->ort[$m] = $scale * $this->ort[$m]; |
|
349
|
|
|
$this->H[$m][$m-1] = $scale * $g; |
|
350
|
|
|
} |
|
351
|
|
|
} |
|
352
|
|
|
|
|
353
|
|
|
// Accumulate transformations (Algol's ortran). |
|
354
|
|
|
for ($i = 0; $i < $this->n; ++$i) { |
|
355
|
|
|
for ($j = 0; $j < $this->n; ++$j) { |
|
356
|
|
|
$this->V[$i][$j] = ($i == $j ? 1.0 : 0.0); |
|
357
|
|
|
} |
|
358
|
|
|
} |
|
359
|
|
|
for ($m = $high-1; $m >= $low+1; --$m) { |
|
360
|
|
|
if ($this->H[$m][$m-1] != 0.0) { |
|
361
|
|
View Code Duplication |
for ($i = $m+1; $i <= $high; ++$i) { |
|
|
|
|
|
|
362
|
|
|
$this->ort[$i] = $this->H[$i][$m-1]; |
|
363
|
|
|
} |
|
364
|
|
|
for ($j = $m; $j <= $high; ++$j) { |
|
365
|
|
|
$g = 0.0; |
|
366
|
|
|
for ($i = $m; $i <= $high; ++$i) { |
|
367
|
|
|
$g += $this->ort[$i] * $this->V[$i][$j]; |
|
368
|
|
|
} |
|
369
|
|
|
// Double division avoids possible underflow |
|
370
|
|
|
$g = ($g / $this->ort[$m]) / $this->H[$m][$m-1]; |
|
371
|
|
|
for ($i = $m; $i <= $high; ++$i) { |
|
372
|
|
|
$this->V[$i][$j] += $g * $this->ort[$i]; |
|
373
|
|
|
} |
|
374
|
|
|
} |
|
375
|
|
|
} |
|
376
|
|
|
} |
|
377
|
|
|
} |
|
378
|
|
|
|
|
379
|
|
|
|
|
380
|
|
|
/** |
|
381
|
|
|
* Performs complex division. |
|
382
|
|
|
*/ |
|
383
|
|
|
private function cdiv($xr, $xi, $yr, $yi) |
|
384
|
|
|
{ |
|
385
|
|
|
if (abs($yr) > abs($yi)) { |
|
386
|
|
|
$r = $yi / $yr; |
|
387
|
|
|
$d = $yr + $r * $yi; |
|
388
|
|
|
$this->cdivr = ($xr + $r * $xi) / $d; |
|
389
|
|
|
$this->cdivi = ($xi - $r * $xr) / $d; |
|
390
|
|
|
} else { |
|
391
|
|
|
$r = $yr / $yi; |
|
392
|
|
|
$d = $yi + $r * $yr; |
|
393
|
|
|
$this->cdivr = ($r * $xr + $xi) / $d; |
|
394
|
|
|
$this->cdivi = ($r * $xi - $xr) / $d; |
|
395
|
|
|
} |
|
396
|
|
|
} |
|
397
|
|
|
|
|
398
|
|
|
|
|
399
|
|
|
/** |
|
400
|
|
|
* Nonsymmetric reduction from Hessenberg to real Schur form. |
|
401
|
|
|
* |
|
402
|
|
|
* Code is derived from the Algol procedure hqr2, |
|
403
|
|
|
* by Martin and Wilkinson, Handbook for Auto. Comp., |
|
404
|
|
|
* Vol.ii-Linear Algebra, and the corresponding |
|
405
|
|
|
* Fortran subroutine in EISPACK. |
|
406
|
|
|
*/ |
|
407
|
|
|
private function hqr2() |
|
408
|
|
|
{ |
|
409
|
|
|
// Initialize |
|
410
|
|
|
$nn = $this->n; |
|
411
|
|
|
$n = $nn - 1; |
|
412
|
|
|
$low = 0; |
|
413
|
|
|
$high = $nn - 1; |
|
414
|
|
|
$eps = pow(2.0, -52.0); |
|
415
|
|
|
$exshift = 0.0; |
|
416
|
|
|
$p = $q = $r = $s = $z = 0; |
|
417
|
|
|
// Store roots isolated by balanc and compute matrix norm |
|
418
|
|
|
$norm = 0.0; |
|
419
|
|
|
|
|
420
|
|
|
for ($i = 0; $i < $nn; ++$i) { |
|
421
|
|
|
if (($i < $low) or ($i > $high)) { |
|
|
|
|
|
|
422
|
|
|
$this->d[$i] = $this->H[$i][$i]; |
|
423
|
|
|
$this->e[$i] = 0.0; |
|
424
|
|
|
} |
|
425
|
|
|
for ($j = max($i-1, 0); $j < $nn; ++$j) { |
|
426
|
|
|
$norm = $norm + abs($this->H[$i][$j]); |
|
427
|
|
|
} |
|
428
|
|
|
} |
|
429
|
|
|
|
|
430
|
|
|
// Outer loop over eigenvalue index |
|
431
|
|
|
$iter = 0; |
|
432
|
|
|
while ($n >= $low) { |
|
433
|
|
|
// Look for single small sub-diagonal element |
|
434
|
|
|
$l = $n; |
|
435
|
|
|
while ($l > $low) { |
|
436
|
|
|
$s = abs($this->H[$l-1][$l-1]) + abs($this->H[$l][$l]); |
|
437
|
|
|
if ($s == 0.0) { |
|
438
|
|
|
$s = $norm; |
|
439
|
|
|
} |
|
440
|
|
|
if (abs($this->H[$l][$l-1]) < $eps * $s) { |
|
441
|
|
|
break; |
|
442
|
|
|
} |
|
443
|
|
|
--$l; |
|
444
|
|
|
} |
|
445
|
|
|
// Check for convergence |
|
446
|
|
|
// One root found |
|
447
|
|
|
if ($l == $n) { |
|
448
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
|
449
|
|
|
$this->d[$n] = $this->H[$n][$n]; |
|
450
|
|
|
$this->e[$n] = 0.0; |
|
451
|
|
|
--$n; |
|
452
|
|
|
$iter = 0; |
|
453
|
|
|
// Two roots found |
|
454
|
|
|
} elseif ($l == $n-1) { |
|
455
|
|
|
$w = $this->H[$n][$n-1] * $this->H[$n-1][$n]; |
|
456
|
|
|
$p = ($this->H[$n-1][$n-1] - $this->H[$n][$n]) / 2.0; |
|
457
|
|
|
$q = $p * $p + $w; |
|
458
|
|
|
$z = sqrt(abs($q)); |
|
459
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
|
460
|
|
|
$this->H[$n-1][$n-1] = $this->H[$n-1][$n-1] + $exshift; |
|
461
|
|
|
$x = $this->H[$n][$n]; |
|
462
|
|
|
// Real pair |
|
463
|
|
|
if ($q >= 0) { |
|
464
|
|
|
if ($p >= 0) { |
|
465
|
|
|
$z = $p + $z; |
|
466
|
|
|
} else { |
|
467
|
|
|
$z = $p - $z; |
|
468
|
|
|
} |
|
469
|
|
|
$this->d[$n-1] = $x + $z; |
|
470
|
|
|
$this->d[$n] = $this->d[$n-1]; |
|
471
|
|
|
if ($z != 0.0) { |
|
472
|
|
|
$this->d[$n] = $x - $w / $z; |
|
473
|
|
|
} |
|
474
|
|
|
$this->e[$n-1] = 0.0; |
|
475
|
|
|
$this->e[$n] = 0.0; |
|
476
|
|
|
$x = $this->H[$n][$n-1]; |
|
477
|
|
|
$s = abs($x) + abs($z); |
|
478
|
|
|
$p = $x / $s; |
|
479
|
|
|
$q = $z / $s; |
|
480
|
|
|
$r = sqrt($p * $p + $q * $q); |
|
481
|
|
|
$p = $p / $r; |
|
482
|
|
|
$q = $q / $r; |
|
483
|
|
|
// Row modification |
|
484
|
|
View Code Duplication |
for ($j = $n-1; $j < $nn; ++$j) { |
|
|
|
|
|
|
485
|
|
|
$z = $this->H[$n-1][$j]; |
|
486
|
|
|
$this->H[$n-1][$j] = $q * $z + $p * $this->H[$n][$j]; |
|
487
|
|
|
$this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z; |
|
488
|
|
|
} |
|
489
|
|
|
// Column modification |
|
490
|
|
View Code Duplication |
for ($i = 0; $i <= $n; ++$i) { |
|
|
|
|
|
|
491
|
|
|
$z = $this->H[$i][$n-1]; |
|
492
|
|
|
$this->H[$i][$n-1] = $q * $z + $p * $this->H[$i][$n]; |
|
493
|
|
|
$this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z; |
|
494
|
|
|
} |
|
495
|
|
|
// Accumulate transformations |
|
496
|
|
View Code Duplication |
for ($i = $low; $i <= $high; ++$i) { |
|
|
|
|
|
|
497
|
|
|
$z = $this->V[$i][$n-1]; |
|
498
|
|
|
$this->V[$i][$n-1] = $q * $z + $p * $this->V[$i][$n]; |
|
499
|
|
|
$this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z; |
|
500
|
|
|
} |
|
501
|
|
|
// Complex pair |
|
502
|
|
|
} else { |
|
503
|
|
|
$this->d[$n-1] = $x + $p; |
|
504
|
|
|
$this->d[$n] = $x + $p; |
|
505
|
|
|
$this->e[$n-1] = $z; |
|
506
|
|
|
$this->e[$n] = -$z; |
|
507
|
|
|
} |
|
508
|
|
|
$n = $n - 2; |
|
509
|
|
|
$iter = 0; |
|
510
|
|
|
// No convergence yet |
|
511
|
|
|
} else { |
|
512
|
|
|
// Form shift |
|
513
|
|
|
$x = $this->H[$n][$n]; |
|
514
|
|
|
$y = 0.0; |
|
515
|
|
|
$w = 0.0; |
|
516
|
|
|
if ($l < $n) { |
|
517
|
|
|
$y = $this->H[$n-1][$n-1]; |
|
518
|
|
|
$w = $this->H[$n][$n-1] * $this->H[$n-1][$n]; |
|
519
|
|
|
} |
|
520
|
|
|
// Wilkinson's original ad hoc shift |
|
521
|
|
|
if ($iter == 10) { |
|
522
|
|
|
$exshift += $x; |
|
523
|
|
|
for ($i = $low; $i <= $n; ++$i) { |
|
524
|
|
|
$this->H[$i][$i] -= $x; |
|
525
|
|
|
} |
|
526
|
|
|
$s = abs($this->H[$n][$n-1]) + abs($this->H[$n-1][$n-2]); |
|
527
|
|
|
$x = $y = 0.75 * $s; |
|
528
|
|
|
$w = -0.4375 * $s * $s; |
|
529
|
|
|
} |
|
530
|
|
|
// MATLAB's new ad hoc shift |
|
531
|
|
|
if ($iter == 30) { |
|
532
|
|
|
$s = ($y - $x) / 2.0; |
|
533
|
|
|
$s = $s * $s + $w; |
|
534
|
|
|
if ($s > 0) { |
|
535
|
|
|
$s = sqrt($s); |
|
536
|
|
|
if ($y < $x) { |
|
537
|
|
|
$s = -$s; |
|
538
|
|
|
} |
|
539
|
|
|
$s = $x - $w / (($y - $x) / 2.0 + $s); |
|
540
|
|
|
for ($i = $low; $i <= $n; ++$i) { |
|
541
|
|
|
$this->H[$i][$i] -= $s; |
|
542
|
|
|
} |
|
543
|
|
|
$exshift += $s; |
|
544
|
|
|
$x = $y = $w = 0.964; |
|
545
|
|
|
} |
|
546
|
|
|
} |
|
547
|
|
|
// Could check iteration count here. |
|
548
|
|
|
$iter = $iter + 1; |
|
549
|
|
|
// Look for two consecutive small sub-diagonal elements |
|
550
|
|
|
$m = $n - 2; |
|
551
|
|
|
while ($m >= $l) { |
|
552
|
|
|
$z = $this->H[$m][$m]; |
|
553
|
|
|
$r = $x - $z; |
|
554
|
|
|
$s = $y - $z; |
|
555
|
|
|
$p = ($r * $s - $w) / $this->H[$m+1][$m] + $this->H[$m][$m+1]; |
|
556
|
|
|
$q = $this->H[$m+1][$m+1] - $z - $r - $s; |
|
557
|
|
|
$r = $this->H[$m+2][$m+1]; |
|
558
|
|
|
$s = abs($p) + abs($q) + abs($r); |
|
559
|
|
|
$p = $p / $s; |
|
560
|
|
|
$q = $q / $s; |
|
561
|
|
|
$r = $r / $s; |
|
562
|
|
|
if ($m == $l) { |
|
563
|
|
|
break; |
|
564
|
|
|
} |
|
565
|
|
|
if (abs($this->H[$m][$m-1]) * (abs($q) + abs($r)) < |
|
566
|
|
|
$eps * (abs($p) * (abs($this->H[$m-1][$m-1]) + abs($z) + abs($this->H[$m+1][$m+1])))) { |
|
567
|
|
|
break; |
|
568
|
|
|
} |
|
569
|
|
|
--$m; |
|
570
|
|
|
} |
|
571
|
|
|
for ($i = $m + 2; $i <= $n; ++$i) { |
|
572
|
|
|
$this->H[$i][$i-2] = 0.0; |
|
573
|
|
|
if ($i > $m+2) { |
|
574
|
|
|
$this->H[$i][$i-3] = 0.0; |
|
575
|
|
|
} |
|
576
|
|
|
} |
|
577
|
|
|
// Double QR step involving rows l:n and columns m:n |
|
578
|
|
|
for ($k = $m; $k <= $n-1; ++$k) { |
|
579
|
|
|
$notlast = ($k != $n-1); |
|
580
|
|
|
if ($k != $m) { |
|
581
|
|
|
$p = $this->H[$k][$k-1]; |
|
582
|
|
|
$q = $this->H[$k+1][$k-1]; |
|
583
|
|
|
$r = ($notlast ? $this->H[$k+2][$k-1] : 0.0); |
|
584
|
|
|
$x = abs($p) + abs($q) + abs($r); |
|
585
|
|
|
if ($x != 0.0) { |
|
586
|
|
|
$p = $p / $x; |
|
587
|
|
|
$q = $q / $x; |
|
588
|
|
|
$r = $r / $x; |
|
589
|
|
|
} |
|
590
|
|
|
} |
|
591
|
|
|
if ($x == 0.0) { |
|
592
|
|
|
break; |
|
593
|
|
|
} |
|
594
|
|
|
$s = sqrt($p * $p + $q * $q + $r * $r); |
|
595
|
|
|
if ($p < 0) { |
|
596
|
|
|
$s = -$s; |
|
597
|
|
|
} |
|
598
|
|
|
if ($s != 0) { |
|
599
|
|
|
if ($k != $m) { |
|
600
|
|
|
$this->H[$k][$k-1] = -$s * $x; |
|
601
|
|
|
} elseif ($l != $m) { |
|
602
|
|
|
$this->H[$k][$k-1] = -$this->H[$k][$k-1]; |
|
603
|
|
|
} |
|
604
|
|
|
$p = $p + $s; |
|
605
|
|
|
$x = $p / $s; |
|
606
|
|
|
$y = $q / $s; |
|
607
|
|
|
$z = $r / $s; |
|
608
|
|
|
$q = $q / $p; |
|
609
|
|
|
$r = $r / $p; |
|
610
|
|
|
// Row modification |
|
611
|
|
View Code Duplication |
for ($j = $k; $j < $nn; ++$j) { |
|
|
|
|
|
|
612
|
|
|
$p = $this->H[$k][$j] + $q * $this->H[$k+1][$j]; |
|
613
|
|
|
if ($notlast) { |
|
614
|
|
|
$p = $p + $r * $this->H[$k+2][$j]; |
|
615
|
|
|
$this->H[$k+2][$j] = $this->H[$k+2][$j] - $p * $z; |
|
616
|
|
|
} |
|
617
|
|
|
$this->H[$k][$j] = $this->H[$k][$j] - $p * $x; |
|
618
|
|
|
$this->H[$k+1][$j] = $this->H[$k+1][$j] - $p * $y; |
|
619
|
|
|
} |
|
620
|
|
|
// Column modification |
|
621
|
|
View Code Duplication |
for ($i = 0; $i <= min($n, $k+3); ++$i) { |
|
|
|
|
|
|
622
|
|
|
$p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k+1]; |
|
623
|
|
|
if ($notlast) { |
|
624
|
|
|
$p = $p + $z * $this->H[$i][$k+2]; |
|
625
|
|
|
$this->H[$i][$k+2] = $this->H[$i][$k+2] - $p * $r; |
|
626
|
|
|
} |
|
627
|
|
|
$this->H[$i][$k] = $this->H[$i][$k] - $p; |
|
628
|
|
|
$this->H[$i][$k+1] = $this->H[$i][$k+1] - $p * $q; |
|
629
|
|
|
} |
|
630
|
|
|
// Accumulate transformations |
|
631
|
|
View Code Duplication |
for ($i = $low; $i <= $high; ++$i) { |
|
|
|
|
|
|
632
|
|
|
$p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k+1]; |
|
633
|
|
|
if ($notlast) { |
|
634
|
|
|
$p = $p + $z * $this->V[$i][$k+2]; |
|
635
|
|
|
$this->V[$i][$k+2] = $this->V[$i][$k+2] - $p * $r; |
|
636
|
|
|
} |
|
637
|
|
|
$this->V[$i][$k] = $this->V[$i][$k] - $p; |
|
638
|
|
|
$this->V[$i][$k+1] = $this->V[$i][$k+1] - $p * $q; |
|
639
|
|
|
} |
|
640
|
|
|
} // ($s != 0) |
|
|
|
|
|
|
641
|
|
|
} // k loop |
|
642
|
|
|
} // check convergence |
|
643
|
|
|
} // while ($n >= $low) |
|
|
|
|
|
|
644
|
|
|
|
|
645
|
|
|
// Backsubstitute to find vectors of upper triangular form |
|
646
|
|
|
if ($norm == 0.0) { |
|
647
|
|
|
return; |
|
648
|
|
|
} |
|
649
|
|
|
|
|
650
|
|
|
for ($n = $nn-1; $n >= 0; --$n) { |
|
651
|
|
|
$p = $this->d[$n]; |
|
652
|
|
|
$q = $this->e[$n]; |
|
653
|
|
|
// Real vector |
|
654
|
|
|
if ($q == 0) { |
|
655
|
|
|
$l = $n; |
|
656
|
|
|
$this->H[$n][$n] = 1.0; |
|
657
|
|
|
for ($i = $n-1; $i >= 0; --$i) { |
|
658
|
|
|
$w = $this->H[$i][$i] - $p; |
|
659
|
|
|
$r = 0.0; |
|
660
|
|
|
for ($j = $l; $j <= $n; ++$j) { |
|
661
|
|
|
$r = $r + $this->H[$i][$j] * $this->H[$j][$n]; |
|
662
|
|
|
} |
|
663
|
|
|
if ($this->e[$i] < 0.0) { |
|
664
|
|
|
$z = $w; |
|
665
|
|
|
$s = $r; |
|
666
|
|
|
} else { |
|
667
|
|
|
$l = $i; |
|
668
|
|
|
if ($this->e[$i] == 0.0) { |
|
669
|
|
|
if ($w != 0.0) { |
|
670
|
|
|
$this->H[$i][$n] = -$r / $w; |
|
671
|
|
|
} else { |
|
672
|
|
|
$this->H[$i][$n] = -$r / ($eps * $norm); |
|
673
|
|
|
} |
|
674
|
|
|
// Solve real equations |
|
675
|
|
|
} else { |
|
676
|
|
|
$x = $this->H[$i][$i+1]; |
|
677
|
|
|
$y = $this->H[$i+1][$i]; |
|
678
|
|
|
$q = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i]; |
|
679
|
|
|
$t = ($x * $s - $z * $r) / $q; |
|
680
|
|
|
$this->H[$i][$n] = $t; |
|
681
|
|
|
if (abs($x) > abs($z)) { |
|
682
|
|
|
$this->H[$i+1][$n] = (-$r - $w * $t) / $x; |
|
683
|
|
|
} else { |
|
684
|
|
|
$this->H[$i+1][$n] = (-$s - $y * $t) / $z; |
|
685
|
|
|
} |
|
686
|
|
|
} |
|
687
|
|
|
// Overflow control |
|
688
|
|
|
$t = abs($this->H[$i][$n]); |
|
689
|
|
|
if (($eps * $t) * $t > 1) { |
|
690
|
|
View Code Duplication |
for ($j = $i; $j <= $n; ++$j) { |
|
|
|
|
|
|
691
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t; |
|
692
|
|
|
} |
|
693
|
|
|
} |
|
694
|
|
|
} |
|
695
|
|
|
} |
|
696
|
|
|
// Complex vector |
|
697
|
|
|
} elseif ($q < 0) { |
|
698
|
|
|
$l = $n-1; |
|
699
|
|
|
// Last vector component imaginary so matrix is triangular |
|
700
|
|
|
if (abs($this->H[$n][$n-1]) > abs($this->H[$n-1][$n])) { |
|
701
|
|
|
$this->H[$n-1][$n-1] = $q / $this->H[$n][$n-1]; |
|
702
|
|
|
$this->H[$n-1][$n] = -($this->H[$n][$n] - $p) / $this->H[$n][$n-1]; |
|
703
|
|
|
} else { |
|
704
|
|
|
$this->cdiv(0.0, -$this->H[$n-1][$n], $this->H[$n-1][$n-1] - $p, $q); |
|
705
|
|
|
$this->H[$n-1][$n-1] = $this->cdivr; |
|
706
|
|
|
$this->H[$n-1][$n] = $this->cdivi; |
|
707
|
|
|
} |
|
708
|
|
|
$this->H[$n][$n-1] = 0.0; |
|
709
|
|
|
$this->H[$n][$n] = 1.0; |
|
710
|
|
|
for ($i = $n-2; $i >= 0; --$i) { |
|
711
|
|
|
// double ra,sa,vr,vi; |
|
|
|
|
|
|
712
|
|
|
$ra = 0.0; |
|
713
|
|
|
$sa = 0.0; |
|
714
|
|
|
for ($j = $l; $j <= $n; ++$j) { |
|
715
|
|
|
$ra = $ra + $this->H[$i][$j] * $this->H[$j][$n-1]; |
|
716
|
|
|
$sa = $sa + $this->H[$i][$j] * $this->H[$j][$n]; |
|
717
|
|
|
} |
|
718
|
|
|
$w = $this->H[$i][$i] - $p; |
|
719
|
|
|
if ($this->e[$i] < 0.0) { |
|
720
|
|
|
$z = $w; |
|
721
|
|
|
$r = $ra; |
|
722
|
|
|
$s = $sa; |
|
723
|
|
|
} else { |
|
724
|
|
|
$l = $i; |
|
725
|
|
|
if ($this->e[$i] == 0) { |
|
726
|
|
|
$this->cdiv(-$ra, -$sa, $w, $q); |
|
727
|
|
|
$this->H[$i][$n-1] = $this->cdivr; |
|
728
|
|
|
$this->H[$i][$n] = $this->cdivi; |
|
729
|
|
|
} else { |
|
730
|
|
|
// Solve complex equations |
|
731
|
|
|
$x = $this->H[$i][$i+1]; |
|
732
|
|
|
$y = $this->H[$i+1][$i]; |
|
733
|
|
|
$vr = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i] - $q * $q; |
|
734
|
|
|
$vi = ($this->d[$i] - $p) * 2.0 * $q; |
|
735
|
|
|
if ($vr == 0.0 & $vi == 0.0) { |
|
|
|
|
|
|
736
|
|
|
$vr = $eps * $norm * (abs($w) + abs($q) + abs($x) + abs($y) + abs($z)); |
|
737
|
|
|
} |
|
738
|
|
|
$this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi); |
|
739
|
|
|
$this->H[$i][$n-1] = $this->cdivr; |
|
740
|
|
|
$this->H[$i][$n] = $this->cdivi; |
|
741
|
|
|
if (abs($x) > (abs($z) + abs($q))) { |
|
742
|
|
|
$this->H[$i+1][$n-1] = (-$ra - $w * $this->H[$i][$n-1] + $q * $this->H[$i][$n]) / $x; |
|
743
|
|
|
$this->H[$i+1][$n] = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n-1]) / $x; |
|
744
|
|
|
} else { |
|
745
|
|
|
$this->cdiv(-$r - $y * $this->H[$i][$n-1], -$s - $y * $this->H[$i][$n], $z, $q); |
|
746
|
|
|
$this->H[$i+1][$n-1] = $this->cdivr; |
|
747
|
|
|
$this->H[$i+1][$n] = $this->cdivi; |
|
748
|
|
|
} |
|
749
|
|
|
} |
|
750
|
|
|
// Overflow control |
|
751
|
|
|
$t = max(abs($this->H[$i][$n-1]), abs($this->H[$i][$n])); |
|
752
|
|
|
if (($eps * $t) * $t > 1) { |
|
753
|
|
|
for ($j = $i; $j <= $n; ++$j) { |
|
754
|
|
|
$this->H[$j][$n-1] = $this->H[$j][$n-1] / $t; |
|
755
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t; |
|
756
|
|
|
} |
|
757
|
|
|
} |
|
758
|
|
|
} // end else |
|
759
|
|
|
} // end for |
|
760
|
|
|
} // end else for complex case |
|
761
|
|
|
} // end for |
|
762
|
|
|
|
|
763
|
|
|
// Vectors of isolated roots |
|
764
|
|
|
for ($i = 0; $i < $nn; ++$i) { |
|
765
|
|
|
if ($i < $low | $i > $high) { |
|
766
|
|
|
for ($j = $i; $j < $nn; ++$j) { |
|
767
|
|
|
$this->V[$i][$j] = $this->H[$i][$j]; |
|
768
|
|
|
} |
|
769
|
|
|
} |
|
770
|
|
|
} |
|
771
|
|
|
|
|
772
|
|
|
// Back transformation to get eigenvectors of original matrix |
|
773
|
|
|
for ($j = $nn-1; $j >= $low; --$j) { |
|
774
|
|
|
for ($i = $low; $i <= $high; ++$i) { |
|
775
|
|
|
$z = 0.0; |
|
776
|
|
|
for ($k = $low; $k <= min($j, $high); ++$k) { |
|
777
|
|
|
$z = $z + $this->V[$i][$k] * $this->H[$k][$j]; |
|
778
|
|
|
} |
|
779
|
|
|
$this->V[$i][$j] = $z; |
|
780
|
|
|
} |
|
781
|
|
|
} |
|
782
|
|
|
} // end hqr2 |
|
783
|
|
|
|
|
784
|
|
|
|
|
785
|
|
|
/** |
|
786
|
|
|
* Constructor: Check for symmetry, then construct the eigenvalue decomposition |
|
787
|
|
|
* |
|
788
|
|
|
* @param array $Arg |
|
789
|
|
|
*/ |
|
790
|
|
|
public function __construct(array $Arg) |
|
791
|
|
|
{ |
|
792
|
|
|
$this->A = $Arg; |
|
|
|
|
|
|
793
|
|
|
$this->n = count($Arg[0]); |
|
794
|
|
|
|
|
795
|
|
|
$issymmetric = true; |
|
796
|
|
|
for ($j = 0; ($j < $this->n) & $issymmetric; ++$j) { |
|
797
|
|
View Code Duplication |
for ($i = 0; ($i < $this->n) & $issymmetric; ++$i) { |
|
|
|
|
|
|
798
|
|
|
$issymmetric = ($this->A[$i][$j] == $this->A[$j][$i]); |
|
799
|
|
|
} |
|
800
|
|
|
} |
|
801
|
|
|
|
|
802
|
|
|
if ($issymmetric) { |
|
803
|
|
|
$this->V = $this->A; |
|
804
|
|
|
// Tridiagonalize. |
|
805
|
|
|
$this->tred2(); |
|
806
|
|
|
// Diagonalize. |
|
807
|
|
|
$this->tql2(); |
|
808
|
|
|
} else { |
|
809
|
|
|
$this->H = $this->A; |
|
810
|
|
|
$this->ort = []; |
|
811
|
|
|
// Reduce to Hessenberg form. |
|
812
|
|
|
$this->orthes(); |
|
813
|
|
|
// Reduce Hessenberg to real Schur form. |
|
814
|
|
|
$this->hqr2(); |
|
815
|
|
|
} |
|
816
|
|
|
} |
|
817
|
|
|
|
|
818
|
|
|
/** |
|
819
|
|
|
* Return the eigenvector matrix |
|
820
|
|
|
* |
|
821
|
|
|
* @access public |
|
822
|
|
|
* @return array |
|
823
|
|
|
*/ |
|
824
|
|
|
public function getEigenvectors() |
|
825
|
|
|
{ |
|
826
|
|
|
$vectors = $this->V; |
|
827
|
|
|
|
|
828
|
|
|
// Always return the eigenvectors of length 1.0 |
|
829
|
|
|
$vectors = new Matrix($vectors); |
|
830
|
|
|
$vectors = array_map(function ($vect) { |
|
831
|
|
|
$sum = 0; |
|
832
|
|
View Code Duplication |
for ($i=0; $i<count($vect); $i++) { |
|
|
|
|
|
|
833
|
|
|
$sum += $vect[$i] ** 2; |
|
834
|
|
|
} |
|
835
|
|
|
$sum = sqrt($sum); |
|
836
|
|
View Code Duplication |
for ($i=0; $i<count($vect); $i++) { |
|
|
|
|
|
|
837
|
|
|
$vect[$i] /= $sum; |
|
838
|
|
|
} |
|
839
|
|
|
return $vect; |
|
840
|
|
|
}, $vectors->transpose()->toArray()); |
|
841
|
|
|
|
|
842
|
|
|
// Re-index eigenvectors so that their indices will be the same with eigenvalues |
|
843
|
|
|
$vectors2 = []; |
|
844
|
|
|
foreach (array_keys($this->d) as $index) { |
|
845
|
|
|
$vectors2[$index] = $vectors[$index]; |
|
846
|
|
|
} |
|
847
|
|
|
|
|
848
|
|
|
return $vectors2; |
|
849
|
|
|
} |
|
850
|
|
|
|
|
851
|
|
|
|
|
852
|
|
|
/** |
|
853
|
|
|
* Return the real parts of the eigenvalues<br> |
|
854
|
|
|
* d = real(diag(D)); |
|
855
|
|
|
* |
|
856
|
|
|
* @return array |
|
857
|
|
|
*/ |
|
858
|
|
|
public function getRealEigenvalues() |
|
859
|
|
|
{ |
|
860
|
|
|
return $this->d; |
|
861
|
|
|
} |
|
862
|
|
|
|
|
863
|
|
|
|
|
864
|
|
|
/** |
|
865
|
|
|
* Return the imaginary parts of the eigenvalues <br> |
|
866
|
|
|
* d = imag(diag(D)) |
|
867
|
|
|
* |
|
868
|
|
|
* @return array |
|
869
|
|
|
*/ |
|
870
|
|
|
public function getImagEigenvalues() |
|
871
|
|
|
{ |
|
872
|
|
|
return $this->e; |
|
873
|
|
|
} |
|
874
|
|
|
|
|
875
|
|
|
|
|
876
|
|
|
/** |
|
877
|
|
|
* Return the block diagonal eigenvalue matrix |
|
878
|
|
|
* |
|
879
|
|
|
* @return array |
|
880
|
|
|
*/ |
|
881
|
|
|
public function getDiagonalEigenvalues() |
|
882
|
|
|
{ |
|
883
|
|
|
for ($i = 0; $i < $this->n; ++$i) { |
|
884
|
|
|
$D[$i] = array_fill(0, $this->n, 0.0); |
|
|
|
|
|
|
885
|
|
|
$D[$i][$i] = $this->d[$i]; |
|
|
|
|
|
|
886
|
|
|
if ($this->e[$i] == 0) { |
|
887
|
|
|
continue; |
|
888
|
|
|
} |
|
889
|
|
|
$o = ($this->e[$i] > 0) ? $i + 1 : $i - 1; |
|
890
|
|
|
$D[$i][$o] = $this->e[$i]; |
|
891
|
|
|
} |
|
892
|
|
|
return $D; |
|
893
|
|
|
} |
|
894
|
|
|
} // class EigenvalueDecomposition |
|
895
|
|
|
|
php-ai /
php-ml
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Duplicated code is one of the most pungent code smells. If you need to duplicate the same code in three or more different places, we strongly encourage you to look into extracting the code into a single class or operation.
You can also find more detailed suggestions in the “Code” section of your repository.