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<?php |
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declare(strict_types=1); |
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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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* |
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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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* |
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* @author Mustafa Karabulut |
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*/ |
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namespace Phpml\Math\LinearAlgebra; |
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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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* |
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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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* |
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* @var bool |
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*/ |
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private $symmetric; |
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/** |
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* Arrays for internal storage of eigenvalues. |
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* |
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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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* |
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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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* |
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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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* |
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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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* |
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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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* Constructor: Check for symmetry, then construct the eigenvalue decomposition |
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*/ |
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public function __construct(array $Arg) |
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{ |
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$this->A = $Arg; |
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$this->n = count($Arg[0]); |
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$this->symmetric = true; |
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for ($j = 0; ($j < $this->n) && $this->symmetric; ++$j) { |
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for ($i = 0; ($i < $this->n) & $this->symmetric; ++$i) { |
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$this->symmetric = ($this->A[$i][$j] == $this->A[$j][$i]); |
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} |
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} |
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if ($this->symmetric) { |
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$this->V = $this->A; |
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// Tridiagonalize. |
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$this->tred2(); |
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// Diagonalize. |
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$this->tql2(); |
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} else { |
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$this->H = $this->A; |
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$this->ort = []; |
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// Reduce to Hessenberg form. |
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$this->orthes(); |
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// Reduce Hessenberg to real Schur form. |
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$this->hqr2(); |
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} |
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} |
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/** |
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* Return the eigenvector matrix |
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*/ |
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public function getEigenvectors(): array |
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{ |
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$vectors = $this->V; |
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// Always return the eigenvectors of length 1.0 |
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$vectors = new Matrix($vectors); |
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$vectors = array_map(function ($vect) { |
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$sum = 0; |
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View Code Duplication |
for ($i = 0; $i < count($vect); ++$i) { |
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$sum += $vect[$i] ** 2; |
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} |
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$sum = sqrt($sum); |
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View Code Duplication |
for ($i = 0; $i < count($vect); ++$i) { |
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$vect[$i] /= $sum; |
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} |
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return $vect; |
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}, $vectors->transpose()->toArray()); |
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return $vectors; |
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} |
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/** |
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* Return the real parts of the eigenvalues<br> |
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* d = real(diag(D)); |
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*/ |
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public function getRealEigenvalues(): array |
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{ |
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return $this->d; |
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} |
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/** |
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* Return the imaginary parts of the eigenvalues <br> |
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* d = imag(diag(D)) |
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*/ |
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public function getImagEigenvalues(): array |
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{ |
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return $this->e; |
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} |
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/** |
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* Return the block diagonal eigenvalue matrix |
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*/ |
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public function getDiagonalEigenvalues(): array |
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{ |
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$D = []; |
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for ($i = 0; $i < $this->n; ++$i) { |
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$D[$i] = array_fill(0, $this->n, 0.0); |
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$D[$i][$i] = $this->d[$i]; |
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if ($this->e[$i] == 0) { |
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continue; |
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} |
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$o = ($this->e[$i] > 0) ? $i + 1 : $i - 1; |
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$D[$i][$o] = $this->e[$i]; |
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} |
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return $D; |
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} |
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/** |
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* Symmetric Householder reduction to tridiagonal form. |
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*/ |
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private function tred2(): void |
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{ |
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// This is derived from the Algol procedures tred2 by |
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// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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// Fortran subroutine in EISPACK. |
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$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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if ($h === 0 || $h < 1e-32) { |
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$h = 1e-32; |
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} |
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for ($j = 0; $j < $i; ++$j) { |
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$this->e[$j] /= $h; |
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$f += $this->e[$j] * $this->d[$j]; |
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} |
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$hh = $f / (2 * $h); |
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for ($j = 0; $j < $i; ++$j) { |
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$this->e[$j] -= $hh * $this->d[$j]; |
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} |
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for ($j = 0; $j < $i; ++$j) { |
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$f = $this->d[$j]; |
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$g = $this->e[$j]; |
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for ($k = $j; $k <= $i_; ++$k) { |
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$this->V[$k][$j] -= ($f * $this->e[$k] + $g * $this->d[$k]); |
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} |
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$this->d[$j] = $this->V[$i - 1][$j]; |
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$this->V[$i][$j] = 0.0; |
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} |
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} |
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$this->d[$i] = $h; |
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} |
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// Accumulate transformations. |
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for ($i = 0; $i < $this->n - 1; ++$i) { |
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$this->V[$this->n - 1][$i] = $this->V[$i][$i]; |
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$this->V[$i][$i] = 1.0; |
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$h = $this->d[$i + 1]; |
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if ($h != 0.0) { |
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View Code Duplication |
for ($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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$this->e[0] = 0.0; |
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} |
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/** |
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* Symmetric tridiagonal QL algorithm. |
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* |
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* This is derived from the Algol procedures tql2, by |
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* Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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* Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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* Fortran subroutine in EISPACK. |
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*/ |
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private function tql2(): void |
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{ |
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for ($i = 1; $i < $this->n; ++$i) { |
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$this->e[$i - 1] = $this->e[$i]; |
|
319
|
|
|
} |
|
320
|
|
|
|
|
321
|
|
|
$this->e[$this->n - 1] = 0.0; |
|
322
|
|
|
$f = 0.0; |
|
323
|
|
|
$tst1 = 0.0; |
|
324
|
|
|
$eps = pow(2.0, -52.0); |
|
325
|
|
|
|
|
326
|
|
|
for ($l = 0; $l < $this->n; ++$l) { |
|
327
|
|
|
// Find small subdiagonal element |
|
328
|
|
|
$tst1 = max($tst1, abs($this->d[$l]) + abs($this->e[$l])); |
|
329
|
|
|
$m = $l; |
|
330
|
|
|
while ($m < $this->n) { |
|
331
|
|
|
if (abs($this->e[$m]) <= $eps * $tst1) { |
|
332
|
|
|
break; |
|
333
|
|
|
} |
|
334
|
|
|
|
|
335
|
|
|
++$m; |
|
336
|
|
|
} |
|
337
|
|
|
|
|
338
|
|
|
// If m == l, $this->d[l] is an eigenvalue, |
|
339
|
|
|
// otherwise, iterate. |
|
340
|
|
|
if ($m > $l) { |
|
341
|
|
|
$iter = 0; |
|
342
|
|
|
do { |
|
343
|
|
|
// Could check iteration count here. |
|
344
|
|
|
$iter += 1; |
|
345
|
|
|
// Compute implicit shift |
|
346
|
|
|
$g = $this->d[$l]; |
|
347
|
|
|
$p = ($this->d[$l + 1] - $g) / (2.0 * $this->e[$l]); |
|
348
|
|
|
$r = hypot($p, 1.0); |
|
349
|
|
|
if ($p < 0) { |
|
350
|
|
|
$r *= -1; |
|
351
|
|
|
} |
|
352
|
|
|
|
|
353
|
|
|
$this->d[$l] = $this->e[$l] / ($p + $r); |
|
354
|
|
|
$this->d[$l + 1] = $this->e[$l] * ($p + $r); |
|
355
|
|
|
$dl1 = $this->d[$l + 1]; |
|
356
|
|
|
$h = $g - $this->d[$l]; |
|
357
|
|
|
for ($i = $l + 2; $i < $this->n; ++$i) { |
|
358
|
|
|
$this->d[$i] -= $h; |
|
359
|
|
|
} |
|
360
|
|
|
|
|
361
|
|
|
$f += $h; |
|
362
|
|
|
// Implicit QL transformation. |
|
363
|
|
|
$p = $this->d[$m]; |
|
364
|
|
|
$c = 1.0; |
|
365
|
|
|
$c2 = $c3 = $c; |
|
366
|
|
|
$el1 = $this->e[$l + 1]; |
|
367
|
|
|
$s = $s2 = 0.0; |
|
368
|
|
|
for ($i = $m - 1; $i >= $l; --$i) { |
|
369
|
|
|
$c3 = $c2; |
|
370
|
|
|
$c2 = $c; |
|
371
|
|
|
$s2 = $s; |
|
372
|
|
|
$g = $c * $this->e[$i]; |
|
373
|
|
|
$h = $c * $p; |
|
374
|
|
|
$r = hypot($p, $this->e[$i]); |
|
375
|
|
|
$this->e[$i + 1] = $s * $r; |
|
376
|
|
|
$s = $this->e[$i] / $r; |
|
377
|
|
|
$c = $p / $r; |
|
378
|
|
|
$p = $c * $this->d[$i] - $s * $g; |
|
379
|
|
|
$this->d[$i + 1] = $h + $s * ($c * $g + $s * $this->d[$i]); |
|
380
|
|
|
// Accumulate transformation. |
|
381
|
|
|
for ($k = 0; $k < $this->n; ++$k) { |
|
382
|
|
|
$h = $this->V[$k][$i + 1]; |
|
383
|
|
|
$this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h; |
|
384
|
|
|
$this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h; |
|
385
|
|
|
} |
|
386
|
|
|
} |
|
387
|
|
|
|
|
388
|
|
|
$p = -$s * $s2 * $c3 * $el1 * $this->e[$l] / $dl1; |
|
389
|
|
|
$this->e[$l] = $s * $p; |
|
390
|
|
|
$this->d[$l] = $c * $p; |
|
391
|
|
|
// Check for convergence. |
|
392
|
|
|
} while (abs($this->e[$l]) > $eps * $tst1); |
|
393
|
|
|
} |
|
394
|
|
|
|
|
395
|
|
|
$this->d[$l] = $this->d[$l] + $f; |
|
396
|
|
|
$this->e[$l] = 0.0; |
|
397
|
|
|
} |
|
398
|
|
|
|
|
399
|
|
|
// Sort eigenvalues and corresponding vectors. |
|
400
|
|
|
for ($i = 0; $i < $this->n - 1; ++$i) { |
|
401
|
|
|
$k = $i; |
|
402
|
|
|
$p = $this->d[$i]; |
|
403
|
|
|
for ($j = $i + 1; $j < $this->n; ++$j) { |
|
404
|
|
|
if ($this->d[$j] < $p) { |
|
405
|
|
|
$k = $j; |
|
406
|
|
|
$p = $this->d[$j]; |
|
407
|
|
|
} |
|
408
|
|
|
} |
|
409
|
|
|
|
|
410
|
|
|
if ($k != $i) { |
|
411
|
|
|
$this->d[$k] = $this->d[$i]; |
|
412
|
|
|
$this->d[$i] = $p; |
|
413
|
|
View Code Duplication |
for ($j = 0; $j < $this->n; ++$j) { |
|
|
|
|
|
|
414
|
|
|
$p = $this->V[$j][$i]; |
|
415
|
|
|
$this->V[$j][$i] = $this->V[$j][$k]; |
|
416
|
|
|
$this->V[$j][$k] = $p; |
|
417
|
|
|
} |
|
418
|
|
|
} |
|
419
|
|
|
} |
|
420
|
|
|
} |
|
421
|
|
|
|
|
422
|
|
|
/** |
|
423
|
|
|
* Nonsymmetric reduction to Hessenberg form. |
|
424
|
|
|
* |
|
425
|
|
|
* This is derived from the Algol procedures orthes and ortran, |
|
426
|
|
|
* by Martin and Wilkinson, Handbook for Auto. Comp., |
|
427
|
|
|
* Vol.ii-Linear Algebra, and the corresponding |
|
428
|
|
|
* Fortran subroutines in EISPACK. |
|
429
|
|
|
*/ |
|
430
|
|
|
private function orthes(): void |
|
431
|
|
|
{ |
|
432
|
|
|
$low = 0; |
|
433
|
|
|
$high = $this->n - 1; |
|
434
|
|
|
|
|
435
|
|
|
for ($m = $low + 1; $m <= $high - 1; ++$m) { |
|
436
|
|
|
// Scale column. |
|
437
|
|
|
$scale = 0.0; |
|
438
|
|
View Code Duplication |
for ($i = $m; $i <= $high; ++$i) { |
|
|
|
|
|
|
439
|
|
|
$scale = $scale + abs($this->H[$i][$m - 1]); |
|
440
|
|
|
} |
|
441
|
|
|
|
|
442
|
|
|
if ($scale != 0.0) { |
|
443
|
|
|
// Compute Householder transformation. |
|
444
|
|
|
$h = 0.0; |
|
445
|
|
|
for ($i = $high; $i >= $m; --$i) { |
|
446
|
|
|
$this->ort[$i] = $this->H[$i][$m - 1] / $scale; |
|
447
|
|
|
$h += $this->ort[$i] * $this->ort[$i]; |
|
448
|
|
|
} |
|
449
|
|
|
|
|
450
|
|
|
$g = sqrt($h); |
|
451
|
|
|
if ($this->ort[$m] > 0) { |
|
452
|
|
|
$g *= -1; |
|
453
|
|
|
} |
|
454
|
|
|
|
|
455
|
|
|
$h -= $this->ort[$m] * $g; |
|
456
|
|
|
$this->ort[$m] -= $g; |
|
457
|
|
|
// Apply Householder similarity transformation |
|
458
|
|
|
// H = (I -u * u' / h) * H * (I -u * u') / h) |
|
459
|
|
View Code Duplication |
for ($j = $m; $j < $this->n; ++$j) { |
|
|
|
|
|
|
460
|
|
|
$f = 0.0; |
|
461
|
|
|
for ($i = $high; $i >= $m; --$i) { |
|
462
|
|
|
$f += $this->ort[$i] * $this->H[$i][$j]; |
|
463
|
|
|
} |
|
464
|
|
|
|
|
465
|
|
|
$f /= $h; |
|
466
|
|
|
for ($i = $m; $i <= $high; ++$i) { |
|
467
|
|
|
$this->H[$i][$j] -= $f * $this->ort[$i]; |
|
468
|
|
|
} |
|
469
|
|
|
} |
|
470
|
|
|
|
|
471
|
|
View Code Duplication |
for ($i = 0; $i <= $high; ++$i) { |
|
|
|
|
|
|
472
|
|
|
$f = 0.0; |
|
473
|
|
|
for ($j = $high; $j >= $m; --$j) { |
|
474
|
|
|
$f += $this->ort[$j] * $this->H[$i][$j]; |
|
475
|
|
|
} |
|
476
|
|
|
|
|
477
|
|
|
$f = $f / $h; |
|
478
|
|
|
for ($j = $m; $j <= $high; ++$j) { |
|
479
|
|
|
$this->H[$i][$j] -= $f * $this->ort[$j]; |
|
480
|
|
|
} |
|
481
|
|
|
} |
|
482
|
|
|
|
|
483
|
|
|
$this->ort[$m] = $scale * $this->ort[$m]; |
|
484
|
|
|
$this->H[$m][$m - 1] = $scale * $g; |
|
485
|
|
|
} |
|
486
|
|
|
} |
|
487
|
|
|
|
|
488
|
|
|
// Accumulate transformations (Algol's ortran). |
|
489
|
|
|
for ($i = 0; $i < $this->n; ++$i) { |
|
490
|
|
|
for ($j = 0; $j < $this->n; ++$j) { |
|
491
|
|
|
$this->V[$i][$j] = ($i == $j ? 1.0 : 0.0); |
|
492
|
|
|
} |
|
493
|
|
|
} |
|
494
|
|
|
|
|
495
|
|
|
for ($m = $high - 1; $m >= $low + 1; --$m) { |
|
496
|
|
|
if ($this->H[$m][$m - 1] != 0.0) { |
|
497
|
|
View Code Duplication |
for ($i = $m + 1; $i <= $high; ++$i) { |
|
|
|
|
|
|
498
|
|
|
$this->ort[$i] = $this->H[$i][$m - 1]; |
|
499
|
|
|
} |
|
500
|
|
|
|
|
501
|
|
|
for ($j = $m; $j <= $high; ++$j) { |
|
502
|
|
|
$g = 0.0; |
|
503
|
|
|
for ($i = $m; $i <= $high; ++$i) { |
|
504
|
|
|
$g += $this->ort[$i] * $this->V[$i][$j]; |
|
505
|
|
|
} |
|
506
|
|
|
|
|
507
|
|
|
// Double division avoids possible underflow |
|
508
|
|
|
$g = ($g / $this->ort[$m]) / $this->H[$m][$m - 1]; |
|
509
|
|
|
for ($i = $m; $i <= $high; ++$i) { |
|
510
|
|
|
$this->V[$i][$j] += $g * $this->ort[$i]; |
|
511
|
|
|
} |
|
512
|
|
|
} |
|
513
|
|
|
} |
|
514
|
|
|
} |
|
515
|
|
|
} |
|
516
|
|
|
|
|
517
|
|
|
/** |
|
518
|
|
|
* Performs complex division. |
|
519
|
|
|
* |
|
520
|
|
|
* @param int|float $xr |
|
521
|
|
|
* @param int|float $xi |
|
522
|
|
|
* @param int|float $yr |
|
523
|
|
|
* @param int|float $yi |
|
524
|
|
|
*/ |
|
525
|
|
|
private function cdiv($xr, $xi, $yr, $yi): void |
|
526
|
|
|
{ |
|
527
|
|
|
if (abs($yr) > abs($yi)) { |
|
528
|
|
|
$r = $yi / $yr; |
|
529
|
|
|
$d = $yr + $r * $yi; |
|
530
|
|
|
$this->cdivr = ($xr + $r * $xi) / $d; |
|
531
|
|
|
$this->cdivi = ($xi - $r * $xr) / $d; |
|
532
|
|
|
} else { |
|
533
|
|
|
$r = $yr / $yi; |
|
534
|
|
|
$d = $yi + $r * $yr; |
|
535
|
|
|
$this->cdivr = ($r * $xr + $xi) / $d; |
|
536
|
|
|
$this->cdivi = ($r * $xi - $xr) / $d; |
|
537
|
|
|
} |
|
538
|
|
|
} |
|
539
|
|
|
|
|
540
|
|
|
/** |
|
541
|
|
|
* Nonsymmetric reduction from Hessenberg to real Schur form. |
|
542
|
|
|
* |
|
543
|
|
|
* Code is derived from the Algol procedure hqr2, |
|
544
|
|
|
* by Martin and Wilkinson, Handbook for Auto. Comp., |
|
545
|
|
|
* Vol.ii-Linear Algebra, and the corresponding |
|
546
|
|
|
* Fortran subroutine in EISPACK. |
|
547
|
|
|
*/ |
|
548
|
|
|
private function hqr2(): void |
|
549
|
|
|
{ |
|
550
|
|
|
// Initialize |
|
551
|
|
|
$nn = $this->n; |
|
552
|
|
|
$n = $nn - 1; |
|
553
|
|
|
$low = 0; |
|
554
|
|
|
$high = $nn - 1; |
|
555
|
|
|
$eps = pow(2.0, -52.0); |
|
556
|
|
|
$exshift = 0.0; |
|
557
|
|
|
$p = $q = $r = $s = $z = 0; |
|
558
|
|
|
// Store roots isolated by balanc and compute matrix norm |
|
559
|
|
|
$norm = 0.0; |
|
560
|
|
|
|
|
561
|
|
|
for ($i = 0; $i < $nn; ++$i) { |
|
562
|
|
|
if (($i < $low) or ($i > $high)) { |
|
|
|
|
|
|
563
|
|
|
$this->d[$i] = $this->H[$i][$i]; |
|
564
|
|
|
$this->e[$i] = 0.0; |
|
565
|
|
|
} |
|
566
|
|
|
|
|
567
|
|
|
for ($j = max($i - 1, 0); $j < $nn; ++$j) { |
|
568
|
|
|
$norm = $norm + abs($this->H[$i][$j]); |
|
569
|
|
|
} |
|
570
|
|
|
} |
|
571
|
|
|
|
|
572
|
|
|
// Outer loop over eigenvalue index |
|
573
|
|
|
$iter = 0; |
|
574
|
|
|
while ($n >= $low) { |
|
575
|
|
|
// Look for single small sub-diagonal element |
|
576
|
|
|
$l = $n; |
|
577
|
|
|
while ($l > $low) { |
|
578
|
|
|
$s = abs($this->H[$l - 1][$l - 1]) + abs($this->H[$l][$l]); |
|
579
|
|
|
if ($s == 0.0) { |
|
580
|
|
|
$s = $norm; |
|
581
|
|
|
} |
|
582
|
|
|
|
|
583
|
|
|
if (abs($this->H[$l][$l - 1]) < $eps * $s) { |
|
584
|
|
|
break; |
|
585
|
|
|
} |
|
586
|
|
|
|
|
587
|
|
|
--$l; |
|
588
|
|
|
} |
|
589
|
|
|
|
|
590
|
|
|
// Check for convergence |
|
591
|
|
|
// One root found |
|
592
|
|
|
if ($l == $n) { |
|
593
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
|
594
|
|
|
$this->d[$n] = $this->H[$n][$n]; |
|
595
|
|
|
$this->e[$n] = 0.0; |
|
596
|
|
|
--$n; |
|
597
|
|
|
$iter = 0; |
|
598
|
|
|
// Two roots found |
|
599
|
|
|
} elseif ($l == $n - 1) { |
|
600
|
|
|
$w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
|
601
|
|
|
$p = ($this->H[$n - 1][$n - 1] - $this->H[$n][$n]) / 2.0; |
|
602
|
|
|
$q = $p * $p + $w; |
|
603
|
|
|
$z = sqrt(abs($q)); |
|
604
|
|
|
$this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
|
605
|
|
|
$this->H[$n - 1][$n - 1] = $this->H[$n - 1][$n - 1] + $exshift; |
|
606
|
|
|
$x = $this->H[$n][$n]; |
|
607
|
|
|
// Real pair |
|
608
|
|
|
if ($q >= 0) { |
|
609
|
|
|
if ($p >= 0) { |
|
610
|
|
|
$z = $p + $z; |
|
611
|
|
|
} else { |
|
612
|
|
|
$z = $p - $z; |
|
613
|
|
|
} |
|
614
|
|
|
|
|
615
|
|
|
$this->d[$n - 1] = $x + $z; |
|
616
|
|
|
$this->d[$n] = $this->d[$n - 1]; |
|
617
|
|
|
if ($z != 0.0) { |
|
618
|
|
|
$this->d[$n] = $x - $w / $z; |
|
619
|
|
|
} |
|
620
|
|
|
|
|
621
|
|
|
$this->e[$n - 1] = 0.0; |
|
622
|
|
|
$this->e[$n] = 0.0; |
|
623
|
|
|
$x = $this->H[$n][$n - 1]; |
|
624
|
|
|
$s = abs($x) + abs($z); |
|
625
|
|
|
$p = $x / $s; |
|
626
|
|
|
$q = $z / $s; |
|
627
|
|
|
$r = sqrt($p * $p + $q * $q); |
|
628
|
|
|
$p = $p / $r; |
|
629
|
|
|
$q = $q / $r; |
|
630
|
|
|
// Row modification |
|
631
|
|
View Code Duplication |
for ($j = $n - 1; $j < $nn; ++$j) { |
|
|
|
|
|
|
632
|
|
|
$z = $this->H[$n - 1][$j]; |
|
633
|
|
|
$this->H[$n - 1][$j] = $q * $z + $p * $this->H[$n][$j]; |
|
634
|
|
|
$this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z; |
|
635
|
|
|
} |
|
636
|
|
|
|
|
637
|
|
|
// Column modification |
|
638
|
|
View Code Duplication |
for ($i = 0; $i <= $n; ++$i) { |
|
|
|
|
|
|
639
|
|
|
$z = $this->H[$i][$n - 1]; |
|
640
|
|
|
$this->H[$i][$n - 1] = $q * $z + $p * $this->H[$i][$n]; |
|
641
|
|
|
$this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z; |
|
642
|
|
|
} |
|
643
|
|
|
|
|
644
|
|
|
// Accumulate transformations |
|
645
|
|
View Code Duplication |
for ($i = $low; $i <= $high; ++$i) { |
|
|
|
|
|
|
646
|
|
|
$z = $this->V[$i][$n - 1]; |
|
647
|
|
|
$this->V[$i][$n - 1] = $q * $z + $p * $this->V[$i][$n]; |
|
648
|
|
|
$this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z; |
|
649
|
|
|
} |
|
650
|
|
|
|
|
651
|
|
|
// Complex pair |
|
652
|
|
|
} else { |
|
653
|
|
|
$this->d[$n - 1] = $x + $p; |
|
654
|
|
|
$this->d[$n] = $x + $p; |
|
655
|
|
|
$this->e[$n - 1] = $z; |
|
656
|
|
|
$this->e[$n] = -$z; |
|
657
|
|
|
} |
|
658
|
|
|
|
|
659
|
|
|
$n = $n - 2; |
|
660
|
|
|
$iter = 0; |
|
661
|
|
|
// No convergence yet |
|
662
|
|
|
} else { |
|
663
|
|
|
// Form shift |
|
664
|
|
|
$x = $this->H[$n][$n]; |
|
665
|
|
|
$y = 0.0; |
|
666
|
|
|
$w = 0.0; |
|
667
|
|
|
if ($l < $n) { |
|
668
|
|
|
$y = $this->H[$n - 1][$n - 1]; |
|
669
|
|
|
$w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
|
670
|
|
|
} |
|
671
|
|
|
|
|
672
|
|
|
// Wilkinson's original ad hoc shift |
|
673
|
|
|
if ($iter == 10) { |
|
674
|
|
|
$exshift += $x; |
|
675
|
|
|
for ($i = $low; $i <= $n; ++$i) { |
|
676
|
|
|
$this->H[$i][$i] -= $x; |
|
677
|
|
|
} |
|
678
|
|
|
|
|
679
|
|
|
$s = abs($this->H[$n][$n - 1]) + abs($this->H[$n - 1][$n - 2]); |
|
680
|
|
|
$x = $y = 0.75 * $s; |
|
681
|
|
|
$w = -0.4375 * $s * $s; |
|
682
|
|
|
} |
|
683
|
|
|
|
|
684
|
|
|
// MATLAB's new ad hoc shift |
|
685
|
|
|
if ($iter == 30) { |
|
686
|
|
|
$s = ($y - $x) / 2.0; |
|
687
|
|
|
$s = $s * $s + $w; |
|
688
|
|
|
if ($s > 0) { |
|
689
|
|
|
$s = sqrt($s); |
|
690
|
|
|
if ($y < $x) { |
|
691
|
|
|
$s = -$s; |
|
692
|
|
|
} |
|
693
|
|
|
|
|
694
|
|
|
$s = $x - $w / (($y - $x) / 2.0 + $s); |
|
695
|
|
|
for ($i = $low; $i <= $n; ++$i) { |
|
696
|
|
|
$this->H[$i][$i] -= $s; |
|
697
|
|
|
} |
|
698
|
|
|
|
|
699
|
|
|
$exshift += $s; |
|
700
|
|
|
$x = $y = $w = 0.964; |
|
701
|
|
|
} |
|
702
|
|
|
} |
|
703
|
|
|
|
|
704
|
|
|
// Could check iteration count here. |
|
705
|
|
|
$iter = $iter + 1; |
|
706
|
|
|
// Look for two consecutive small sub-diagonal elements |
|
707
|
|
|
$m = $n - 2; |
|
708
|
|
|
while ($m >= $l) { |
|
709
|
|
|
$z = $this->H[$m][$m]; |
|
710
|
|
|
$r = $x - $z; |
|
711
|
|
|
$s = $y - $z; |
|
712
|
|
|
$p = ($r * $s - $w) / $this->H[$m + 1][$m] + $this->H[$m][$m + 1]; |
|
713
|
|
|
$q = $this->H[$m + 1][$m + 1] - $z - $r - $s; |
|
714
|
|
|
$r = $this->H[$m + 2][$m + 1]; |
|
715
|
|
|
$s = abs($p) + abs($q) + abs($r); |
|
716
|
|
|
$p = $p / $s; |
|
717
|
|
|
$q = $q / $s; |
|
718
|
|
|
$r = $r / $s; |
|
719
|
|
|
if ($m == $l) { |
|
720
|
|
|
break; |
|
721
|
|
|
} |
|
722
|
|
|
|
|
723
|
|
|
if (abs($this->H[$m][$m - 1]) * (abs($q) + abs($r)) < |
|
724
|
|
|
$eps * (abs($p) * (abs($this->H[$m - 1][$m - 1]) + abs($z) + abs($this->H[$m + 1][$m + 1])))) { |
|
725
|
|
|
break; |
|
726
|
|
|
} |
|
727
|
|
|
|
|
728
|
|
|
--$m; |
|
729
|
|
|
} |
|
730
|
|
|
|
|
731
|
|
|
for ($i = $m + 2; $i <= $n; ++$i) { |
|
732
|
|
|
$this->H[$i][$i - 2] = 0.0; |
|
733
|
|
|
if ($i > $m + 2) { |
|
734
|
|
|
$this->H[$i][$i - 3] = 0.0; |
|
735
|
|
|
} |
|
736
|
|
|
} |
|
737
|
|
|
|
|
738
|
|
|
// Double QR step involving rows l:n and columns m:n |
|
739
|
|
|
for ($k = $m; $k <= $n - 1; ++$k) { |
|
740
|
|
|
$notlast = ($k != $n - 1); |
|
741
|
|
|
if ($k != $m) { |
|
742
|
|
|
$p = $this->H[$k][$k - 1]; |
|
743
|
|
|
$q = $this->H[$k + 1][$k - 1]; |
|
744
|
|
|
$r = ($notlast ? $this->H[$k + 2][$k - 1] : 0.0); |
|
745
|
|
|
$x = abs($p) + abs($q) + abs($r); |
|
746
|
|
|
if ($x != 0.0) { |
|
747
|
|
|
$p = $p / $x; |
|
748
|
|
|
$q = $q / $x; |
|
749
|
|
|
$r = $r / $x; |
|
750
|
|
|
} |
|
751
|
|
|
} |
|
752
|
|
|
|
|
753
|
|
|
if ($x == 0.0) { |
|
754
|
|
|
break; |
|
755
|
|
|
} |
|
756
|
|
|
|
|
757
|
|
|
$s = sqrt($p * $p + $q * $q + $r * $r); |
|
758
|
|
|
if ($p < 0) { |
|
759
|
|
|
$s = -$s; |
|
760
|
|
|
} |
|
761
|
|
|
|
|
762
|
|
|
if ($s != 0) { |
|
763
|
|
|
if ($k != $m) { |
|
764
|
|
|
$this->H[$k][$k - 1] = -$s * $x; |
|
765
|
|
|
} elseif ($l != $m) { |
|
766
|
|
|
$this->H[$k][$k - 1] = -$this->H[$k][$k - 1]; |
|
767
|
|
|
} |
|
768
|
|
|
|
|
769
|
|
|
$p = $p + $s; |
|
770
|
|
|
$x = $p / $s; |
|
771
|
|
|
$y = $q / $s; |
|
772
|
|
|
$z = $r / $s; |
|
773
|
|
|
$q = $q / $p; |
|
774
|
|
|
$r = $r / $p; |
|
775
|
|
|
// Row modification |
|
776
|
|
View Code Duplication |
for ($j = $k; $j < $nn; ++$j) { |
|
|
|
|
|
|
777
|
|
|
$p = $this->H[$k][$j] + $q * $this->H[$k + 1][$j]; |
|
778
|
|
|
if ($notlast) { |
|
779
|
|
|
$p = $p + $r * $this->H[$k + 2][$j]; |
|
780
|
|
|
$this->H[$k + 2][$j] = $this->H[$k + 2][$j] - $p * $z; |
|
781
|
|
|
} |
|
782
|
|
|
|
|
783
|
|
|
$this->H[$k][$j] = $this->H[$k][$j] - $p * $x; |
|
784
|
|
|
$this->H[$k + 1][$j] = $this->H[$k + 1][$j] - $p * $y; |
|
785
|
|
|
} |
|
786
|
|
|
|
|
787
|
|
|
// Column modification |
|
788
|
|
View Code Duplication |
for ($i = 0; $i <= min($n, $k + 3); ++$i) { |
|
|
|
|
|
|
789
|
|
|
$p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k + 1]; |
|
790
|
|
|
if ($notlast) { |
|
791
|
|
|
$p = $p + $z * $this->H[$i][$k + 2]; |
|
792
|
|
|
$this->H[$i][$k + 2] = $this->H[$i][$k + 2] - $p * $r; |
|
793
|
|
|
} |
|
794
|
|
|
|
|
795
|
|
|
$this->H[$i][$k] = $this->H[$i][$k] - $p; |
|
796
|
|
|
$this->H[$i][$k + 1] = $this->H[$i][$k + 1] - $p * $q; |
|
797
|
|
|
} |
|
798
|
|
|
|
|
799
|
|
|
// Accumulate transformations |
|
800
|
|
View Code Duplication |
for ($i = $low; $i <= $high; ++$i) { |
|
|
|
|
|
|
801
|
|
|
$p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k + 1]; |
|
802
|
|
|
if ($notlast) { |
|
803
|
|
|
$p = $p + $z * $this->V[$i][$k + 2]; |
|
804
|
|
|
$this->V[$i][$k + 2] = $this->V[$i][$k + 2] - $p * $r; |
|
805
|
|
|
} |
|
806
|
|
|
|
|
807
|
|
|
$this->V[$i][$k] = $this->V[$i][$k] - $p; |
|
808
|
|
|
$this->V[$i][$k + 1] = $this->V[$i][$k + 1] - $p * $q; |
|
809
|
|
|
} |
|
810
|
|
|
} // ($s != 0) |
|
|
|
|
|
|
811
|
|
|
} // k loop |
|
812
|
|
|
} // check convergence |
|
813
|
|
|
} // while ($n >= $low) |
|
|
|
|
|
|
814
|
|
|
|
|
815
|
|
|
// Backsubstitute to find vectors of upper triangular form |
|
816
|
|
|
if ($norm == 0.0) { |
|
817
|
|
|
return; |
|
818
|
|
|
} |
|
819
|
|
|
|
|
820
|
|
|
for ($n = $nn - 1; $n >= 0; --$n) { |
|
821
|
|
|
$p = $this->d[$n]; |
|
822
|
|
|
$q = $this->e[$n]; |
|
823
|
|
|
// Real vector |
|
824
|
|
|
if ($q == 0) { |
|
825
|
|
|
$l = $n; |
|
826
|
|
|
$this->H[$n][$n] = 1.0; |
|
827
|
|
|
for ($i = $n - 1; $i >= 0; --$i) { |
|
828
|
|
|
$w = $this->H[$i][$i] - $p; |
|
829
|
|
|
$r = 0.0; |
|
830
|
|
|
for ($j = $l; $j <= $n; ++$j) { |
|
831
|
|
|
$r = $r + $this->H[$i][$j] * $this->H[$j][$n]; |
|
832
|
|
|
} |
|
833
|
|
|
|
|
834
|
|
|
if ($this->e[$i] < 0.0) { |
|
835
|
|
|
$z = $w; |
|
836
|
|
|
$s = $r; |
|
837
|
|
|
} else { |
|
838
|
|
|
$l = $i; |
|
839
|
|
|
if ($this->e[$i] == 0.0) { |
|
840
|
|
|
if ($w != 0.0) { |
|
841
|
|
|
$this->H[$i][$n] = -$r / $w; |
|
842
|
|
|
} else { |
|
843
|
|
|
$this->H[$i][$n] = -$r / ($eps * $norm); |
|
844
|
|
|
} |
|
845
|
|
|
|
|
846
|
|
|
// Solve real equations |
|
847
|
|
|
} else { |
|
848
|
|
|
$x = $this->H[$i][$i + 1]; |
|
849
|
|
|
$y = $this->H[$i + 1][$i]; |
|
850
|
|
|
$q = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i]; |
|
851
|
|
|
$t = ($x * $s - $z * $r) / $q; |
|
852
|
|
|
$this->H[$i][$n] = $t; |
|
853
|
|
|
if (abs($x) > abs($z)) { |
|
854
|
|
|
$this->H[$i + 1][$n] = (-$r - $w * $t) / $x; |
|
855
|
|
|
} else { |
|
856
|
|
|
$this->H[$i + 1][$n] = (-$s - $y * $t) / $z; |
|
857
|
|
|
} |
|
858
|
|
|
} |
|
859
|
|
|
|
|
860
|
|
|
// Overflow control |
|
861
|
|
|
$t = abs($this->H[$i][$n]); |
|
862
|
|
|
if (($eps * $t) * $t > 1) { |
|
863
|
|
|
for ($j = $i; $j <= $n; ++$j) { |
|
864
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t; |
|
865
|
|
|
} |
|
866
|
|
|
} |
|
867
|
|
|
} |
|
868
|
|
|
} |
|
869
|
|
|
|
|
870
|
|
|
// Complex vector |
|
871
|
|
|
} elseif ($q < 0) { |
|
872
|
|
|
$l = $n - 1; |
|
873
|
|
|
// Last vector component imaginary so matrix is triangular |
|
874
|
|
|
if (abs($this->H[$n][$n - 1]) > abs($this->H[$n - 1][$n])) { |
|
875
|
|
|
$this->H[$n - 1][$n - 1] = $q / $this->H[$n][$n - 1]; |
|
876
|
|
|
$this->H[$n - 1][$n] = -($this->H[$n][$n] - $p) / $this->H[$n][$n - 1]; |
|
877
|
|
|
} else { |
|
878
|
|
|
$this->cdiv(0.0, -$this->H[$n - 1][$n], $this->H[$n - 1][$n - 1] - $p, $q); |
|
879
|
|
|
$this->H[$n - 1][$n - 1] = $this->cdivr; |
|
880
|
|
|
$this->H[$n - 1][$n] = $this->cdivi; |
|
881
|
|
|
} |
|
882
|
|
|
|
|
883
|
|
|
$this->H[$n][$n - 1] = 0.0; |
|
884
|
|
|
$this->H[$n][$n] = 1.0; |
|
885
|
|
|
for ($i = $n - 2; $i >= 0; --$i) { |
|
886
|
|
|
// double ra,sa,vr,vi; |
|
|
|
|
|
|
887
|
|
|
$ra = 0.0; |
|
888
|
|
|
$sa = 0.0; |
|
889
|
|
|
for ($j = $l; $j <= $n; ++$j) { |
|
890
|
|
|
$ra = $ra + $this->H[$i][$j] * $this->H[$j][$n - 1]; |
|
891
|
|
|
$sa = $sa + $this->H[$i][$j] * $this->H[$j][$n]; |
|
892
|
|
|
} |
|
893
|
|
|
|
|
894
|
|
|
$w = $this->H[$i][$i] - $p; |
|
895
|
|
|
if ($this->e[$i] < 0.0) { |
|
896
|
|
|
$z = $w; |
|
897
|
|
|
$r = $ra; |
|
898
|
|
|
$s = $sa; |
|
899
|
|
|
} else { |
|
900
|
|
|
$l = $i; |
|
901
|
|
|
if ($this->e[$i] == 0) { |
|
902
|
|
|
$this->cdiv(-$ra, -$sa, $w, $q); |
|
903
|
|
|
$this->H[$i][$n - 1] = $this->cdivr; |
|
904
|
|
|
$this->H[$i][$n] = $this->cdivi; |
|
905
|
|
|
} else { |
|
906
|
|
|
// Solve complex equations |
|
907
|
|
|
$x = $this->H[$i][$i + 1]; |
|
908
|
|
|
$y = $this->H[$i + 1][$i]; |
|
909
|
|
|
$vr = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i] - $q * $q; |
|
910
|
|
|
$vi = ($this->d[$i] - $p) * 2.0 * $q; |
|
911
|
|
|
if ($vr == 0.0 & $vi == 0.0) { |
|
|
|
|
|
|
912
|
|
|
$vr = $eps * $norm * (abs($w) + abs($q) + abs($x) + abs($y) + abs($z)); |
|
913
|
|
|
} |
|
914
|
|
|
|
|
915
|
|
|
$this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi); |
|
916
|
|
|
$this->H[$i][$n - 1] = $this->cdivr; |
|
917
|
|
|
$this->H[$i][$n] = $this->cdivi; |
|
918
|
|
|
if (abs($x) > (abs($z) + abs($q))) { |
|
919
|
|
|
$this->H[$i + 1][$n - 1] = (-$ra - $w * $this->H[$i][$n - 1] + $q * $this->H[$i][$n]) / $x; |
|
920
|
|
|
$this->H[$i + 1][$n] = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n - 1]) / $x; |
|
921
|
|
|
} else { |
|
922
|
|
|
$this->cdiv(-$r - $y * $this->H[$i][$n - 1], -$s - $y * $this->H[$i][$n], $z, $q); |
|
923
|
|
|
$this->H[$i + 1][$n - 1] = $this->cdivr; |
|
924
|
|
|
$this->H[$i + 1][$n] = $this->cdivi; |
|
925
|
|
|
} |
|
926
|
|
|
} |
|
927
|
|
|
|
|
928
|
|
|
// Overflow control |
|
929
|
|
|
$t = max(abs($this->H[$i][$n - 1]), abs($this->H[$i][$n])); |
|
930
|
|
|
if (($eps * $t) * $t > 1) { |
|
931
|
|
|
for ($j = $i; $j <= $n; ++$j) { |
|
932
|
|
|
$this->H[$j][$n - 1] = $this->H[$j][$n - 1] / $t; |
|
933
|
|
|
$this->H[$j][$n] = $this->H[$j][$n] / $t; |
|
934
|
|
|
} |
|
935
|
|
|
} |
|
936
|
|
|
} // end else |
|
937
|
|
|
} // end for |
|
938
|
|
|
} // end else for complex case |
|
939
|
|
|
} // end for |
|
940
|
|
|
|
|
941
|
|
|
// Vectors of isolated roots |
|
942
|
|
|
for ($i = 0; $i < $nn; ++$i) { |
|
943
|
|
|
if ($i < $low | $i > $high) { |
|
944
|
|
|
for ($j = $i; $j < $nn; ++$j) { |
|
945
|
|
|
$this->V[$i][$j] = $this->H[$i][$j]; |
|
946
|
|
|
} |
|
947
|
|
|
} |
|
948
|
|
|
} |
|
949
|
|
|
|
|
950
|
|
|
// Back transformation to get eigenvectors of original matrix |
|
951
|
|
|
for ($j = $nn - 1; $j >= $low; --$j) { |
|
952
|
|
|
for ($i = $low; $i <= $high; ++$i) { |
|
953
|
|
|
$z = 0.0; |
|
954
|
|
|
for ($k = $low; $k <= min($j, $high); ++$k) { |
|
955
|
|
|
$z = $z + $this->V[$i][$k] * $this->H[$k][$j]; |
|
956
|
|
|
} |
|
957
|
|
|
|
|
958
|
|
|
$this->V[$i][$j] = $z; |
|
959
|
|
|
} |
|
960
|
|
|
} |
|
961
|
|
|
} |
|
962
|
|
|
} |
|
963
|
|
|
|
php-ai /
php-ml
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In PHP it is possible to write to properties without declaring them. For example, the following is perfectly valid PHP code:
Generally, it is a good practice to explictly declare properties to avoid accidental typos and provide IDE auto-completion: