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<?php |
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declare(strict_types=1); |
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/** |
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* @package JAMA |
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* |
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* For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n |
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* unit lower triangular matrix L, an n-by-n upper triangular matrix U, |
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* and a permutation vector piv of length m so that A(piv,:) = L*U. |
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* If m < n, then L is m-by-m and U is m-by-n. |
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* |
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* The LU decompostion with pivoting always exists, even if the matrix is |
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* singular, so the constructor will never fail. The primary use of the |
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* LU decomposition is in the solution of square systems of simultaneous |
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* linear equations. This will fail if isNonsingular() returns false. |
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* |
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* @author Paul Meagher |
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* @author Bartosz Matosiuk |
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* @author Michael Bommarito |
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* |
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* @version 1.1 |
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* |
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* @license PHP v3.0 |
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* |
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* Slightly changed to adapt the original code to PHP-ML library |
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* @date 2017/04/24 |
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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\Exception\MatrixException; |
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use Phpml\Math\Matrix; |
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class LUDecomposition |
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{ |
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/** |
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* Decomposition storage |
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* |
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* @var array |
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*/ |
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private $LU = []; |
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/** |
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* Row dimension. |
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* |
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* @var int |
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*/ |
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private $m; |
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/** |
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* Column dimension. |
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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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* Pivot sign. |
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* |
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* @var int |
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*/ |
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private $pivsign; |
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/** |
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* Internal storage of pivot vector. |
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* |
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* @var array |
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*/ |
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private $piv = []; |
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/** |
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* Constructs Structure to access L, U and piv. |
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* |
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* @param Matrix $A Rectangular matrix |
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* |
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* @throws MatrixException |
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*/ |
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public function __construct(Matrix $A) |
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{ |
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if ($A->getRows() !== $A->getColumns()) { |
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throw new MatrixException('Matrix is not square matrix'); |
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} |
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// Use a "left-looking", dot-product, Crout/Doolittle algorithm. |
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$this->LU = $A->toArray(); |
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$this->m = $A->getRows(); |
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$this->n = $A->getColumns(); |
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for ($i = 0; $i < $this->m; ++$i) { |
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$this->piv[$i] = $i; |
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} |
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$this->pivsign = 1; |
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$LUcolj = []; |
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// Outer loop. |
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for ($j = 0; $j < $this->n; ++$j) { |
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// Make a copy of the j-th column to localize references. |
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for ($i = 0; $i < $this->m; ++$i) { |
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$LUcolj[$i] = &$this->LU[$i][$j]; |
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} |
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// Apply previous transformations. |
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for ($i = 0; $i < $this->m; ++$i) { |
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$LUrowi = $this->LU[$i]; |
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// Most of the time is spent in the following dot product. |
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$kmax = min($i, $j); |
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$s = 0.0; |
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for ($k = 0; $k < $kmax; ++$k) { |
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$s += $LUrowi[$k] * $LUcolj[$k]; |
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} |
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$LUrowi[$j] = $LUcolj[$i] -= $s; |
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} |
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// Find pivot and exchange if necessary. |
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$p = $j; |
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for ($i = $j + 1; $i < $this->m; ++$i) { |
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if (abs($LUcolj[$i] ?? 0) > abs($LUcolj[$p] ?? 0)) { |
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$p = $i; |
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} |
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} |
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if ($p != $j) { |
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for ($k = 0; $k < $this->n; ++$k) { |
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$t = $this->LU[$p][$k]; |
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$this->LU[$p][$k] = $this->LU[$j][$k]; |
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$this->LU[$j][$k] = $t; |
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} |
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$k = $this->piv[$p]; |
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$this->piv[$p] = $this->piv[$j]; |
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$this->piv[$j] = $k; |
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$this->pivsign *= -1; |
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} |
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// Compute multipliers. |
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if (($j < $this->m) && ($this->LU[$j][$j] != 0.0)) { |
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for ($i = $j + 1; $i < $this->m; ++$i) { |
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$this->LU[$i][$j] /= $this->LU[$j][$j]; |
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} |
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} |
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} |
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} |
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/** |
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* Get lower triangular factor. |
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* |
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* @return Matrix Lower triangular factor |
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*/ |
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public function getL(): Matrix |
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{ |
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$L = []; |
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for ($i = 0; $i < $this->m; ++$i) { |
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for ($j = 0; $j < $this->n; ++$j) { |
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if ($i > $j) { |
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$L[$i][$j] = $this->LU[$i][$j]; |
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} elseif ($i == $j) { |
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$L[$i][$j] = 1.0; |
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} else { |
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$L[$i][$j] = 0.0; |
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} |
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} |
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} |
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return new Matrix($L); |
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} |
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/** |
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* Get upper triangular factor. |
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* |
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* @return Matrix Upper triangular factor |
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*/ |
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public function getU(): Matrix |
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{ |
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$U = []; |
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for ($i = 0; $i < $this->n; ++$i) { |
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for ($j = 0; $j < $this->n; ++$j) { |
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if ($i <= $j) { |
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$U[$i][$j] = $this->LU[$i][$j]; |
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} else { |
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$U[$i][$j] = 0.0; |
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} |
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} |
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} |
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return new Matrix($U); |
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} |
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/** |
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* Return pivot permutation vector. |
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* |
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* @return array Pivot vector |
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*/ |
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public function getPivot(): array |
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{ |
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return $this->piv; |
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} |
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/** |
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* Alias for getPivot |
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* |
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* @see getPivot |
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*/ |
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public function getDoublePivot(): array |
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{ |
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return $this->getPivot(); |
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} |
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/** |
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* Is the matrix nonsingular? |
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* |
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* @return bool true if U, and hence A, is nonsingular. |
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*/ |
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public function isNonsingular(): bool |
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{ |
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for ($j = 0; $j < $this->n; ++$j) { |
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if ($this->LU[$j][$j] == 0) { |
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return false; |
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} |
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} |
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return true; |
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} |
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public function det(): float |
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{ |
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$d = $this->pivsign; |
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for ($j = 0; $j < $this->n; ++$j) { |
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$d *= $this->LU[$j][$j]; |
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} |
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return (float) $d; |
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} |
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/** |
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* Solve A*X = B |
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* |
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* @param Matrix $B A Matrix with as many rows as A and any number of columns. |
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* |
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* @return array X so that L*U*X = B(piv,:) |
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* |
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* @throws MatrixException |
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*/ |
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public function solve(Matrix $B): array |
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{ |
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if ($B->getRows() != $this->m) { |
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throw new MatrixException('Matrix is not square matrix'); |
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} |
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if (!$this->isNonsingular()) { |
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throw new MatrixException('Matrix is singular'); |
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} |
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// Copy right hand side with pivoting |
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$nx = $B->getColumns(); |
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$X = $this->getSubMatrix($B->toArray(), $this->piv, 0, $nx - 1); |
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// Solve L*Y = B(piv,:) |
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for ($k = 0; $k < $this->n; ++$k) { |
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for ($i = $k + 1; $i < $this->n; ++$i) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$i][$j] -= $X[$k][$j] * $this->LU[$i][$k]; |
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} |
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} |
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} |
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// Solve U*X = Y; |
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for ($k = $this->n - 1; $k >= 0; --$k) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$k][$j] /= $this->LU[$k][$k]; |
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} |
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for ($i = 0; $i < $k; ++$i) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$i][$j] -= $X[$k][$j] * $this->LU[$i][$k]; |
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} |
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} |
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} |
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return $X; |
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} |
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protected function getSubMatrix(array $matrix, array $RL, int $j0, int $jF): array |
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{ |
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$m = count($RL); |
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$n = $jF - $j0; |
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$R = array_fill(0, $m, array_fill(0, $n + 1, 0.0)); |
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for ($i = 0; $i < $m; ++$i) { |
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for ($j = $j0; $j <= $jF; ++$j) { |
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$R[$i][$j - $j0] = $matrix[$RL[$i]][$j]; |
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} |
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} |
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return $R; |
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} |
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} |
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