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<?php namespace PhpOffice\PhpSpreadsheet\Shared\JAMA; use PhpOffice\PhpSpreadsheet\Calculation\Exception as CalculationException; /** * For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n * orthogonal matrix Q and an n-by-n upper triangular matrix R so that * A = Q*R. * * The QR decompostion always exists, even if the matrix does not have * full rank, so the constructor will never fail. The primary use of the * QR decomposition is in the least squares solution of nonsquare systems * of simultaneous linear equations. This will fail if isFullRank() * returns false. * * @author Paul Meagher * * @version 1.1 */ class QRDecomposition { const MATRIX_RANK_EXCEPTION = 'Can only perform operation on full-rank matrix.'; /** * Array for internal storage of decomposition. * * @var array */ private $QR = []; /** * Row dimension. * * @var int */ private $m; /** * Column dimension. * * @var int */ private $n; /** * Array for internal storage of diagonal of R. * * @var array */ private $Rdiag = []; /** * QR Decomposition computed by Householder reflections. * * @param Matrix $A Rectangular matrix */ public function __construct(Matrix $A) { // Initialize. $this->QR = $A->getArray(); $this->m = $A->getRowDimension(); $this->n = $A->getColumnDimension(); // Main loop. for ($k = 0; $k < $this->n; ++$k) { // Compute 2-norm of k-th column without under/overflow. $nrm = 0.0; for ($i = $k; $i < $this->m; ++$i) { $nrm = hypo($nrm, $this->QR[$i][$k]); } if ($nrm != 0.0) { // Form k-th Householder vector. if ($this->QR[$k][$k] < 0) { $nrm = -$nrm; } for ($i = $k; $i < $this->m; ++$i) { $this->QR[$i][$k] /= $nrm; } $this->QR[$k][$k] += 1.0; // Apply transformation to remaining columns. for ($j = $k + 1; $j < $this->n; ++$j) { $s = 0.0; for ($i = $k; $i < $this->m; ++$i) { $s += $this->QR[$i][$k] * $this->QR[$i][$j]; } $s = -$s / $this->QR[$k][$k]; for ($i = $k; $i < $this->m; ++$i) { $this->QR[$i][$j] += $s * $this->QR[$i][$k]; } } } $this->Rdiag[$k] = -$nrm; } } // function __construct() /** * Is the matrix full rank? * * @return bool true if R, and hence A, has full rank, else false */ public function isFullRank() { for ($j = 0; $j < $this->n; ++$j) { if ($this->Rdiag[$j] == 0) { return false; } } return true; } // function isFullRank() /** * Return the Householder vectors. * * @return Matrix Lower trapezoidal matrix whose columns define the reflections */ public function getH() { $H = []; for ($i = 0; $i < $this->m; ++$i) { for ($j = 0; $j < $this->n; ++$j) { if ($i >= $j) { $H[$i][$j] = $this->QR[$i][$j]; } else { $H[$i][$j] = 0.0; } } } return new Matrix($H); } // function getH() /** * Return the upper triangular factor. * * @return Matrix upper triangular factor */ public function getR() { $R = []; for ($i = 0; $i < $this->n; ++$i) { for ($j = 0; $j < $this->n; ++$j) { if ($i < $j) { $R[$i][$j] = $this->QR[$i][$j]; } elseif ($i == $j) { $R[$i][$j] = $this->Rdiag[$i]; } else { $R[$i][$j] = 0.0; } } } return new Matrix($R); } // function getR() /** * Generate and return the (economy-sized) orthogonal factor. * * @return Matrix orthogonal factor */ public function getQ() { $Q = []; for ($k = $this->n - 1; $k >= 0; --$k) { for ($i = 0; $i < $this->m; ++$i) { $Q[$i][$k] = 0.0; } $Q[$k][$k] = 1.0; for ($j = $k; $j < $this->n; ++$j) { if ($this->QR[$k][$k] != 0) { $s = 0.0; for ($i = $k; $i < $this->m; ++$i) { $s += $this->QR[$i][$k] * $Q[$i][$j]; } $s = -$s / $this->QR[$k][$k]; for ($i = $k; $i < $this->m; ++$i) { $Q[$i][$j] += $s * $this->QR[$i][$k]; } } } } return new Matrix($Q); } // function getQ() /** * Least squares solution of A*X = B. * * @param Matrix $B a Matrix with as many rows as A and any number of columns * * @return Matrix matrix that minimizes the two norm of Q*R*X-B */ public function solve(Matrix $B) { if ($B->getRowDimension() == $this->m) { if ($this->isFullRank()) { // Copy right hand side $nx = $B->getColumnDimension(); $X = $B->getArray(); // Compute Y = transpose(Q)*B for ($k = 0; $k < $this->n; ++$k) { for ($j = 0; $j < $nx; ++$j) { $s = 0.0; for ($i = $k; $i < $this->m; ++$i) { $s += $this->QR[$i][$k] * $X[$i][$j]; } $s = -$s / $this->QR[$k][$k]; for ($i = $k; $i < $this->m; ++$i) { $X[$i][$j] += $s * $this->QR[$i][$k]; } } } // Solve R*X = Y; for ($k = $this->n - 1; $k >= 0; --$k) { for ($j = 0; $j < $nx; ++$j) { $X[$k][$j] /= $this->Rdiag[$k]; } for ($i = 0; $i < $k; ++$i) { for ($j = 0; $j < $nx; ++$j) { $X[$i][$j] -= $X[$k][$j] * $this->QR[$i][$k]; } } } $X = new Matrix($X); return $X->getMatrix(0, $this->n - 1, 0, $nx); } throw new CalculationException(self::MATRIX_RANK_EXCEPTION); } throw new CalculationException(Matrix::MATRIX_DIMENSION_EXCEPTION); } }