JAMA

# Source for file SingularValueDecomposition.php Documentation is available at SingularValueDecomposition.php <?php /** * @package JAMA * * For an m-by-n matrix A with m >= n, the singular value decomposition is * an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and * an n-by-n orthogonal matrix V so that A = U*S*V'. * * The singular values, sigma[$k] = S[$k][$k], are ordered so that * sigma[0] >= sigma[1] >= ... >= sigma[n-1]. * * The singular value decompostion always exists, so the constructor will * never fail. The matrix condition number and the effective numerical * rank can be computed from this decomposition. * * @author Paul Meagher * @license PHP v3.0 * @version 1.1 */ class SingularValueDecomposition { /** * Internal storage of U. * @var array */ private $U = array(); /** * Internal storage of V. * @var array */ private $V = array(); /** * Internal storage of singular values. * @var array */ private $s = array(); /** * Row dimension. * @var int */ private $m; /** * Column dimension. * @var int */ private $n; /** * Construct the singular value decomposition * * Derived from LINPACK code. * * @param $A Rectangular matrix * @return Structure to access U, S and V. */ public function __construct($Arg) { // Initialize. $A = $Arg->getArrayCopy(); $this->m = $Arg->getRowDimension(); $this->n = $Arg->getColumnDimension(); $nu = min($this->m, $this->n); $e = array(); $work = array(); $wantu = true; $wantv = true; $nct = min($this->m - 1, $this->n); $nrt = max(0, min($this->n - 2, $this->m)); // Reduce A to bidiagonal form, storing the diagonal elements // in s and the super-diagonal elements in e. for ($k = 0; $k < max($nct,$nrt); ++$k) { if ($k < $nct) { // Compute the transformation for the k-th column and // place the k-th diagonal in s[$k]. // Compute 2-norm of k-th column without under/overflow. $this->s[$k] = 0; for ($i = $k; $i < $this->m; ++$i) { $this->s[$k] = hypo($this->s[$k], $A[$i][$k]); } if ($this->s[$k] != 0.0) { if ($A[$k][$k] < 0.0) { $this->s[$k] = -$this->s[$k]; } for ($i = $k; $i < $this->m; ++$i) { $A[$i][$k] /= $this->s[$k]; } $A[$k][$k] += 1.0; } $this->s[$k] = -$this->s[$k]; } for ($j = $k + 1; $j < $this->n; ++$j) { if (($k < $nct) & ($this->s[$k] != 0.0)) { // Apply the transformation. $t = 0; for ($i = $k; $i < $this->m; ++$i) { $t += $A[$i][$k] * $A[$i][$j]; } $t = -$t / $A[$k][$k]; for ($i = $k; $i < $this->m; ++$i) { $A[$i][$j] += $t * $A[$i][$k]; } // Place the k-th row of A into e for the // subsequent calculation of the row transformation. $e[$j] = $A[$k][$j]; } } if ($wantu AND ($k < $nct)) { // Place the transformation in U for subsequent back // multiplication. for ($i = $k; $i < $this->m; ++$i) { $this->U[$i][$k] = $A[$i][$k]; } } if ($k < $nrt) { // Compute the k-th row transformation and place the // k-th super-diagonal in e[$k]. // Compute 2-norm without under/overflow. $e[$k] = 0; for ($i = $k + 1; $i < $this->n; ++$i) { $e[$k] = hypo($e[$k], $e[$i]); } if ($e[$k] != 0.0) { if ($e[$k+1] < 0.0) { $e[$k] = -$e[$k]; } for ($i = $k + 1; $i < $this->n; ++$i) { $e[$i] /= $e[$k]; } $e[$k+1] += 1.0; } $e[$k] = -$e[$k]; if (($k+1 < $this->m) AND ($e[$k] != 0.0)) { // Apply the transformation. for ($i = $k+1; $i < $this->m; ++$i) { $work[$i] = 0.0; } for ($j = $k+1; $j < $this->n; ++$j) { for ($i = $k+1; $i < $this->m; ++$i) { $work[$i] += $e[$j] * $A[$i][$j]; } } for ($j = $k + 1; $j < $this->n; ++$j) { $t = -$e[$j] / $e[$k+1]; for ($i = $k + 1; $i < $this->m; ++$i) { $A[$i][$j] += $t * $work[$i]; } } } if ($wantv) { // Place the transformation in V for subsequent // back multiplication. for ($i = $k + 1; $i < $this->n; ++$i) { $this->V[$i][$k] = $e[$i]; } } } } // Set up the final bidiagonal matrix or order p. $p = min($this->n, $this->m + 1); if ($nct < $this->n) { $this->s[$nct] = $A[$nct][$nct]; } if ($this->m < $p) { $this->s[$p-1] = 0.0; } if ($nrt + 1 < $p) { $e[$nrt] = $A[$nrt][$p-1]; } $e[$p-1] = 0.0; // If required, generate U. if ($wantu) { for ($j = $nct; $j < $nu; ++$j) { for ($i = 0; $i < $this->m; ++$i) { $this->U[$i][$j] = 0.0; } $this->U[$j][$j] = 1.0; } for ($k = $nct - 1; $k >= 0; --$k) { if ($this->s[$k] != 0.0) { for ($j = $k + 1; $j < $nu; ++$j) { $t = 0; for ($i = $k; $i < $this->m; ++$i) { $t += $this->U[$i][$k] * $this->U[$i][$j]; } $t = -$t / $this->U[$k][$k]; for ($i = $k; $i < $this->m; ++$i) { $this->U[$i][$j] += $t * $this->U[$i][$k]; } } for ($i = $k; $i < $this->m; ++$i ) { $this->U[$i][$k] = -$this->U[$i][$k]; } $this->U[$k][$k] = 1.0 + $this->U[$k][$k]; for ($i = 0; $i < $k - 1; ++$i) { $this->U[$i][$k] = 0.0; } } else { for ($i = 0; $i < $this->m; ++$i) { $this->U[$i][$k] = 0.0; } $this->U[$k][$k] = 1.0; } } } // If required, generate V. if ($wantv) { for ($k = $this->n - 1; $k >= 0; --$k) { if (($k < $nrt) AND ($e[$k] != 0.0)) { for ($j = $k + 1; $j < $nu; ++$j) { $t = 0; for ($i = $k + 1; $i < $this->n; ++$i) { $t += $this->V[$i][$k]* $this->V[$i][$j]; } $t = -$t / $this->V[$k+1][$k]; for ($i = $k + 1; $i < $this->n; ++$i) { $this->V[$i][$j] += $t * $this->V[$i][$k]; } } } for ($i = 0; $i < $this->n; ++$i) { $this->V[$i][$k] = 0.0; } $this->V[$k][$k] = 1.0; } } // Main iteration loop for the singular values. $pp = $p - 1; $iter = 0; $eps = pow(2.0, -52.0); while ($p > 0) { // Here is where a test for too many iterations would go. // This section of the program inspects for negligible // elements in the s and e arrays. On completion the // variables kase and k are set as follows: // kase = 1 if s(p) and e[k-1] are negligible and k<p // kase = 2 if s(k) is negligible and k<p // kase = 3 if e[k-1] is negligible, k<p, and // s(k), ..., s(p) are not negligible (qr step). // kase = 4 if e(p-1) is negligible (convergence). for ($k = $p - 2; $k >= -1; --$k) { if ($k == -1) { break; } if (abs($e[$k]) <= $eps * (abs($this->s[$k]) + abs($this->s[$k+1]))) { $e[$k] = 0.0; break; } } if ($k == $p - 2) { $kase = 4; } else { for ($ks = $p - 1; $ks >= $k; --$ks) { if ($ks == $k) { break; } $t = ($ks != $p ? abs($e[$ks]) : 0.) + ($ks != $k + 1 ? abs($e[$ks-1]) : 0.); if (abs($this->s[$ks]) <= $eps * $t) { $this->s[$ks] = 0.0; break; } } if ($ks == $k) { $kase = 3; } else if ($ks == $p-1) { $kase = 1; } else { $kase = 2; $k = $ks; } } ++$k; // Perform the task indicated by kase. switch ($kase) { // Deflate negligible s(p). case 1: $f = $e[$p-2]; $e[$p-2] = 0.0; for ($j = $p - 2; $j >= $k; --$j) { $t = hypo($this->s[$j],$f); $cs = $this->s[$j] / $t; $sn = $f / $t; $this->s[$j] = $t; if ($j != $k) { $f = -$sn * $e[$j-1]; $e[$j-1] = $cs * $e[$j-1]; } if ($wantv) { for ($i = 0; $i < $this->n; ++$i) { $t = $cs * $this->V[$i][$j] + $sn * $this->V[$i][$p-1]; $this->V[$i][$p-1] = -$sn * $this->V[$i][$j] + $cs * $this->V[$i][$p-1]; $this->V[$i][$j] = $t; } } } break; // Split at negligible s(k). case 2: $f = $e[$k-1]; $e[$k-1] = 0.0; for ($j = $k; $j < $p; ++$j) { $t = hypo($this->s[$j], $f); $cs = $this->s[$j] / $t; $sn = $f / $t; $this->s[$j] = $t; $f = -$sn * $e[$j]; $e[$j] = $cs * $e[$j]; if ($wantu) { for ($i = 0; $i < $this->m; ++$i) { $t = $cs * $this->U[$i][$j] + $sn * $this->U[$i][$k-1]; $this->U[$i][$k-1] = -$sn * $this->U[$i][$j] + $cs * $this->U[$i][$k-1]; $this->U[$i][$j] = $t; } } } break; // Perform one qr step. case 3: // Calculate the shift. $scale = max(max(max(max( abs($this->s[$p-1]),abs($this->s[$p-2])),abs($e[$p-2])), abs($this->s[$k])), abs($e[$k])); $sp = $this->s[$p-1] / $scale; $spm1 = $this->s[$p-2] / $scale; $epm1 = $e[$p-2] / $scale; $sk = $this->s[$k] / $scale; $ek = $e[$k] / $scale; $b = (($spm1 + $sp) * ($spm1 - $sp) + $epm1 * $epm1) / 2.0; $c = ($sp * $epm1) * ($sp * $epm1); $shift = 0.0; if (($b != 0.0) || ($c != 0.0)) { $shift = sqrt($b * $b + $c); if ($b < 0.0) { $shift = -$shift; } $shift = $c / ($b + $shift); } $f = ($sk + $sp) * ($sk - $sp) + $shift; $g = $sk * $ek; // Chase zeros. for ($j = $k; $j < $p-1; ++$j) { $t = hypo($f,$g); $cs = $f/$t; $sn = $g/$t; if ($j != $k) { $e[$j-1] = $t; } $f = $cs * $this->s[$j] + $sn * $e[$j]; $e[$j] = $cs * $e[$j] - $sn * $this->s[$j]; $g = $sn * $this->s[$j+1]; $this->s[$j+1] = $cs * $this->s[$j+1]; if ($wantv) { for ($i = 0; $i < $this->n; ++$i) { $t = $cs * $this->V[$i][$j] + $sn * $this->V[$i][$j+1]; $this->V[$i][$j+1] = -$sn * $this->V[$i][$j] + $cs * $this->V[$i][$j+1]; $this->V[$i][$j] = $t; } } $t = hypo($f,$g); $cs = $f/$t; $sn = $g/$t; $this->s[$j] = $t; $f = $cs * $e[$j] + $sn * $this->s[$j+1]; $this->s[$j+1] = -$sn * $e[$j] + $cs * $this->s[$j+1]; $g = $sn * $e[$j+1]; $e[$j+1] = $cs * $e[$j+1]; if ($wantu && ($j < $this->m - 1)) { for ($i = 0; $i < $this->m; ++$i) { $t = $cs * $this->U[$i][$j] + $sn * $this->U[$i][$j+1]; $this->U[$i][$j+1] = -$sn * $this->U[$i][$j] + $cs * $this->U[$i][$j+1]; $this->U[$i][$j] = $t; } } } $e[$p-2] = $f; $iter = $iter + 1; break; // Convergence. case 4: // Make the singular values positive. if ($this->s[$k] <= 0.0) { $this->s[$k] = ($this->s[$k] < 0.0 ? -$this->s[$k] : 0.0); if ($wantv) { for ($i = 0; $i <= $pp; ++$i) { $this->V[$i][$k] = -$this->V[$i][$k]; } } } // Order the singular values. while ($k < $pp) { if ($this->s[$k] >= $this->s[$k+1]) { break; } $t = $this->s[$k]; $this->s[$k] = $this->s[$k+1]; $this->s[$k+1] = $t; if ($wantv AND ($k < $this->n - 1)) { for ($i = 0; $i < $this->n; ++$i) { $t = $this->V[$i][$k+1]; $this->V[$i][$k+1] = $this->V[$i][$k]; $this->V[$i][$k] = $t; } } if ($wantu AND ($k < $this->m-1)) { for ($i = 0; $i < $this->m; ++$i) { $t = $this->U[$i][$k+1]; $this->U[$i][$k+1] = $this->U[$i][$k]; $this->U[$i][$k] = $t; } } ++$k; } $iter = 0; --$p; break; } // end switch } // end while } // end constructor /** * Return the left singular vectors * * @access public * @return U */ public function getU() { return new Matrix($this->U, $this->m, min($this->m + 1, $this->n)); } /** * Return the right singular vectors * * @access public * @return V */ public function getV() { return new Matrix($this->V); } /** * Return the one-dimensional array of singular values * * @access public * @return diagonal of S. */ public function getSingularValues() { return $this->s; } /** * Return the diagonal matrix of singular values * * @access public * @return S */ public function getS() { for ($i = 0; $i < $this->n; ++$i) { for ($j = 0; $j < $this->n; ++$j) { $S[$i][$j] = 0.0; } $S[$i][$i] = $this->s[$i]; } return new Matrix($S); } /** * Two norm * * @access public * @return max(S) */ public function norm2() { return $this->s[0]; } /** * Two norm condition number * * @access public * @return max(S)/min(S) */ public function cond() { return $this->s[0] / $this->s[min($this->m, $this->n) - 1]; } /** * Effective numerical matrix rank * * @access public * @return Number of nonnegligible singular values. */ public function rank() { $eps = pow(2.0, -52.0); $tol = max($this->m, $this->n) * $this->s[0] * $eps; $r = 0; for ($i = 0; $i < count($this->s); ++$i) { if ($this->s[$i] > $tol) { ++$r; } } return $r; } } // class SingularValueDecomposition Documentation generated on Mon, 11 Jan 2010 08:13:52 +0100 by phpDocumentor 1.4.1