2010-08-26 19:14:53 +00:00
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<?php
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/**
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2015-05-12 09:22:06 +00:00
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* @package JAMA
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* For an m-by-n matrix A with m >= n, the singular value decomposition is
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* an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
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* an n-by-n orthogonal matrix V so that A = U*S*V'.
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* The singular values, sigma[$k] = S[$k][$k], are ordered so that
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* sigma[0] >= sigma[1] >= ... >= sigma[n-1].
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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* The singular value decompostion always exists, so the constructor will
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* never fail. The matrix condition number and the effective numerical
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* rank can be computed from this decomposition.
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2010-08-26 19:14:53 +00:00
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*
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2015-05-12 09:22:06 +00:00
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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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2010-08-26 19:14:53 +00:00
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*/
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class SingularValueDecomposition {
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2015-05-12 09:22:06 +00:00
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/**
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* Internal storage of U.
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* @var array
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*/
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private $U = array();
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/**
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* Internal storage of V.
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* @var array
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*/
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private $V = array();
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/**
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* Internal storage of singular values.
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* @var array
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*/
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private $s = array();
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/**
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* Row dimension.
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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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* @var int
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*/
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private $n;
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/**
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* Construct the singular value decomposition
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*
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* Derived from LINPACK code.
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*
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* @param $A Rectangular matrix
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* @return Structure to access U, S and V.
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*/
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public function __construct($Arg) {
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// Initialize.
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$A = $Arg->getArrayCopy();
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$this->m = $Arg->getRowDimension();
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$this->n = $Arg->getColumnDimension();
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$nu = min($this->m, $this->n);
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$e = array();
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$work = array();
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$wantu = true;
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$wantv = true;
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$nct = min($this->m - 1, $this->n);
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$nrt = max(0, min($this->n - 2, $this->m));
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// Reduce A to bidiagonal form, storing the diagonal elements
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// in s and the super-diagonal elements in e.
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2015-05-13 10:27:01 +00:00
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for ($k = 0; $k < max($nct, $nrt); ++$k) {
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2015-05-12 09:22:06 +00:00
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if ($k < $nct) {
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// Compute the transformation for the k-th column and
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// place the k-th diagonal in s[$k].
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// Compute 2-norm of k-th column without under/overflow.
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$this->s[$k] = 0;
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for ($i = $k; $i < $this->m; ++$i) {
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$this->s[$k] = hypo($this->s[$k], $A[$i][$k]);
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}
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if ($this->s[$k] != 0.0) {
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if ($A[$k][$k] < 0.0) {
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$this->s[$k] = -$this->s[$k];
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}
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for ($i = $k; $i < $this->m; ++$i) {
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$A[$i][$k] /= $this->s[$k];
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}
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$A[$k][$k] += 1.0;
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}
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$this->s[$k] = -$this->s[$k];
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}
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for ($j = $k + 1; $j < $this->n; ++$j) {
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if (($k < $nct) & ($this->s[$k] != 0.0)) {
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// Apply the transformation.
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$t = 0;
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for ($i = $k; $i < $this->m; ++$i) {
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$t += $A[$i][$k] * $A[$i][$j];
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}
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$t = -$t / $A[$k][$k];
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for ($i = $k; $i < $this->m; ++$i) {
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$A[$i][$j] += $t * $A[$i][$k];
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}
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// Place the k-th row of A into e for the
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// subsequent calculation of the row transformation.
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$e[$j] = $A[$k][$j];
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}
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}
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if ($wantu AND ($k < $nct)) {
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// Place the transformation in U for subsequent back
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// multiplication.
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for ($i = $k; $i < $this->m; ++$i) {
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$this->U[$i][$k] = $A[$i][$k];
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}
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}
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if ($k < $nrt) {
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// Compute the k-th row transformation and place the
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// k-th super-diagonal in e[$k].
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// Compute 2-norm without under/overflow.
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$e[$k] = 0;
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for ($i = $k + 1; $i < $this->n; ++$i) {
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$e[$k] = hypo($e[$k], $e[$i]);
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}
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if ($e[$k] != 0.0) {
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if ($e[$k+1] < 0.0) {
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$e[$k] = -$e[$k];
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}
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for ($i = $k + 1; $i < $this->n; ++$i) {
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$e[$i] /= $e[$k];
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}
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$e[$k+1] += 1.0;
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}
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$e[$k] = -$e[$k];
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if (($k+1 < $this->m) AND ($e[$k] != 0.0)) {
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// Apply the transformation.
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for ($i = $k+1; $i < $this->m; ++$i) {
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$work[$i] = 0.0;
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}
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for ($j = $k+1; $j < $this->n; ++$j) {
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for ($i = $k+1; $i < $this->m; ++$i) {
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$work[$i] += $e[$j] * $A[$i][$j];
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}
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}
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for ($j = $k + 1; $j < $this->n; ++$j) {
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$t = -$e[$j] / $e[$k+1];
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for ($i = $k + 1; $i < $this->m; ++$i) {
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$A[$i][$j] += $t * $work[$i];
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}
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}
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}
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if ($wantv) {
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// Place the transformation in V for subsequent
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// back multiplication.
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for ($i = $k + 1; $i < $this->n; ++$i) {
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$this->V[$i][$k] = $e[$i];
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}
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}
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}
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}
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// Set up the final bidiagonal matrix or order p.
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$p = min($this->n, $this->m + 1);
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if ($nct < $this->n) {
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$this->s[$nct] = $A[$nct][$nct];
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}
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if ($this->m < $p) {
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$this->s[$p-1] = 0.0;
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}
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if ($nrt + 1 < $p) {
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$e[$nrt] = $A[$nrt][$p-1];
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}
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$e[$p-1] = 0.0;
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// If required, generate U.
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if ($wantu) {
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for ($j = $nct; $j < $nu; ++$j) {
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for ($i = 0; $i < $this->m; ++$i) {
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$this->U[$i][$j] = 0.0;
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}
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$this->U[$j][$j] = 1.0;
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}
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for ($k = $nct - 1; $k >= 0; --$k) {
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if ($this->s[$k] != 0.0) {
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for ($j = $k + 1; $j < $nu; ++$j) {
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$t = 0;
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for ($i = $k; $i < $this->m; ++$i) {
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$t += $this->U[$i][$k] * $this->U[$i][$j];
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}
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$t = -$t / $this->U[$k][$k];
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for ($i = $k; $i < $this->m; ++$i) {
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$this->U[$i][$j] += $t * $this->U[$i][$k];
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}
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}
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for ($i = $k; $i < $this->m; ++$i ) {
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$this->U[$i][$k] = -$this->U[$i][$k];
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}
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$this->U[$k][$k] = 1.0 + $this->U[$k][$k];
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for ($i = 0; $i < $k - 1; ++$i) {
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$this->U[$i][$k] = 0.0;
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}
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} else {
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for ($i = 0; $i < $this->m; ++$i) {
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$this->U[$i][$k] = 0.0;
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}
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$this->U[$k][$k] = 1.0;
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}
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}
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}
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// If required, generate V.
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if ($wantv) {
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for ($k = $this->n - 1; $k >= 0; --$k) {
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if (($k < $nrt) AND ($e[$k] != 0.0)) {
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for ($j = $k + 1; $j < $nu; ++$j) {
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$t = 0;
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for ($i = $k + 1; $i < $this->n; ++$i) {
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$t += $this->V[$i][$k]* $this->V[$i][$j];
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}
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$t = -$t / $this->V[$k+1][$k];
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for ($i = $k + 1; $i < $this->n; ++$i) {
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$this->V[$i][$j] += $t * $this->V[$i][$k];
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}
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}
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}
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for ($i = 0; $i < $this->n; ++$i) {
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$this->V[$i][$k] = 0.0;
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}
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$this->V[$k][$k] = 1.0;
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}
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}
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// Main iteration loop for the singular values.
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$pp = $p - 1;
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$iter = 0;
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$eps = pow(2.0, -52.0);
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while ($p > 0) {
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// Here is where a test for too many iterations would go.
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// This section of the program inspects for negligible
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// elements in the s and e arrays. On completion the
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// variables kase and k are set as follows:
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// kase = 1 if s(p) and e[k-1] are negligible and k<p
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// kase = 2 if s(k) is negligible and k<p
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// kase = 3 if e[k-1] is negligible, k<p, and
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// s(k), ..., s(p) are not negligible (qr step).
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// kase = 4 if e(p-1) is negligible (convergence).
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for ($k = $p - 2; $k >= -1; --$k) {
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if ($k == -1) {
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break;
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}
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if (abs($e[$k]) <= $eps * (abs($this->s[$k]) + abs($this->s[$k+1]))) {
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$e[$k] = 0.0;
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break;
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}
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}
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if ($k == $p - 2) {
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$kase = 4;
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} else {
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for ($ks = $p - 1; $ks >= $k; --$ks) {
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if ($ks == $k) {
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break;
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}
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$t = ($ks != $p ? abs($e[$ks]) : 0.) + ($ks != $k + 1 ? abs($e[$ks-1]) : 0.);
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if (abs($this->s[$ks]) <= $eps * $t) {
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$this->s[$ks] = 0.0;
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break;
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}
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}
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if ($ks == $k) {
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$kase = 3;
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} else if ($ks == $p-1) {
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$kase = 1;
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} else {
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$kase = 2;
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$k = $ks;
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}
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}
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++$k;
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// Perform the task indicated by kase.
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switch ($kase) {
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// Deflate negligible s(p).
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case 1:
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$f = $e[$p-2];
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$e[$p-2] = 0.0;
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for ($j = $p - 2; $j >= $k; --$j) {
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2015-05-13 10:27:01 +00:00
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$t = hypo($this->s[$j], $f);
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2015-05-12 09:22:06 +00:00
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$cs = $this->s[$j] / $t;
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$sn = $f / $t;
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$this->s[$j] = $t;
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if ($j != $k) {
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$f = -$sn * $e[$j-1];
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$e[$j-1] = $cs * $e[$j-1];
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}
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if ($wantv) {
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for ($i = 0; $i < $this->n; ++$i) {
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$t = $cs * $this->V[$i][$j] + $sn * $this->V[$i][$p-1];
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$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) {
|
2015-05-13 10:27:01 +00:00
|
|
|
$t = hypo($f, $g);
|
2015-05-12 09:22:06 +00:00
|
|
|
$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;
|
|
|
|
|
}
|
|
|
|
|
}
|
2015-05-13 10:27:01 +00:00
|
|
|
$t = hypo($f, $g);
|
2015-05-12 09:22:06 +00:00
|
|
|
$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
|
