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Diffstat (limited to 'admin/survey/excel/PHPExcel/Shared/JAMA/SingularValueDecomposition.php')
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diff --git a/admin/survey/excel/PHPExcel/Shared/JAMA/SingularValueDecomposition.php b/admin/survey/excel/PHPExcel/Shared/JAMA/SingularValueDecomposition.php deleted file mode 100644 index ecfb0cd..0000000 --- a/admin/survey/excel/PHPExcel/Shared/JAMA/SingularValueDecomposition.php +++ /dev/null @@ -1,526 +0,0 @@ -<?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
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