class SingularValueDecomposition in Loft Data Grids 6.2
Same name and namespace in other branches
- 7.2 vendor/phpoffice/phpexcel/Classes/PHPExcel/Shared/JAMA/SingularValueDecomposition.php \SingularValueDecomposition
@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
Hierarchy
- class \SingularValueDecomposition
Expanded class hierarchy of SingularValueDecomposition
File
- vendor/
phpoffice/ phpexcel/ Classes/ PHPExcel/ Shared/ JAMA/ SingularValueDecomposition.php, line 20
View source
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[$ks]) : 0.0) + ($ks != $k + 1 ? abs($e[$ks - 1]) : 0.0);
if (abs($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;
}
}Members
Name |
Modifiers | Type | Description | Overrides |
|---|---|---|---|---|
|
SingularValueDecomposition:: |
private | property | * Row dimension. * | |
|
SingularValueDecomposition:: |
private | property | * Column dimension. * | |
|
SingularValueDecomposition:: |
private | property | * Internal storage of singular values. * | |
|
SingularValueDecomposition:: |
private | property | * Internal storage of U. * | |
|
SingularValueDecomposition:: |
private | property | * Internal storage of V. * | |
|
SingularValueDecomposition:: |
public | function | * Two norm condition number * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Return the diagonal matrix of singular values * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Return the one-dimensional array of singular values * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Return the left singular vectors * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Return the right singular vectors * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Two norm * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Effective numerical matrix rank * * @access public * | |
|
SingularValueDecomposition:: |
public | function | * Construct the singular value decomposition * * Derived from LINPACK code. * * |