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<?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) {
$kase = 3;
} elseif ($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;
}
}
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