Source Listing:
<?php
/**
* @package JAMA
*
* For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
* orthogonal matrix Q and an n-by-n upper triangular matrix R so that
* A = Q*R.
*
* The QR decompostion always exists, even if the matrix does not have
* full rank, so the constructor will never fail. The primary use of the
* QR decomposition is in the least squares solution of nonsquare systems
* of simultaneous linear equations. This will fail if isFullRank()
* returns false.
*
* @author Paul Meagher
* @license PHP v3.0
* @version 1.1
*/
class QRDecomposition {
/**
* Array for internal storage of decomposition.
* @var array
*/
var $QR = array();
/**
* Row dimension.
* @var integer
*/
var $m;
/**
* Column dimension.
* @var integer
*/
var $n;
/**
* Array for internal storage of diagonal of R.
* @var array
*/
var $Rdiag = array();
/**
* QR Decomposition computed by Householder reflections.
* @param matrix $A Rectangular matrix
* @return Structure to access R and the Householder vectors and compute Q.
*/
function QRDecomposition($A) {
if( is_a($A, 'Matrix') ) {
// Initialize.
$this->QR = $A->getArrayCopy();
$this->m = $A->getRowDimension();
$this->n = $A->getColumnDimension();
// Main loop.
for ($k = 0; $k < $this->n; $k++) {
// Compute 2-norm of k-th column without under/overflow.
$nrm = 0.0;
for ($i = $k; $i < $this->m; $i++)
$nrm = hypo($nrm, $this->QR[$i][$k]);
if ($nrm != 0.0) {
// Form k-th Householder vector.
if ($this->QR[$k][$k] < 0)
$nrm = -$nrm;
for ($i = $k; $i < $this->m; $i++)
$this->QR[$i][$k] /= $nrm;
$this->QR[$k][$k] += 1.0;
// Apply transformation to remaining columns.
for ($j = $k+1; $j < $this->n; $j++) {
$s = 0.0;
for ($i = $k; $i < $this->m; $i++)
$s += $this->QR[$i][$k] * $this->QR[$i][$j];
$s = -$s/$this->QR[$k][$k];
for ($i = $k; $i < $this->m; $i++)
$this->QR[$i][$j] += $s * $this->QR[$i][$k];
}
}
$this->Rdiag[$k] = -$nrm;
}
} else
trigger_error(ArgumentTypeException, ERROR);
}
/**
* Is the matrix full rank?
* @return boolean true if R, and hence A, has full rank, else false.
*/
function isFullRank() {
for ($j = 0; $j < $this->n; $j++)
if ($this->Rdiag[$j] == 0)
return false;
return true;
}
/**
* Return the Householder vectors
* @return Matrix Lower trapezoidal matrix whose columns define the reflections
*/
function getH() {
for ($i = 0; $i < $this->m; $i++) {
for ($j = 0; $j < $this->n; $j++) {
if ($i >= $j)
$H[$i][$j] = $this->QR[$i][$j];
else
$H[$i][$j] = 0.0;
}
}
return new Matrix($H);
}
/**
* Return the upper triangular factor
* @return Matrix upper triangular factor
*/
function getR() {
for ($i = 0; $i < $this->n; $i++) {
for ($j = 0; $j < $this->n; $j++) {
if ($i < $j)
$R[$i][$j] = $this->QR[$i][$j];
else if ($i == $j)
$R[$i][$j] = $this->Rdiag[$i];
else
$R[$i][$j] = 0.0;
}
}
return new Matrix($R);
}
/**
* Generate and return the (economy-sized) orthogonal factor
* @return Matrix orthogonal factor
*/
function getQ() {
for ($k = $this->n-1; $k >= 0; $k--) {
for ($i = 0; $i < $this->m; $i++)
$Q[$i][$k] = 0.0;
$Q[$k][$k] = 1.0;
for ($j = $k; $j < $this->n; $j++) {
if ($this->QR[$k][$k] != 0) {
$s = 0.0;
for ($i = $k; $i < $this->m; $i++)
$s += $this->QR[$i][$k] * $Q[$i][$j];
$s = -$s/$this->QR[$k][$k];
for ($i = $k; $i < $this->m; $i++)
$Q[$i][$j] += $s * $this->QR[$i][$k];
}
}
}
/*
for( $i = 0; $i < count($Q); $i++ )
for( $j = 0; $j < count($Q); $j++ )
if(! isset($Q[$i][$j]) )
$Q[$i][$j] = 0;
*/
return new Matrix($Q);
}
/**
* Least squares solution of A*X = B
* @param Matrix $B A Matrix with as many rows as A and any number of columns.
* @return Matrix Matrix that minimizes the two norm of Q*R*X-B.
*/
function solve($B) {
if ($B->getRowDimension() == $this->m) {
if ($this->isFullRank()) {
// Copy right hand side
$nx = $B->getColumnDimension();
$X = $B->getArrayCopy();
// Compute Y = transpose(Q)*B
for ($k = 0; $k < $this->n; $k++) {
for ($j = 0; $j < $nx; $j++) {
$s = 0.0;
for ($i = $k; $i < $this->m; $i++)
$s += $this->QR[$i][$k] * $X[$i][$j];
$s = -$s/$this->QR[$k][$k];
for ($i = $k; $i < $this->m; $i++)
$X[$i][$j] += $s * $this->QR[$i][$k];
}
}
// Solve R*X = Y;
for ($k = $this->n-1; $k >= 0; $k--) {
for ($j = 0; $j < $nx; $j++)
$X[$k][$j] /= $this->Rdiag[$k];
for ($i = 0; $i < $k; $i++)
for ($j = 0; $j < $nx; $j++)
$X[$i][$j] -= $X[$k][$j]* $this->QR[$i][$k];
}
$X = new Matrix($X);
return ($X->getMatrix(0, $this->n-1, 0, $nx));
} else
trigger_error(MatrixRankException, ERROR);
} else
trigger_error(MatrixDimensionException, ERROR);
}
}