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<?php
namespace PhpOffice\PhpSpreadsheet\Shared\JAMA;
use PhpOffice\PhpSpreadsheet\Calculation\Exception as CalculationException;
/**
* 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
*
* @version 1.1
*/
class QRDecomposition
{
const MATRIX_RANK_EXCEPTION = 'Can only perform operation on full-rank matrix.';
/**
* Array for internal storage of decomposition.
*
* @var array
*/
private $QR = [];
/**
* Row dimension.
*
* @var int
*/
private $m;
/**
* Column dimension.
*
* @var int
*/
private $n;
/**
* Array for internal storage of diagonal of R.
*
* @var array
*/
private $Rdiag = [];
/**
* QR Decomposition computed by Householder reflections.
*
* @param Matrix $A Rectangular matrix
*/
public function __construct(Matrix $A)
{
// Initialize.
$this->QR = $A->getArray();
$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] = -$nrm;
}
}
// function __construct()
/**
* Is the matrix full rank?
*
* @return bool true if R, and hence A, has full rank, else false
*/
public function isFullRank()
{
for ($j = 0; $j < $this->n; ++$j) {
if ($this->Rdiag[$j] == 0) {
return false;
}
}
return true;
}
// function isFullRank()
/**
* Return the Householder vectors.
*
* @return Matrix Lower trapezoidal matrix whose columns define the reflections
*/
public function getH()
{
$H = [];
for ($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);
}
// function getH()
/**
* Return the upper triangular factor.
*
* @return Matrix upper triangular factor
*/
public function getR()
{
$R = [];
for ($i = 0; $i < $this->n; ++$i) {
for ($j = 0; $j < $this->n; ++$j) {
if ($i < $j) {
$R[$i][$j] = $this->QR[$i][$j];
} elseif ($i == $j) {
$R[$i][$j] = $this->Rdiag[$i];
} else {
$R[$i][$j] = 0.0;
}
}
}
return new Matrix($R);
}
// function getR()
/**
* Generate and return the (economy-sized) orthogonal factor.
*
* @return Matrix orthogonal factor
*/
public function getQ()
{
$Q = [];
for ($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][$k];
}
}
}
}
return new Matrix($Q);
}
// function getQ()
/**
* 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
*/
public function solve(Matrix $B)
{
if ($B->getRowDimension() == $this->m) {
if ($this->isFullRank()) {
// Copy right hand side
$nx = $B->getColumnDimension();
$X = $B->getArray();
// Compute Y = transpose(Q)*B
for ($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);
}
throw new CalculationException(self::MATRIX_RANK_EXCEPTION);
}
throw new CalculationException(Matrix::MATRIX_DIMENSION_EXCEPTION);
}
}
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