/**
* Calculates lower triangular matrix L of given symmetric positive definite matrix
*
* @param NumArray $squareMatrix symmetric positive definite matrix
*
* @return NumArray
*
* @throws InvalidArgumentException will be thrown, when `$squareMatrix` is not positive definite
* @throws NoSquareMatrixException will be thrown, when `$squareMatrix` is not square
* @throws NoSymmetricMatrixException will be thrown, when `$squareMatrix` is not symmetric
*
* @since 1.0.2
*/
public static function cholesky(NumArray $squareMatrix)
{
if (!Helper::isSquareMatrix($squareMatrix)) {
throw new NoSquareMatrixException(sprintf("Matrix with shape (%s) given, matrix has to be square", implode(', ', $squareMatrix->getShape())));
}
$shape = $squareMatrix->getShape();
$size = $shape[0];
$aMatrix = clone $squareMatrix;
$lMatrix = NumPHP::zerosLike($aMatrix);
for ($k = 0; $k < $size; $k++) {
$diaElem = $aMatrix->get($k, $k)->getData();
if ($diaElem <= 0) {
throw new InvalidArgumentException("Matrix is not positive definite");
}
$diaElem = sqrt($diaElem);
$lMatrix->set($k, $k, $diaElem);
for ($i = $k + 1; $i < $size; $i++) {
if ($squareMatrix->get($i, $k) != $squareMatrix->get($k, $i)) {
throw new NoSymmetricMatrixException("Matrix is not symmetric");
}
$lMatrix->set($i, $k, $aMatrix->get($i, $k)->div($diaElem));
for ($j = $k + 1; $j <= $i; $j++) {
$aMatrix->set($i, $j, $aMatrix->get($i, $j)->sub($lMatrix->get($i, $k)->mult($lMatrix->get($j, $k))));
}
}
}
return $lMatrix;
}
/**
* Factors a matrix into a pivot matrix, a lower and upper triangular matrix
*
* @param NumArray $array matrix
*
* @throws NoMatrixException will be thrown, if `array` is no matrix
*
* @return array
*
* @since 1.0.0
*/
public static function lud(NumArray $array)
{
if (!Helper::isMatrix($array)) {
throw new NoMatrixException(sprintf("NumArray with dimension %d given, NumArray should have 2 dimensions", $array->getNDim()));
}
$numArray = clone $array;
$shape = $numArray->getShape();
$mAxis = $shape[0];
$nAxis = $shape[1];
$size = min($mAxis, $nAxis);
$pArray = range(0, $mAxis - 1);
$lMatrix = NumPHP::zeros($mAxis, $size);
for ($i = 0; $i < $size; $i++) {
// pivoting
$maxIndex = self::getPivotIndex($numArray, $i);
if ($maxIndex !== $i) {
$temp = $numArray->get($i);
$numArray->set($i, $numArray->get($maxIndex));
$numArray->set($maxIndex, $temp);
$temp = $lMatrix->get($i);
$lMatrix->set($i, $lMatrix->get($maxIndex));
$lMatrix->set($maxIndex, $temp);
$temp = $pArray[$i];
$pArray[$i] = $pArray[$maxIndex];
$pArray[$maxIndex] = $temp;
}
// elimination
for ($j = $i + 1; $j < $mAxis; $j++) {
$fac = 0;
$facNumerator = $numArray->get($j, $i)->getData();
$facDenominator = $numArray->get($i, $i)->getData();
if ($facDenominator) {
$fac = $facNumerator / $facDenominator;
}
$lMatrix->set($j, $i, $fac);
$slice = sprintf("%d:", $i + 1);
$numArray->set($j, $slice, $numArray->get($j, $slice)->sub($numArray->get($i, $slice)->mult($fac)));
}
}
return [self::buildPivotMatrix($pArray), self::buildLMatrix($lMatrix), self::buildUMatrix($numArray)];
}
/**
* Calculates the inverse of a not singular square matrix
*
* @param mixed $squareMatrix not singular matrix
*
* @throws SingularMatrixException will be thrown, when `$squareMatrix` is singular
*
* @api
* @since 1.0.0
*
* @return NumArray
*/
public static function inv($squareMatrix)
{
if (!$squareMatrix instanceof NumArray) {
$squareMatrix = new NumArray($squareMatrix);
} elseif ($squareMatrix->inCache(self::CACHE_KEY_INVERSE)) {
return $squareMatrix->getCache(self::CACHE_KEY_INVERSE);
}
if (!Helper::isNotSingularMatrix($squareMatrix)) {
throw new SingularMatrixException("Matrix is singular");
}
$shape = $squareMatrix->getShape();
$inv = self::solve($squareMatrix, NumPHP::identity($shape[0]));
$squareMatrix->setCache(self::CACHE_KEY_INVERSE, $inv);
return self::inv($squareMatrix);
}
/**
* Tests if Helper::isNotSingularMatrix works with invalid not singular matrix
*/
public function testCheckNotSingularMatrixInvalid()
{
$numArray = NumPHP::identity(4);
$numArray->set(2, 2, 0);
$this->assertFalse(Helper::isNotSingularMatrix($numArray));
}
/**
* Back Substitution solves a linear system with a upper triangular matrix of
* size n*n and a vector of size n or a matrix of size n*m
*
* @param NumArray $uMatrix upper triangular matrix of size n*n
* @param NumArray $numArray vector of size n or matrix of size n*m
*
* @return NumArray
*
* @since 1.0.0
*/
protected static function backSubstitution(NumArray $uMatrix, NumArray $numArray)
{
$shape = $numArray->getShape();
if (Helper::isVector($numArray)) {
$xVector = NumPHP::zerosLike($numArray);
for ($i = $shape[0] - 1; $i >= 0; $i--) {
$slice = sprintf("%d:%d", $i + 1, $shape[0]);
$sum = $uMatrix->get($i, $slice)->dot($xVector->get($slice));
$xVector->set($i, $numArray->get($i)->sub($sum)->div($uMatrix->get($i, $i)));
}
return $xVector;
}
// $numArray is a matrix
$copy = clone $numArray;
for ($i = 0; $i < $shape[1]; $i++) {
$copy->set(':', $i, self::backSubstitution($uMatrix, $copy->get(':', $i)));
}
return $copy;
}
