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<?php |
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|
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namespace Matrix\Decomposition; |
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|
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use Matrix\Exception; |
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use Matrix\Matrix; |
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|
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class LU |
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{ |
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private $luMatrix; |
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private $rows; |
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private $columns; |
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|
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private $pivot = []; |
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|
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public function __construct(Matrix $matrix) |
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{ |
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$this->luMatrix = $matrix->toArray(); |
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$this->rows = $matrix->rows; |
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$this->columns = $matrix->columns; |
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|
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$this->buildPivot(); |
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} |
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|
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/** |
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* Get lower triangular factor. |
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* |
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* @return Matrix Lower triangular factor |
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*/ |
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public function getL() |
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{ |
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$lower = []; |
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|
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$columns = min($this->rows, $this->columns); |
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for ($row = 0; $row < $this->rows; ++$row) { |
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for ($column = 0; $column < $columns; ++$column) { |
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if ($row > $column) { |
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$lower[$row][$column] = $this->luMatrix[$row][$column]; |
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} elseif ($row === $column) { |
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$lower[$row][$column] = 1.0; |
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} else { |
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$lower[$row][$column] = 0.0; |
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} |
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} |
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} |
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|
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return new Matrix($lower); |
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} |
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|
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/** |
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* Get upper triangular factor. |
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* |
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* @return Matrix Upper triangular factor |
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*/ |
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public function getU() |
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{ |
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$upper = []; |
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|
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$rows = min($this->rows, $this->columns); |
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for ($row = 0; $row < $rows; ++$row) { |
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for ($column = 0; $column < $this->columns; ++$column) { |
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if ($row <= $column) { |
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$upper[$row][$column] = $this->luMatrix[$row][$column]; |
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} else { |
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$upper[$row][$column] = 0.0; |
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} |
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} |
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} |
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|
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return new Matrix($upper); |
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} |
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|
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/** |
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* Return pivot permutation vector. |
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* |
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* @return Matrix Pivot matrix |
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*/ |
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public function getP() |
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{ |
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$pMatrix = []; |
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|
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$pivots = $this->pivot; |
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$pivotCount = count($pivots); |
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foreach ($pivots as $row => $pivot) { |
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$pMatrix[$row] = array_fill(0, $pivotCount, 0); |
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$pMatrix[$row][$pivot] = 1; |
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} |
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|
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return new Matrix($pMatrix); |
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} |
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|
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/** |
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* Return pivot permutation vector. |
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* |
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* @return array Pivot vector |
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*/ |
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public function getPivot() |
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{ |
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return $this->pivot; |
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} |
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|
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/** |
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* Is the matrix nonsingular? |
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* |
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* @return bool true if U, and hence A, is nonsingular |
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*/ |
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public function isNonsingular() |
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{ |
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for ($diagonal = 0; $diagonal < $this->columns; ++$diagonal) { |
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if ($this->luMatrix[$diagonal][$diagonal] === 0.0) { |
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return false; |
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} |
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} |
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|
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return true; |
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} |
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|
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private function buildPivot() |
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{ |
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for ($row = 0; $row < $this->rows; ++$row) { |
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$this->pivot[$row] = $row; |
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} |
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|
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for ($column = 0; $column < $this->columns; ++$column) { |
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$luColumn = $this->localisedReferenceColumn($column); |
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|
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$this->applyTransformations($column, $luColumn); |
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|
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$pivot = $this->findPivot($column, $luColumn); |
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if ($pivot !== $column) { |
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$this->pivotExchange($pivot, $column); |
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} |
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|
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$this->computeMultipliers($column); |
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|
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unset($luColumn); |
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} |
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} |
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|
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private function localisedReferenceColumn($column) |
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{ |
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$luColumn = []; |
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|
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for ($row = 0; $row < $this->rows; ++$row) { |
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$luColumn[$row] = &$this->luMatrix[$row][$column]; |
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} |
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|
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return $luColumn; |
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} |
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|
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private function applyTransformations($column, array $luColumn) |
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{ |
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for ($row = 0; $row < $this->rows; ++$row) { |
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$luRow = $this->luMatrix[$row]; |
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// Most of the time is spent in the following dot product. |
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$kmax = min($row, $column); |
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$sValue = 0.0; |
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for ($kValue = 0; $kValue < $kmax; ++$kValue) { |
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$sValue += $luRow[$kValue] * $luColumn[$kValue]; |
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} |
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$luRow[$column] = $luColumn[$row] -= $sValue; |
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} |
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} |
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|
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private function findPivot($column, array $luColumn) |
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{ |
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$pivot = $column; |
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for ($row = $column + 1; $row < $this->rows; ++$row) { |
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if (abs($luColumn[$row]) > abs($luColumn[$pivot])) { |
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$pivot = $row; |
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} |
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} |
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|
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return $pivot; |
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} |
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|
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private function pivotExchange($pivot, $column) |
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{ |
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for ($kValue = 0; $kValue < $this->columns; ++$kValue) { |
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$tValue = $this->luMatrix[$pivot][$kValue]; |
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$this->luMatrix[$pivot][$kValue] = $this->luMatrix[$column][$kValue]; |
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$this->luMatrix[$column][$kValue] = $tValue; |
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} |
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|
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$lValue = $this->pivot[$pivot]; |
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$this->pivot[$pivot] = $this->pivot[$column]; |
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$this->pivot[$column] = $lValue; |
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} |
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|
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private function computeMultipliers($diagonal) |
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{ |
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if (($diagonal < $this->rows) && ($this->luMatrix[$diagonal][$diagonal] != 0.0)) { |
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for ($row = $diagonal + 1; $row < $this->rows; ++$row) { |
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$this->luMatrix[$row][$diagonal] /= $this->luMatrix[$diagonal][$diagonal]; |
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} |
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} |
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} |
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|
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private function pivotB(Matrix $B) |
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{ |
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$X = []; |
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foreach ($this->pivot as $rowId) { |
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$row = $B->getRows($rowId + 1)->toArray(); |
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$X[] = array_pop($row); |
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} |
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|
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return $X; |
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} |
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|
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/** |
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* Solve A*X = B. |
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* |
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* @param Matrix $B a Matrix with as many rows as A and any number of columns |
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* |
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* @throws Exception |
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* |
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* @return Matrix X so that L*U*X = B(piv,:) |
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*/ |
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public function solve(Matrix $B) |
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{ |
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if ($B->rows !== $this->rows) { |
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throw new Exception('Matrix row dimensions are not equal'); |
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} |
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|
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if ($this->rows !== $this->columns) { |
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throw new Exception('LU solve() only works on square matrices'); |
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} |
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|
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if (!$this->isNonsingular()) { |
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throw new Exception('Can only perform operation on singular matrix'); |
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} |
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|
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// Copy right hand side with pivoting |
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$nx = $B->columns; |
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$X = $this->pivotB($B); |
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|
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// Solve L*Y = B(piv,:) |
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for ($k = 0; $k < $this->columns; ++$k) { |
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for ($i = $k + 1; $i < $this->columns; ++$i) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$i][$j] -= $X[$k][$j] * $this->luMatrix[$i][$k]; |
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} |
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} |
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} |
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|
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// Solve U*X = Y; |
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for ($k = $this->columns - 1; $k >= 0; --$k) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$k][$j] /= $this->luMatrix[$k][$k]; |
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} |
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for ($i = 0; $i < $k; ++$i) { |
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for ($j = 0; $j < $nx; ++$j) { |
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$X[$i][$j] -= $X[$k][$j] * $this->luMatrix[$i][$k]; |
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} |
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} |
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} |
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|
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return new Matrix($X); |
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} |
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} |
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|
