Add Moore-Penrose inverse

src/LinearAlgebra/Decomposition/SVD.php

@@ -2,7 +2,9 @@
 
 namespace MathPHP\LinearAlgebra\Decomposition;
 
+use MathPHP\Arithmetic;
 use MathPHP\Exception;
+use MathPHP\Exception\MatrixException;
 use MathPHP\LinearAlgebra\NumericMatrix;
 use MathPHP\LinearAlgebra\MatrixFactory;
 use MathPHP\LinearAlgebra\Vector;
@@ -107,6 +109,19 @@
         // A rectangular diagonal matrix
         $S = $U->transpose()->multiply($M)->multiply($V);
 
+        // If S is non-diagonal, try permuting S to be diagonal
+        if (!$S->isRectangularDiagonal()) {
+            ['sort'=>$sort, 'P'=>$P] = self::diagonalize($S);
+            // Depending on the value of $sort, we either permute the rows or columns of $S 
+            if ($sort === 'm') {
+                $S = $P->multiply($S);            // Permute rows of S
+                $U = $U->multiply($P->inverse()); // Permute corresponding columns of U
+            } elseif ($sort === 'n') {
+                $S = $S->multiply($P);            // Permute columns of S
+                $V = $P->inverse()->multiply($V); // Permute corresponding rows of V
+            }
+        }
+
         $diag = $S->getDiagonalElements();
 
         // If there is a negative singular value, we need to adjust the signs of columns in U
@@ -120,9 +135,226 @@
             $S = $signature->multiply($S);
         }
 
+        // Check the elements are in descending order
+        if (!self::isDiagonalDescending($S)) {
+            $P = self::sortDiagonal($S);
+            /** @var NumericMatrix */
+            $Pᵀ = $P->transpose();
+
+            $S = $Pᵀ->multiply($S)->multiply($P);
+            $U = $P->multiply($U)->multiply($Pᵀ);
+        }
+
         return new SVD($U, $S, $V);
     }
 
+    /**
+     * Returns a permutation matrix, P, such that the product of SP is diagonal
+     * 
+     * @param NumericMatrix $S the matrix to diagonalize
+     * 
+     * @return array{'sort': string, 'P': NumericMatrix} a matrix, P, that will diagonalize S. Multiplication order defined by sort
+     * If 'm', then pre-multiply
+     * If 'n', then post-multiply
+     */
+    private static function diagonalize(NumericMatrix $S): array
+    {
+        if ($S->isRectangularDiagonal()) {
+            return MatrixFactory::identity($S->getN());
+        }
+
+        $sort = '';
+        $vecMethod = '';
+        $max = 0;
+        $min = 0;
+
+        if ($S->getM() >= $S->getN()) {
+            $sort = 'm'; // rows
+            $vecMethod = 'asRowVectors';
+            $max = $S->getM();
+            $min = $S->getN();
+        } else {
+            $sort = 'n'; // columns
+            $vecMethod = 'asVectors';
+            $max = $S->getN();
+            $min = $S->getM();
+        }
+
+        // Create an identity matrix to store permutations in
+        $P = MatrixFactory::identity($max)->{$vecMethod}();
+
+        // Push all zero-columns to the right
+        $vectors = $S->{$vecMethod}();
+
+        $zeroCols = [];
+
+        foreach ($vectors as $i => $vector)
+        {
+            // Each column should contain 1 non-zero element
+            $isZero = Arithmetic::almostEqual((float) $vector->l2Norm(), 0);
+
+            $zeroCols[$i] = $isZero ? 0 : 1;
+        }
+
+        arsort($zeroCols, SORT_NUMERIC);
+
+        $zeroMap = array_keys($zeroCols);
+
+        uksort($P, function ($left, $right) use ($zeroMap) {
+            $leftPos = $zeroMap[$left];
+            $rightPos = $zeroMap[$right];
+
+            return $leftPos >= $rightPos;
+        });
+
+        // Only check the columns that contain diagonal entries
+        $vectors = $S->submatrix(0,0, $min-1, $min-1)->{$vecMethod}();
+
+        $nonDiagonalValues = [];
+
+        /** @var Vector */
+        foreach ($vectors as $i => $vector)
+        {
+            $ε = $S->getError();
+
+            // Each column should contain up to 1 non-zero element
+            if (Arithmetic::almostEqual($vector->l2Norm(), 0, $ε)) {
+                $j = $i;
+            } else {
+                $j = self::getStandardBasisIndex($vector, $ε);
+            }
+
+            if ($j === -1) {
+                throw new MatrixException("S Matrix in SVD is not orthogonal:\n" . (string) $S);
+            }
+
+            if ($i === $j) {
+                continue;
+            } else {
+                $nonDiagonalValues[$i] = ['value' => $vector[$j], 'j' => $j];
+            }
+        }
+
+        // Now create a sort order
+        $order = range(0, $min - 1);
+
+        foreach ($nonDiagonalValues as $i => $elem)
+        {
+            $entry = $elem['j'];
+            $order[$entry] = $i;
+        }
+
+        $map = array_flip($order);
+
+        // Need to make column ($i of $nonDiagonalValues) = row ($j)
+        // order = [1=>2, 2=>1, 3=>3]
+        uksort($P, function ($left, $right) use ($map) {
+            $leftPos = isset($map[$left]) ? $map[$left] : INF; // sorts in ascending order, so just use inf
+            $rightPos = isset($map[$right]) ? $map[$right] : INF;
+
+            return $leftPos <=> $rightPos;
+        });
+
+        $P = MatrixFactory::createFromVectors($P);
+
+        // fromVectors treats the array as column vectors, so the matrix might need to be transposed
+        if ($sort === 'm') {
+            $P = $P->transpose();
+        }
+
+        return ['sort'=>$sort, 'P' => $P];
+    }
+
+    /**
+     * Checks that a vector has a single non-zero entry and returns its index
+     * 
+     * @param Vector $v
+     * 
+     * @return int The index of the non-zero entry or -1 if either:
+     *      1. There are multiple non-zero entries
+     *      2. The vector is a zero vector 
+     */
+    private static function getStandardBasisIndex(Vector $v, float $ε): int
+    {
+        if ($v->l2Norm() === 0) {
+            return false;
+        }
+
+        // Vectors don't have negative indices
+        $index = -1;
+
+        foreach ($v->getVector() as $i => $component)
+        {
+            if (!Arithmetic::almostEqual($component, 0, $ε)) {
+                if ($index === -1) {
+                    $index = $i;
+                } else { // If we already found a non-zero component, then return -1
+                    return -1;
+                }
+            }
+        }
+
+        return $index;
+    }
+
+    /**
+     * Returns a permutation matrix that sorts its diagonal values in descending order
+     * 
+     * @param NumericMatrix $S singular matrix
+     * 
+     * @return NumericMatrix a permutation matrix such that PᵀSP is diagonal
+     */
+    private static function sortDiagonal(NumericMatrix $S): NumericMatrix
+    {
+        // Get diagonal, pad it by columns, and sort it
+        $diagonal = $S->getDiagonalElements();
+
+        // Pad
+        $padLength = $S->getN() - count($diagonal);
+
+        $diagonal = array_merge($diagonal, array_fill(0, $padLength, 0)); // Pick 0 because the numbers should all be positive
+
+        // arsort preserves the indices
+        arsort($diagonal, SORT_NUMERIC);
+
+        // ... so we can create a position map from the keys
+        $map = array_keys($diagonal);
+
+        $P = MatrixFactory::identity($S->getM())->asVectors();
+
+        uksort($P, function ($left, $right) use ($map) {
+            $leftPos = $map[$left];
+            $rightPos = $map[$right];
+
+            return $leftPos >= $rightPos;
+        });
+
+        return MatrixFactory::createFromVectors($P);
+    }
+
+    /**
+     * Checks if the elements of a diagonal matrix are in descending order
+     * 
+     * @param NumericMatrix $S the matrix to check
+     * 
+     * @return bool
+     */
+    private static function isDiagonalDescending(NumericMatrix $S): bool
+    {
+        $diagonal = $S->getDiagonalElements();
+        $sorted = array_values($diagonal); rsort($sorted, SORT_NUMERIC);
+
+        // Compare sorted using matrix error (in case duplicate, floating-point eigenvalues) 
+        $n = count($diagonal);
+        for ($i = 0; $i < $n; $i++) {
+            if (!Arithmetic::almostEqual($diagonal[$i], $sorted[$i], $S->getError())) {
+                return false;
+            }
+        }
+
+        return true;
+    }
+
     /**
      * Get U, S, or V matrix, or D vector
      *

src/LinearAlgebra/Eigenvector.php

@@ -2,6 +2,7 @@
 
 namespace MathPHP\LinearAlgebra;
 
+use MathPHP\Arithmetic;
 use MathPHP\Exception;
 use MathPHP\Exception\MatrixException;
 use MathPHP\Functions\Map\Single;
@@ -66,8 +67,14 @@ public static function eigenvectors(NumericMatrix $A, array $eigenvalues = []):
         foreach ($eigenvalues as $eigenvalue) {
             // If this is a duplicate eigenvalue, and this is the second instance, the first
             // pass already found all the vectors.
-            $key = \array_search($eigenvalue, \array_column($solution_array, 'eigenvalue'));
-            if (!$key) {
+            $key = false;
+            foreach (\array_column($solution_array, 'eigenvalue') as $i => $v) {
+                if (Arithmetic::almostEqual($v, $eigenvalue, $A->getError())) {
+                    $key = $i;
+                    break;
+                }
+            }
+            if ($key === false) {
                 $Iλ = MatrixFactory::identity($number)->scalarMultiply($eigenvalue);
                 $T = $A->subtract($Iλ);
 

src/LinearAlgebra/MatrixCatalog.php

@@ -15,6 +15,9 @@
     /** @var Matrix<T> inverse */
     private $A⁻¹;
 
+    /** @var NumericMatrix pseudoInverse */
+    private $A⁺;
+
     /** @var Reduction\RowEchelonForm */
     private $REF;
 
@@ -43,6 +46,7 @@
      * DERIVED MATRICES
      *  - transpose
      *  - inverse
+     *  - pseudo-inverse
      **************************************************************************/
 
     // TRANSPOSE
@@ -99,6 +103,32 @@
         return $this->A⁻¹;
     }
 
+    // PSEUDO-INVERSE
+
+    /**
+     * @param Matrix $A⁺
+     */
+    public function addPseudoInverse(Matrix $A⁺): void
+    {
+        $this->A⁺ = $A⁺;
+    }
+
+    /**
+     * @return bool
+     */
+    public function hasPseudoInverse(): bool
+    {
+        return isset($this->A⁺);
+    }
+
+    /**
+     * @return Matrix
+     */
+    public function getPseudoInverse(): Matrix
+    {
+        return $this->A⁺;
+    }
+
     /**************************************************************************
      * MATRIX REDUCTIONS
      *  - ref (row echelon form)

src/LinearAlgebra/NumericDiagonalMatrix.php

@@ -80,4 +80,14 @@ public function inverse(): NumericMatrix
     {
         return MatrixFactory::diagonal(Single::reciprocal($this->getDiagonalElements()));
     }
+
+    /**
+     * pseudoInverse identical to inverse
+     *
+     * @return NumericMatrix
+     */
+    public function pseudoInverse(): NumericMatrix
+    {
+        return $this->inverse();
+    }
 }