commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/linear/LUDecomposition.java

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 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
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 * See the License for the specific language governing permissions and
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package org.apache.commons.math4.legacy.linear;

import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.core.jdkmath.AccurateMath;

/**
 * Calculates the LUP-decomposition of a square matrix.
 * <p>The LUP-decomposition of a matrix A consists of three matrices L, U and
 * P that satisfy: P&times;A = L&times;U. L is lower triangular (with unit
 * diagonal terms), U is upper triangular and P is a permutation matrix. All
 * matrices are m&times;m.</p>
 * <p>As shown by the presence of the P matrix, this decomposition is
 * implemented using partial pivoting.</p>
 * <p>This class is based on the class with similar name from the
 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library.</p>
 * <ul>
 *   <li>a {@link #getP() getP} method has been added,</li>
 *   <li>the {@code det} method has been renamed as {@link #getDeterminant()
 *   getDeterminant},</li>
 *   <li>the {@code getDoublePivot} method has been removed (but the int based
 *   {@link #getPivot() getPivot} method has been kept),</li>
 *   <li>the {@code solve} and {@code isNonSingular} methods have been replaced
 *   by a {@link #getSolver() getSolver} method and the equivalent methods
 *   provided by the returned {@link DecompositionSolver}.</li>
 * </ul>
 *
 * @see <a href="http://mathworld.wolfram.com/LUDecomposition.html">MathWorld</a>
 * @see <a href="http://en.wikipedia.org/wiki/LU_decomposition">Wikipedia</a>
 * @since 2.0 (changed to concrete class in 3.0)
 */
public class LUDecomposition {
    /** Default bound to determine effective singularity in LU decomposition. */
    private static final double DEFAULT_TOO_SMALL = 1e-11;
    /** Entries of LU decomposition. */
    private final double[][] lu;
    /** Pivot permutation associated with LU decomposition. */
    private final int[] pivot;
    /** Parity of the permutation associated with the LU decomposition. */
    private boolean even;
    /** Singularity indicator. */
    private boolean singular;
    /** Cached value of L. */
    private RealMatrix cachedL;
    /** Cached value of U. */
    private RealMatrix cachedU;
    /** Cached value of P. */
    private RealMatrix cachedP;

    /**
     * Calculates the LU-decomposition of the given matrix.
     * This constructor uses 1e-11 as default value for the singularity
     * threshold.
     *
     * @param matrix Matrix to decompose.
     * @throws NonSquareMatrixException if matrix is not square.
     */
    public LUDecomposition(RealMatrix matrix) {
        this(matrix, DEFAULT_TOO_SMALL);
    }

    /**
     * Calculates the LU-decomposition of the given matrix.
     * @param matrix The matrix to decompose.
     * @param singularityThreshold threshold (based on partial row norm)
     * under which a matrix is considered singular
     * @throws NonSquareMatrixException if matrix is not square
     */
    public LUDecomposition(RealMatrix matrix, double singularityThreshold) {
        if (!matrix.isSquare()) {
            throw new NonSquareMatrixException(matrix.getRowDimension(),
                                               matrix.getColumnDimension());
        }

        final int m = matrix.getColumnDimension();
        lu = matrix.getData();
        pivot = new int[m];
        cachedL = null;
        cachedU = null;
        cachedP = null;

        // Initialize permutation array and parity
        for (int row = 0; row < m; row++) {
            pivot[row] = row;
        }
        even     = true;
        singular = false;

        // Loop over columns
        for (int col = 0; col < m; col++) {

            // upper
            for (int row = 0; row < col; row++) {
                final double[] luRow = lu[row];
                double sum = luRow[col];
                for (int i = 0; i < row; i++) {
                    sum -= luRow[i] * lu[i][col];
                }
                luRow[col] = sum;
            }

            // lower
            int max = col; // permutation row
            double largest = Double.NEGATIVE_INFINITY;
            for (int row = col; row < m; row++) {
                final double[] luRow = lu[row];
                double sum = luRow[col];
                for (int i = 0; i < col; i++) {
                    sum -= luRow[i] * lu[i][col];
                }
                luRow[col] = sum;

                // maintain best permutation choice
                if (AccurateMath.abs(sum) > largest) {
                    largest = AccurateMath.abs(sum);
                    max = row;
                }
            }

            // Singularity check
            if (AccurateMath.abs(lu[max][col]) < singularityThreshold) {
                singular = true;
                return;
            }

            // Pivot if necessary
            if (max != col) {
                double tmp = 0;
                final double[] luMax = lu[max];
                final double[] luCol = lu[col];
                for (int i = 0; i < m; i++) {
                    tmp = luMax[i];
                    luMax[i] = luCol[i];
                    luCol[i] = tmp;
                }
                int temp = pivot[max];
                pivot[max] = pivot[col];
                pivot[col] = temp;
                even = !even;
            }

            // Divide the lower elements by the "winning" diagonal elt.
            final double luDiag = lu[col][col];
            for (int row = col + 1; row < m; row++) {
                lu[row][col] /= luDiag;
            }
        }
    }

    /**
     * Returns the matrix L of the decomposition.
     * <p>L is a lower-triangular matrix</p>
     * @return the L matrix (or null if decomposed matrix is singular)
     */
    public RealMatrix getL() {
        if ((cachedL == null) && !singular) {
            final int m = pivot.length;
            cachedL = MatrixUtils.createRealMatrix(m, m);
            for (int i = 0; i < m; ++i) {
                final double[] luI = lu[i];
                for (int j = 0; j < i; ++j) {
                    cachedL.setEntry(i, j, luI[j]);
                }
                cachedL.setEntry(i, i, 1.0);
            }
        }
        return cachedL;
    }

    /**
     * Returns the matrix U of the decomposition.
     * <p>U is an upper-triangular matrix</p>
     * @return the U matrix (or null if decomposed matrix is singular)
     */
    public RealMatrix getU() {
        if ((cachedU == null) && !singular) {
            final int m = pivot.length;
            cachedU = MatrixUtils.createRealMatrix(m, m);
            for (int i = 0; i < m; ++i) {
                final double[] luI = lu[i];
                for (int j = i; j < m; ++j) {
                    cachedU.setEntry(i, j, luI[j]);
                }
            }
        }
        return cachedU;
    }

    /**
     * Returns the P rows permutation matrix.
     * <p>P is a sparse matrix with exactly one element set to 1.0 in
     * each row and each column, all other elements being set to 0.0.</p>
     * <p>The positions of the 1 elements are given by the {@link #getPivot()
     * pivot permutation vector}.</p>
     * @return the P rows permutation matrix (or null if decomposed matrix is singular)
     * @see #getPivot()
     */
    public RealMatrix getP() {
        if ((cachedP == null) && !singular) {
            final int m = pivot.length;
            cachedP = MatrixUtils.createRealMatrix(m, m);
            for (int i = 0; i < m; ++i) {
                cachedP.setEntry(i, pivot[i], 1.0);
            }
        }
        return cachedP;
    }

    /**
     * Returns the pivot permutation vector.
     * @return the pivot permutation vector
     * @see #getP()
     */
    public int[] getPivot() {
        return pivot.clone();
    }

    /**
     * Return the determinant of the matrix.
     * @return determinant of the matrix
     */
    public double getDeterminant() {
        if (singular) {
            return 0;
        } else {
            final int m = pivot.length;
            double determinant = even ? 1 : -1;
            for (int i = 0; i < m; i++) {
                determinant *= lu[i][i];
            }
            return determinant;
        }
    }

    /**
     * Get a solver for finding the A &times; X = B solution in exact linear
     * sense.
     * @return a solver
     */
    public DecompositionSolver getSolver() {
        return new Solver(lu, pivot, singular);
    }

    /** Specialized solver. */
    private static final class Solver implements DecompositionSolver {

        /** Entries of LU decomposition. */
        private final double[][] lu;

        /** Pivot permutation associated with LU decomposition. */
        private final int[] pivot;

        /** Singularity indicator. */
        private final boolean singular;

        /**
         * Build a solver from decomposed matrix.
         * @param lu entries of LU decomposition
         * @param pivot pivot permutation associated with LU decomposition
         * @param singular singularity indicator
         */
        private Solver(final double[][] lu, final int[] pivot, final boolean singular) {
            this.lu       = lu;
            this.pivot    = pivot;
            this.singular = singular;
        }

        /** {@inheritDoc} */
        @Override
        public boolean isNonSingular() {
            return !singular;
        }

        /** {@inheritDoc} */
        @Override
        public RealVector solve(RealVector b) {
            final int m = pivot.length;
            if (b.getDimension() != m) {
                throw new DimensionMismatchException(b.getDimension(), m);
            }
            if (singular) {
                throw new SingularMatrixException();
            }

            final double[] bp = new double[m];

            // Apply permutations to b
            for (int row = 0; row < m; row++) {
                bp[row] = b.getEntry(pivot[row]);
            }

            // Solve LY = b
            for (int col = 0; col < m; col++) {
                final double bpCol = bp[col];
                for (int i = col + 1; i < m; i++) {
                    bp[i] -= bpCol * lu[i][col];
                }
            }

            // Solve UX = Y
            for (int col = m - 1; col >= 0; col--) {
                bp[col] /= lu[col][col];
                final double bpCol = bp[col];
                for (int i = 0; i < col; i++) {
                    bp[i] -= bpCol * lu[i][col];
                }
            }

            return new ArrayRealVector(bp, false);
        }

        /** {@inheritDoc} */
        @Override
        public RealMatrix solve(RealMatrix b) {

            final int m = pivot.length;
            if (b.getRowDimension() != m) {
                throw new DimensionMismatchException(b.getRowDimension(), m);
            }
            if (singular) {
                throw new SingularMatrixException();
            }

            final int nColB = b.getColumnDimension();

            // Apply permutations to b
            final double[][] bp = new double[m][nColB];
            for (int row = 0; row < m; row++) {
                final double[] bpRow = bp[row];
                final int pRow = pivot[row];
                for (int col = 0; col < nColB; col++) {
                    bpRow[col] = b.getEntry(pRow, col);
                }
            }

            // Solve LY = b
            for (int col = 0; col < m; col++) {
                final double[] bpCol = bp[col];
                for (int i = col + 1; i < m; i++) {
                    final double[] bpI = bp[i];
                    final double luICol = lu[i][col];
                    for (int j = 0; j < nColB; j++) {
                        bpI[j] -= bpCol[j] * luICol;
                    }
                }
            }

            // Solve UX = Y
            for (int col = m - 1; col >= 0; col--) {
                final double[] bpCol = bp[col];
                final double luDiag = lu[col][col];
                for (int j = 0; j < nColB; j++) {
                    bpCol[j] /= luDiag;
                }
                for (int i = 0; i < col; i++) {
                    final double[] bpI = bp[i];
                    final double luICol = lu[i][col];
                    for (int j = 0; j < nColB; j++) {
                        bpI[j] -= bpCol[j] * luICol;
                    }
                }
            }

            return new Array2DRowRealMatrix(bp, false);
        }

        /**
         * Get the inverse of the decomposed matrix.
         *
         * @return the inverse matrix.
         * @throws SingularMatrixException if the decomposed matrix is singular.
         */
        @Override
        public RealMatrix getInverse() {
            return solve(MatrixUtils.createRealIdentityMatrix(pivot.length));
        }
    }
}