QRDecomposition.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
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package org.apache.commons.math4.legacy.linear;

import java.util.Arrays;

import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.core.jdkmath.JdkMath;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;


/**
 * Calculates the QR-decomposition of a matrix.
 * <p>The QR-decomposition of a matrix A consists of two matrices Q and R
 * that satisfy: A = QR, Q is orthogonal (Q<sup>T</sup>Q = I), and R is
 * upper triangular. If A is m&times;n, Q is m&times;m and R m&times;n.</p>
 * <p>This class compute the decomposition using Householder reflectors.</p>
 * <p>For efficiency purposes, the decomposition in packed form is transposed.
 * This allows inner loop to iterate inside rows, which is much more cache-efficient
 * in Java.</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, with the
 * following changes:</p>
 * <ul>
 *   <li>a {@link #getQT() getQT} method has been added,</li>
 *   <li>the {@code solve} and {@code isFullRank} 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/QRDecomposition.html">MathWorld</a>
 * @see <a href="http://en.wikipedia.org/wiki/QR_decomposition">Wikipedia</a>
 *
 * @since 1.2 (changed to concrete class in 3.0)
 */
public class QRDecomposition {
    /**
     * A packed TRANSPOSED representation of the QR decomposition.
     * <p>The elements BELOW the diagonal are the elements of the UPPER triangular
     * matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors
     * from which an explicit form of Q can be recomputed if desired.</p>
     */
    private double[][] qrt;
    /** The diagonal elements of R. */
    private double[] rDiag;
    /** Cached value of Q. */
    private RealMatrix cachedQ;
    /** Cached value of QT. */
    private RealMatrix cachedQT;
    /** Cached value of R. */
    private RealMatrix cachedR;
    /** Cached value of H. */
    private RealMatrix cachedH;
    /** Singularity threshold. */
    private final double threshold;

    /**
     * Calculates the QR-decomposition of the given matrix.
     * The singularity threshold defaults to zero.
     *
     * @param matrix The matrix to decompose.
     *
     * @see #QRDecomposition(RealMatrix,double)
     */
    public QRDecomposition(RealMatrix matrix) {
        this(matrix, 0d);
    }

    /**
     * Calculates the QR-decomposition of the given matrix.
     *
     * @param matrix The matrix to decompose.
     * @param threshold Singularity threshold.
     * The matrix will be considered singular if the absolute value of
     * any of the diagonal elements of the "R" matrix is smaller than
     * the threshold.
     */
    public QRDecomposition(RealMatrix matrix,
                           double threshold) {
        this.threshold = threshold;

        final int m = matrix.getRowDimension();
        final int n = matrix.getColumnDimension();
        qrt = matrix.transpose().getData();
        rDiag = new double[JdkMath.min(m, n)];
        cachedQ  = null;
        cachedQT = null;
        cachedR  = null;
        cachedH  = null;

        decompose(qrt);
    }

    /** Decompose matrix.
     * @param matrix transposed matrix
     * @since 3.2
     */
    protected void decompose(double[][] matrix) {
        for (int minor = 0; minor < JdkMath.min(matrix.length, matrix[0].length); minor++) {
            performHouseholderReflection(minor, matrix);
        }
    }

    /** Perform Householder reflection for a minor A(minor, minor) of A.
     * @param minor minor index
     * @param matrix transposed matrix
     * @since 3.2
     */
    protected void performHouseholderReflection(int minor, double[][] matrix) {

        final double[] qrtMinor = matrix[minor];

        /*
         * Let x be the first column of the minor, and a^2 = |x|^2.
         * x will be in the positions qr[minor][minor] through qr[m][minor].
         * The first column of the transformed minor will be (a,0,0,..)'
         * The sign of a is chosen to be opposite to the sign of the first
         * component of x. Let's find a:
         */
        double xNormSqr = 0;
        for (int row = minor; row < qrtMinor.length; row++) {
            final double c = qrtMinor[row];
            xNormSqr += c * c;
        }
        final double a = (qrtMinor[minor] > 0) ? -JdkMath.sqrt(xNormSqr) : JdkMath.sqrt(xNormSqr);
        rDiag[minor] = a;

        if (a != 0.0) {

            /*
             * Calculate the normalized reflection vector v and transform
             * the first column. We know the norm of v beforehand: v = x-ae
             * so |v|^2 = <x-ae,x-ae> = <x,x>-2a<x,e>+a^2<e,e> =
             * a^2+a^2-2a<x,e> = 2a*(a - <x,e>).
             * Here <x, e> is now qr[minor][minor].
             * v = x-ae is stored in the column at qr:
             */
            qrtMinor[minor] -= a; // now |v|^2 = -2a*(qr[minor][minor])

            /*
             * Transform the rest of the columns of the minor:
             * They will be transformed by the matrix H = I-2vv'/|v|^2.
             * If x is a column vector of the minor, then
             * Hx = (I-2vv'/|v|^2)x = x-2vv'x/|v|^2 = x - 2<x,v>/|v|^2 v.
             * Therefore the transformation is easily calculated by
             * subtracting the column vector (2<x,v>/|v|^2)v from x.
             *
             * Let 2<x,v>/|v|^2 = alpha. From above we have
             * |v|^2 = -2a*(qr[minor][minor]), so
             * alpha = -<x,v>/(a*qr[minor][minor])
             */
            for (int col = minor+1; col < matrix.length; col++) {
                final double[] qrtCol = matrix[col];
                double alpha = 0;
                for (int row = minor; row < qrtCol.length; row++) {
                    alpha -= qrtCol[row] * qrtMinor[row];
                }
                alpha /= a * qrtMinor[minor];

                // Subtract the column vector alpha*v from x.
                for (int row = minor; row < qrtCol.length; row++) {
                    qrtCol[row] -= alpha * qrtMinor[row];
                }
            }
        }
    }


    /**
     * Returns the matrix R of the decomposition.
     * <p>R is an upper-triangular matrix</p>
     * @return the R matrix
     */
    public RealMatrix getR() {

        if (cachedR == null) {

            // R is supposed to be m x n
            final int n = qrt.length;
            final int m = qrt[0].length;
            double[][] ra = new double[m][n];
            // copy the diagonal from rDiag and the upper triangle of qr
            for (int row = JdkMath.min(m, n) - 1; row >= 0; row--) {
                ra[row][row] = rDiag[row];
                for (int col = row + 1; col < n; col++) {
                    ra[row][col] = qrt[col][row];
                }
            }
            cachedR = MatrixUtils.createRealMatrix(ra);
        }

        // return the cached matrix
        return cachedR;
    }

    /**
     * Returns the matrix Q of the decomposition.
     * <p>Q is an orthogonal matrix</p>
     * @return the Q matrix
     */
    public RealMatrix getQ() {
        if (cachedQ == null) {
            cachedQ = getQT().transpose();
        }
        return cachedQ;
    }

    /**
     * Returns the transpose of the matrix Q of the decomposition.
     * <p>Q is an orthogonal matrix</p>
     * @return the transpose of the Q matrix, Q<sup>T</sup>
     */
    public RealMatrix getQT() {
        if (cachedQT == null) {

            // QT is supposed to be m x m
            final int n = qrt.length;
            final int m = qrt[0].length;
            double[][] qta = new double[m][m];

            /*
             * Q = Q1 Q2 ... Q_m, so Q is formed by first constructing Q_m and then
             * applying the Householder transformations Q_(m-1),Q_(m-2),...,Q1 in
             * succession to the result
             */
            for (int minor = m - 1; minor >= JdkMath.min(m, n); minor--) {
                qta[minor][minor] = 1.0d;
            }

            for (int minor = JdkMath.min(m, n)-1; minor >= 0; minor--){
                final double[] qrtMinor = qrt[minor];
                qta[minor][minor] = 1.0d;
                if (qrtMinor[minor] != 0.0) {
                    for (int col = minor; col < m; col++) {
                        double alpha = 0;
                        for (int row = minor; row < m; row++) {
                            alpha -= qta[col][row] * qrtMinor[row];
                        }
                        alpha /= rDiag[minor] * qrtMinor[minor];

                        for (int row = minor; row < m; row++) {
                            qta[col][row] += -alpha * qrtMinor[row];
                        }
                    }
                }
            }
            cachedQT = MatrixUtils.createRealMatrix(qta);
        }

        // return the cached matrix
        return cachedQT;
    }

    /**
     * Returns the Householder reflector vectors.
     * <p>H is a lower trapezoidal matrix whose columns represent
     * each successive Householder reflector vector. This matrix is used
     * to compute Q.</p>
     * @return a matrix containing the Householder reflector vectors
     */
    public RealMatrix getH() {
        if (cachedH == null) {

            final int n = qrt.length;
            final int m = qrt[0].length;
            double[][] ha = new double[m][n];
            for (int i = 0; i < m; ++i) {
                for (int j = 0; j < JdkMath.min(i + 1, n); ++j) {
                    ha[i][j] = qrt[j][i] / -rDiag[j];
                }
            }
            cachedH = MatrixUtils.createRealMatrix(ha);
        }

        // return the cached matrix
        return cachedH;
    }

    /**
     * Get a solver for finding the A &times; X = B solution in least square sense.
     * <p>
     * Least Square sense means a solver can be computed for an overdetermined system,
     * (i.e. a system with more equations than unknowns, which corresponds to a tall A
     * matrix with more rows than columns). In any case, if the matrix is singular
     * within the tolerance set at {@link QRDecomposition#QRDecomposition(RealMatrix,
     * double) construction}, an error will be triggered when
     * the {@link DecompositionSolver#solve(RealVector) solve} method will be called.
     * </p>
     * @return a solver
     */
    public DecompositionSolver getSolver() {
        return new Solver(qrt, rDiag, threshold);
    }

    /** Specialized solver. */
    private static final class Solver implements DecompositionSolver {
        /**
         * A packed TRANSPOSED representation of the QR decomposition.
         * <p>The elements BELOW the diagonal are the elements of the UPPER triangular
         * matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors
         * from which an explicit form of Q can be recomputed if desired.</p>
         */
        private final double[][] qrt;
        /** The diagonal elements of R. */
        private final double[] rDiag;
        /** Singularity threshold. */
        private final double threshold;

        /**
         * Build a solver from decomposed matrix.
         *
         * @param qrt Packed TRANSPOSED representation of the QR decomposition.
         * @param rDiag Diagonal elements of R.
         * @param threshold Singularity threshold.
         */
        private Solver(final double[][] qrt,
                       final double[] rDiag,
                       final double threshold) {
            this.qrt   = qrt;
            this.rDiag = rDiag;
            this.threshold = threshold;
        }

        /** {@inheritDoc} */
        @Override
        public boolean isNonSingular() {
            return !checkSingular(rDiag, threshold, false);
        }

        /** {@inheritDoc} */
        @Override
        public RealVector solve(RealVector b) {
            final int n = qrt.length;
            final int m = qrt[0].length;
            if (b.getDimension() != m) {
                throw new DimensionMismatchException(b.getDimension(), m);
            }
            checkSingular(rDiag, threshold, true);

            final double[] x = new double[n];
            final double[] y = b.toArray();

            // apply Householder transforms to solve Q.y = b
            for (int minor = 0; minor < JdkMath.min(m, n); minor++) {

                final double[] qrtMinor = qrt[minor];
                double dotProduct = 0;
                for (int row = minor; row < m; row++) {
                    dotProduct += y[row] * qrtMinor[row];
                }
                dotProduct /= rDiag[minor] * qrtMinor[minor];

                for (int row = minor; row < m; row++) {
                    y[row] += dotProduct * qrtMinor[row];
                }
            }

            // solve triangular system R.x = y
            for (int row = rDiag.length - 1; row >= 0; --row) {
                y[row] /= rDiag[row];
                final double yRow = y[row];
                final double[] qrtRow = qrt[row];
                x[row] = yRow;
                for (int i = 0; i < row; i++) {
                    y[i] -= yRow * qrtRow[i];
                }
            }

            return new ArrayRealVector(x, false);
        }

        /** {@inheritDoc} */
        @Override
        public RealMatrix solve(RealMatrix b) {
            final int n = qrt.length;
            final int m = qrt[0].length;
            if (b.getRowDimension() != m) {
                throw new DimensionMismatchException(b.getRowDimension(), m);
            }
            checkSingular(rDiag, threshold, true);

            final int columns        = b.getColumnDimension();
            final int blockSize      = BlockRealMatrix.BLOCK_SIZE;
            final int cBlocks        = (columns + blockSize - 1) / blockSize;
            final double[][] xBlocks = BlockRealMatrix.createBlocksLayout(n, columns);
            final double[][] y       = new double[b.getRowDimension()][blockSize];
            final double[]   alpha   = new double[blockSize];

            for (int kBlock = 0; kBlock < cBlocks; ++kBlock) {
                final int kStart = kBlock * blockSize;
                final int kEnd   = JdkMath.min(kStart + blockSize, columns);
                final int kWidth = kEnd - kStart;

                // get the right hand side vector
                b.copySubMatrix(0, m - 1, kStart, kEnd - 1, y);

                // apply Householder transforms to solve Q.y = b
                for (int minor = 0; minor < JdkMath.min(m, n); minor++) {
                    final double[] qrtMinor = qrt[minor];
                    final double factor     = 1.0 / (rDiag[minor] * qrtMinor[minor]);

                    Arrays.fill(alpha, 0, kWidth, 0.0);
                    for (int row = minor; row < m; ++row) {
                        final double   d    = qrtMinor[row];
                        final double[] yRow = y[row];
                        for (int k = 0; k < kWidth; ++k) {
                            alpha[k] += d * yRow[k];
                        }
                    }
                    for (int k = 0; k < kWidth; ++k) {
                        alpha[k] *= factor;
                    }

                    for (int row = minor; row < m; ++row) {
                        final double   d    = qrtMinor[row];
                        final double[] yRow = y[row];
                        for (int k = 0; k < kWidth; ++k) {
                            yRow[k] += alpha[k] * d;
                        }
                    }
                }

                // solve triangular system R.x = y
                for (int j = rDiag.length - 1; j >= 0; --j) {
                    final int      jBlock = j / blockSize;
                    final int      jStart = jBlock * blockSize;
                    final double   factor = 1.0 / rDiag[j];
                    final double[] yJ     = y[j];
                    final double[] xBlock = xBlocks[jBlock * cBlocks + kBlock];
                    int index = (j - jStart) * kWidth;
                    for (int k = 0; k < kWidth; ++k) {
                        yJ[k]          *= factor;
                        xBlock[index++] = yJ[k];
                    }

                    final double[] qrtJ = qrt[j];
                    for (int i = 0; i < j; ++i) {
                        final double rIJ  = qrtJ[i];
                        final double[] yI = y[i];
                        for (int k = 0; k < kWidth; ++k) {
                            yI[k] -= yJ[k] * rIJ;
                        }
                    }
                }
            }

            return new BlockRealMatrix(n, columns, xBlocks, false);
        }

        /**
         * {@inheritDoc}
         * @throws SingularMatrixException if the decomposed matrix is singular.
         */
        @Override
        public RealMatrix getInverse() {
            return solve(MatrixUtils.createRealIdentityMatrix(qrt[0].length));
        }

        /**
         * Check singularity.
         *
         * @param diag Diagonal elements of the R matrix.
         * @param min Singularity threshold.
         * @param raise Whether to raise a {@link SingularMatrixException}
         * if any element of the diagonal fails the check.
         * @return {@code true} if any element of the diagonal is smaller
         * or equal to {@code min}.
         * @throws SingularMatrixException if the matrix is singular and
         * {@code raise} is {@code true}.
         */
        private static boolean checkSingular(double[] diag,
                                             double min,
                                             boolean raise) {
            final int len = diag.length;
            for (int i = 0; i < len; i++) {
                final double d = diag[i];
                if (JdkMath.abs(d) <= min) {
                    if (raise) {
                        final SingularMatrixException e = new SingularMatrixException();
                        e.getContext().addMessage(LocalizedFormats.NUMBER_TOO_SMALL, d, min);
                        e.getContext().addMessage(LocalizedFormats.INDEX, i);
                        throw e;
                    } else {
                        return true;
                    }
                }
            }
            return false;
        }
    }
}