Java tutorial
/* * Licensed to the Apache Software Foundation (ASF) under one or more * 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, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ package org.apache.commons.math3.linear; import java.util.Arrays; import org.apache.commons.math3.exception.DimensionMismatchException; import org.apache.commons.math3.util.FastMath; /** * 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×n, Q is m×m and R m×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> * * @version $Id: QRDecomposition.java 1416643 2012-12-03 19:37:14Z tn $ * @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. */ 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[FastMath.min(m, n)]; cachedQ = null; cachedQT = null; cachedR = null; cachedH = null; /* * The QR decomposition of a matrix A is calculated using Householder * reflectors by repeating the following operations to each minor * A(minor,minor) of A: */ for (int minor = 0; minor < FastMath.min(m, n); minor++) { final double[] qrtMinor = qrt[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 < m; row++) { final double c = qrtMinor[row]; xNormSqr += c * c; } final double a = (qrtMinor[minor] > 0) ? -FastMath.sqrt(xNormSqr) : FastMath.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 < n; col++) { final double[] qrtCol = qrt[col]; double alpha = 0; for (int row = minor; row < m; row++) { alpha -= qrtCol[row] * qrtMinor[row]; } alpha /= a * qrtMinor[minor]; // Subtract the column vector alpha*v from x. for (int row = minor; row < m; 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 = FastMath.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 >= FastMath.min(m, n); minor--) { qta[minor][minor] = 1.0d; } for (int minor = FastMath.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 < FastMath.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 × X = B solution in least square sense. * @return a solver */ public DecompositionSolver getSolver() { return new Solver(qrt, rDiag, threshold); } /** Specialized solver. */ private static 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} */ public boolean isNonSingular() { for (double diag : rDiag) { if (FastMath.abs(diag) <= threshold) { return false; } } return true; } /** {@inheritDoc} */ 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); } if (!isNonSingular()) { throw new SingularMatrixException(); } final double[] x = new double[n]; final double[] y = b.toArray(); // apply Householder transforms to solve Q.y = b for (int minor = 0; minor < FastMath.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} */ 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); } if (!isNonSingular()) { throw new SingularMatrixException(); } 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 = FastMath.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 < FastMath.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} */ public RealMatrix getInverse() { return solve(MatrixUtils.createRealIdentityMatrix(rDiag.length)); } } }