| /* |
| * 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.math4.legacy.linear; |
| |
| import org.apache.commons.math4.legacy.exception.DimensionMismatchException; |
| import org.apache.commons.math4.core.jdkmath.JdkMath; |
| |
| |
| /** |
| * Calculates the Cholesky decomposition of a matrix. |
| * <p>The Cholesky decomposition of a real symmetric positive-definite |
| * matrix A consists of a lower triangular matrix L with same size such |
| * that: A = LL<sup>T</sup>. In a sense, this is the square root of A.</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 #getLT() getLT} method has been added,</li> |
| * <li>the {@code isspd} method has been removed, since the constructor of |
| * this class throws a {@link NonPositiveDefiniteMatrixException} when a |
| * matrix cannot be decomposed,</li> |
| * <li>a {@link #getDeterminant() getDeterminant} method has been added,</li> |
| * <li>the {@code solve} method has been replaced by a {@link #getSolver() |
| * getSolver} method and the equivalent method provided by the returned |
| * {@link DecompositionSolver}.</li> |
| * </ul> |
| * |
| * @see <a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">MathWorld</a> |
| * @see <a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Wikipedia</a> |
| * @since 2.0 (changed to concrete class in 3.0) |
| */ |
| public class CholeskyDecomposition { |
| /** |
| * Default threshold above which off-diagonal elements are considered too different |
| * and matrix not symmetric. |
| */ |
| public static final double DEFAULT_RELATIVE_SYMMETRY_THRESHOLD = 1.0e-15; |
| /** |
| * Default threshold below which diagonal elements are considered null |
| * and matrix not positive definite. |
| */ |
| public static final double DEFAULT_ABSOLUTE_POSITIVITY_THRESHOLD = 1.0e-10; |
| /** Row-oriented storage for L<sup>T</sup> matrix data. */ |
| private final double[][] lTData; |
| /** Cached value of L. */ |
| private RealMatrix cachedL; |
| /** Cached value of LT. */ |
| private RealMatrix cachedLT; |
| |
| /** |
| * Calculates the Cholesky decomposition of the given matrix. |
| * <p> |
| * Calling this constructor is equivalent to call {@link |
| * #CholeskyDecomposition(RealMatrix, double, double)} with the |
| * thresholds set to the default values {@link |
| * #DEFAULT_RELATIVE_SYMMETRY_THRESHOLD} and {@link |
| * #DEFAULT_ABSOLUTE_POSITIVITY_THRESHOLD} |
| * </p> |
| * @param matrix the matrix to decompose |
| * @throws NonSquareMatrixException if the matrix is not square. |
| * @throws NonSymmetricMatrixException if the matrix is not symmetric. |
| * @throws NonPositiveDefiniteMatrixException if the matrix is not |
| * strictly positive definite. |
| * @see #CholeskyDecomposition(RealMatrix, double, double) |
| * @see #DEFAULT_RELATIVE_SYMMETRY_THRESHOLD |
| * @see #DEFAULT_ABSOLUTE_POSITIVITY_THRESHOLD |
| */ |
| public CholeskyDecomposition(final RealMatrix matrix) { |
| this(matrix, DEFAULT_RELATIVE_SYMMETRY_THRESHOLD, |
| DEFAULT_ABSOLUTE_POSITIVITY_THRESHOLD); |
| } |
| |
| /** |
| * Calculates the Cholesky decomposition of the given matrix. |
| * @param matrix the matrix to decompose |
| * @param relativeSymmetryThreshold threshold above which off-diagonal |
| * elements are considered too different and matrix not symmetric |
| * @param absolutePositivityThreshold threshold below which diagonal |
| * elements are considered null and matrix not positive definite |
| * @throws NonSquareMatrixException if the matrix is not square. |
| * @throws NonSymmetricMatrixException if the matrix is not symmetric. |
| * @throws NonPositiveDefiniteMatrixException if the matrix is not |
| * strictly positive definite. |
| * @see #CholeskyDecomposition(RealMatrix) |
| * @see #DEFAULT_RELATIVE_SYMMETRY_THRESHOLD |
| * @see #DEFAULT_ABSOLUTE_POSITIVITY_THRESHOLD |
| */ |
| public CholeskyDecomposition(final RealMatrix matrix, |
| final double relativeSymmetryThreshold, |
| final double absolutePositivityThreshold) { |
| if (!matrix.isSquare()) { |
| throw new NonSquareMatrixException(matrix.getRowDimension(), |
| matrix.getColumnDimension()); |
| } |
| |
| final int order = matrix.getRowDimension(); |
| lTData = matrix.getData(); |
| cachedL = null; |
| cachedLT = null; |
| |
| // check the matrix before transformation |
| for (int i = 0; i < order; ++i) { |
| final double[] lI = lTData[i]; |
| |
| // check off-diagonal elements (and reset them to 0) |
| for (int j = i + 1; j < order; ++j) { |
| final double[] lJ = lTData[j]; |
| final double lIJ = lI[j]; |
| final double lJI = lJ[i]; |
| final double maxDelta = |
| relativeSymmetryThreshold * JdkMath.max(JdkMath.abs(lIJ), JdkMath.abs(lJI)); |
| if (JdkMath.abs(lIJ - lJI) > maxDelta) { |
| throw new NonSymmetricMatrixException(i, j, relativeSymmetryThreshold); |
| } |
| lJ[i] = 0; |
| } |
| } |
| |
| // transform the matrix |
| for (int i = 0; i < order; ++i) { |
| |
| final double[] ltI = lTData[i]; |
| |
| // check diagonal element |
| if (ltI[i] <= absolutePositivityThreshold) { |
| throw new NonPositiveDefiniteMatrixException(ltI[i], i, absolutePositivityThreshold); |
| } |
| |
| ltI[i] = JdkMath.sqrt(ltI[i]); |
| final double inverse = 1.0 / ltI[i]; |
| |
| for (int q = order - 1; q > i; --q) { |
| ltI[q] *= inverse; |
| final double[] ltQ = lTData[q]; |
| for (int p = q; p < order; ++p) { |
| ltQ[p] -= ltI[q] * ltI[p]; |
| } |
| } |
| } |
| } |
| |
| /** |
| * Returns the matrix L of the decomposition. |
| * <p>L is an lower-triangular matrix</p> |
| * @return the L matrix |
| */ |
| public RealMatrix getL() { |
| if (cachedL == null) { |
| cachedL = getLT().transpose(); |
| } |
| return cachedL; |
| } |
| |
| /** |
| * Returns the transpose of the matrix L of the decomposition. |
| * <p>L<sup>T</sup> is an upper-triangular matrix</p> |
| * @return the transpose of the matrix L of the decomposition |
| */ |
| public RealMatrix getLT() { |
| |
| if (cachedLT == null) { |
| cachedLT = MatrixUtils.createRealMatrix(lTData); |
| } |
| |
| // return the cached matrix |
| return cachedLT; |
| } |
| |
| /** |
| * Return the determinant of the matrix. |
| * @return determinant of the matrix |
| */ |
| public double getDeterminant() { |
| double determinant = 1.0; |
| for (int i = 0; i < lTData.length; ++i) { |
| double lTii = lTData[i][i]; |
| determinant *= lTii * lTii; |
| } |
| return determinant; |
| } |
| |
| /** |
| * Get a solver for finding the A × X = B solution in least square sense. |
| * @return a solver |
| */ |
| public DecompositionSolver getSolver() { |
| return new Solver(lTData); |
| } |
| |
| /** Specialized solver. */ |
| private static final class Solver implements DecompositionSolver { |
| /** Row-oriented storage for L<sup>T</sup> matrix data. */ |
| private final double[][] lTData; |
| |
| /** |
| * Build a solver from decomposed matrix. |
| * @param lTData row-oriented storage for L<sup>T</sup> matrix data |
| */ |
| private Solver(final double[][] lTData) { |
| this.lTData = lTData; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public boolean isNonSingular() { |
| // if we get this far, the matrix was positive definite, hence non-singular |
| return true; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public RealVector solve(final RealVector b) { |
| final int m = lTData.length; |
| if (b.getDimension() != m) { |
| throw new DimensionMismatchException(b.getDimension(), m); |
| } |
| |
| final double[] x = b.toArray(); |
| |
| // Solve LY = b |
| for (int j = 0; j < m; j++) { |
| final double[] lJ = lTData[j]; |
| x[j] /= lJ[j]; |
| final double xJ = x[j]; |
| for (int i = j + 1; i < m; i++) { |
| x[i] -= xJ * lJ[i]; |
| } |
| } |
| |
| // Solve LTX = Y |
| for (int j = m - 1; j >= 0; j--) { |
| x[j] /= lTData[j][j]; |
| final double xJ = x[j]; |
| for (int i = 0; i < j; i++) { |
| x[i] -= xJ * lTData[i][j]; |
| } |
| } |
| |
| return new ArrayRealVector(x, false); |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public RealMatrix solve(RealMatrix b) { |
| final int m = lTData.length; |
| if (b.getRowDimension() != m) { |
| throw new DimensionMismatchException(b.getRowDimension(), m); |
| } |
| |
| final int nColB = b.getColumnDimension(); |
| final double[][] x = b.getData(); |
| |
| // Solve LY = b |
| for (int j = 0; j < m; j++) { |
| final double[] lJ = lTData[j]; |
| final double lJJ = lJ[j]; |
| final double[] xJ = x[j]; |
| for (int k = 0; k < nColB; ++k) { |
| xJ[k] /= lJJ; |
| } |
| for (int i = j + 1; i < m; i++) { |
| final double[] xI = x[i]; |
| final double lJI = lJ[i]; |
| for (int k = 0; k < nColB; ++k) { |
| xI[k] -= xJ[k] * lJI; |
| } |
| } |
| } |
| |
| // Solve LTX = Y |
| for (int j = m - 1; j >= 0; j--) { |
| final double lJJ = lTData[j][j]; |
| final double[] xJ = x[j]; |
| for (int k = 0; k < nColB; ++k) { |
| xJ[k] /= lJJ; |
| } |
| for (int i = 0; i < j; i++) { |
| final double[] xI = x[i]; |
| final double lIJ = lTData[i][j]; |
| for (int k = 0; k < nColB; ++k) { |
| xI[k] -= xJ[k] * lIJ; |
| } |
| } |
| } |
| |
| return new Array2DRowRealMatrix(x); |
| } |
| |
| /** |
| * Get the inverse of the decomposed matrix. |
| * |
| * @return the inverse matrix. |
| */ |
| @Override |
| public RealMatrix getInverse() { |
| return solve(MatrixUtils.createRealIdentityMatrix(lTData.length)); |
| } |
| } |
| } |