001/* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 018package org.apache.commons.math3.linear; 019 020import org.apache.commons.math3.util.FastMath; 021 022/** 023 * Calculates the rectangular Cholesky decomposition of a matrix. 024 * <p>The rectangular Cholesky decomposition of a real symmetric positive 025 * semidefinite matrix A consists of a rectangular matrix B with the same 026 * number of rows such that: A is almost equal to BB<sup>T</sup>, depending 027 * on a user-defined tolerance. In a sense, this is the square root of A.</p> 028 * <p>The difference with respect to the regular {@link CholeskyDecomposition} 029 * is that rows/columns may be permuted (hence the rectangular shape instead 030 * of the traditional triangular shape) and there is a threshold to ignore 031 * small diagonal elements. This is used for example to generate {@link 032 * org.apache.commons.math3.random.CorrelatedRandomVectorGenerator correlated 033 * random n-dimensions vectors} in a p-dimension subspace (p < n). 034 * In other words, it allows generating random vectors from a covariance 035 * matrix that is only positive semidefinite, and not positive definite.</p> 036 * <p>Rectangular Cholesky decomposition is <em>not</em> suited for solving 037 * linear systems, so it does not provide any {@link DecompositionSolver 038 * decomposition solver}.</p> 039 * 040 * @see <a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">MathWorld</a> 041 * @see <a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Wikipedia</a> 042 * @since 2.0 (changed to concrete class in 3.0) 043 */ 044public class RectangularCholeskyDecomposition { 045 046 /** Permutated Cholesky root of the symmetric positive semidefinite matrix. */ 047 private final RealMatrix root; 048 049 /** Rank of the symmetric positive semidefinite matrix. */ 050 private int rank; 051 052 /** 053 * Decompose a symmetric positive semidefinite matrix. 054 * <p> 055 * <b>Note:</b> this constructor follows the linpack method to detect dependent 056 * columns by proceeding with the Cholesky algorithm until a nonpositive diagonal 057 * element is encountered. 058 * 059 * @see <a href="http://eprints.ma.man.ac.uk/1193/01/covered/MIMS_ep2008_56.pdf"> 060 * Analysis of the Cholesky Decomposition of a Semi-definite Matrix</a> 061 * 062 * @param matrix Symmetric positive semidefinite matrix. 063 * @exception NonPositiveDefiniteMatrixException if the matrix is not 064 * positive semidefinite. 065 * @since 3.1 066 */ 067 public RectangularCholeskyDecomposition(RealMatrix matrix) 068 throws NonPositiveDefiniteMatrixException { 069 this(matrix, 0); 070 } 071 072 /** 073 * Decompose a symmetric positive semidefinite matrix. 074 * 075 * @param matrix Symmetric positive semidefinite matrix. 076 * @param small Diagonal elements threshold under which columns are 077 * considered to be dependent on previous ones and are discarded. 078 * @exception NonPositiveDefiniteMatrixException if the matrix is not 079 * positive semidefinite. 080 */ 081 public RectangularCholeskyDecomposition(RealMatrix matrix, double small) 082 throws NonPositiveDefiniteMatrixException { 083 084 final int order = matrix.getRowDimension(); 085 final double[][] c = matrix.getData(); 086 final double[][] b = new double[order][order]; 087 088 int[] index = new int[order]; 089 for (int i = 0; i < order; ++i) { 090 index[i] = i; 091 } 092 093 int r = 0; 094 for (boolean loop = true; loop;) { 095 096 // find maximal diagonal element 097 int swapR = r; 098 for (int i = r + 1; i < order; ++i) { 099 int ii = index[i]; 100 int isr = index[swapR]; 101 if (c[ii][ii] > c[isr][isr]) { 102 swapR = i; 103 } 104 } 105 106 107 // swap elements 108 if (swapR != r) { 109 final int tmpIndex = index[r]; 110 index[r] = index[swapR]; 111 index[swapR] = tmpIndex; 112 final double[] tmpRow = b[r]; 113 b[r] = b[swapR]; 114 b[swapR] = tmpRow; 115 } 116 117 // check diagonal element 118 int ir = index[r]; 119 if (c[ir][ir] <= small) { 120 121 if (r == 0) { 122 throw new NonPositiveDefiniteMatrixException(c[ir][ir], ir, small); 123 } 124 125 // check remaining diagonal elements 126 for (int i = r; i < order; ++i) { 127 if (c[index[i]][index[i]] < -small) { 128 // there is at least one sufficiently negative diagonal element, 129 // the symmetric positive semidefinite matrix is wrong 130 throw new NonPositiveDefiniteMatrixException(c[index[i]][index[i]], i, small); 131 } 132 } 133 134 // all remaining diagonal elements are close to zero, we consider we have 135 // found the rank of the symmetric positive semidefinite matrix 136 loop = false; 137 138 } else { 139 140 // transform the matrix 141 final double sqrt = FastMath.sqrt(c[ir][ir]); 142 b[r][r] = sqrt; 143 final double inverse = 1 / sqrt; 144 final double inverse2 = 1 / c[ir][ir]; 145 for (int i = r + 1; i < order; ++i) { 146 final int ii = index[i]; 147 final double e = inverse * c[ii][ir]; 148 b[i][r] = e; 149 c[ii][ii] -= c[ii][ir] * c[ii][ir] * inverse2; 150 for (int j = r + 1; j < i; ++j) { 151 final int ij = index[j]; 152 final double f = c[ii][ij] - e * b[j][r]; 153 c[ii][ij] = f; 154 c[ij][ii] = f; 155 } 156 } 157 158 // prepare next iteration 159 loop = ++r < order; 160 } 161 } 162 163 // build the root matrix 164 rank = r; 165 root = MatrixUtils.createRealMatrix(order, r); 166 for (int i = 0; i < order; ++i) { 167 for (int j = 0; j < r; ++j) { 168 root.setEntry(index[i], j, b[i][j]); 169 } 170 } 171 172 } 173 174 /** Get the root of the covariance matrix. 175 * The root is the rectangular matrix <code>B</code> such that 176 * the covariance matrix is equal to <code>B.B<sup>T</sup></code> 177 * @return root of the square matrix 178 * @see #getRank() 179 */ 180 public RealMatrix getRootMatrix() { 181 return root; 182 } 183 184 /** Get the rank of the symmetric positive semidefinite matrix. 185 * The r is the number of independent rows in the symmetric positive semidefinite 186 * matrix, it is also the number of columns of the rectangular 187 * matrix of the decomposition. 188 * @return r of the square matrix. 189 * @see #getRootMatrix() 190 */ 191 public int getRank() { 192 return rank; 193 } 194 195}