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 */ 017 018 package org.apache.commons.math.linear; 019 020 021 022 /** 023 * An interface to classes that implement an algorithm to calculate the 024 * Singular Value Decomposition of a real matrix. 025 * <p>The Singular Value Decomposition of matrix A is a set of three matrices: 026 * U, Σ and V such that A = U × Σ × V<sup>T</sup>. 027 * Let A be an m × n matrix, then U is an m × m orthogonal matrix, 028 * Σ is a m × n diagonal matrix with positive diagonal elements, 029 * and V is an n × n orthogonal matrix.</p> 030 * <p>This interface is similar to the class with similar name from the now defunct 031 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the 032 * following changes:</p> 033 * <ul> 034 * <li>the <code>norm2</code> method which has been renamed as {@link #getNorm() 035 * getNorm},</li> 036 * <li>the <code>cond</code> method which has been renamed as {@link 037 * #getConditionNumber() getConditionNumber},</li> 038 * <li>the <code>rank</code> method which has been renamed as {@link #getRank() 039 * getRank},</li> 040 * <li>a {@link #getUT() getUT} method has been added,</li> 041 * <li>a {@link #getVT() getVT} method has been added,</li> 042 * <li>a {@link #getSolver() getSolver} method has been added,</li> 043 * <li>a {@link #getCovariance(double) getCovariance} method has been added.</li> 044 * </ul> 045 * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a> 046 * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a> 047 * @version $Revision: 799857 $ $Date: 2009-08-01 09:07:12 -0400 (Sat, 01 Aug 2009) $ 048 * @since 2.0 049 */ 050 public interface SingularValueDecomposition { 051 052 /** 053 * Returns the matrix U of the decomposition. 054 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 055 * @return the U matrix 056 * @see #getUT() 057 */ 058 RealMatrix getU(); 059 060 /** 061 * Returns the transpose of the matrix U of the decomposition. 062 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 063 * @return the U matrix (or null if decomposed matrix is singular) 064 * @see #getU() 065 */ 066 RealMatrix getUT(); 067 068 /** 069 * Returns the diagonal matrix Σ of the decomposition. 070 * <p>Σ is a diagonal matrix. The singular values are provided in 071 * non-increasing order, for compatibility with Jama.</p> 072 * @return the Σ matrix 073 */ 074 RealMatrix getS(); 075 076 /** 077 * Returns the diagonal elements of the matrix Σ of the decomposition. 078 * <p>The singular values are provided in non-increasing order, for 079 * compatibility with Jama.</p> 080 * @return the diagonal elements of the Σ matrix 081 */ 082 double[] getSingularValues(); 083 084 /** 085 * Returns the matrix V of the decomposition. 086 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 087 * @return the V matrix (or null if decomposed matrix is singular) 088 * @see #getVT() 089 */ 090 RealMatrix getV(); 091 092 /** 093 * Returns the transpose of the matrix V of the decomposition. 094 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 095 * @return the V matrix (or null if decomposed matrix is singular) 096 * @see #getV() 097 */ 098 RealMatrix getVT(); 099 100 /** 101 * Returns the n × n covariance matrix. 102 * <p>The covariance matrix is V × J × V<sup>T</sup> 103 * where J is the diagonal matrix of the inverse of the squares of 104 * the singular values.</p> 105 * @param minSingularValue value below which singular values are ignored 106 * (a 0 or negative value implies all singular value will be used) 107 * @return covariance matrix 108 * @exception IllegalArgumentException if minSingularValue is larger than 109 * the largest singular value, meaning all singular values are ignored 110 */ 111 RealMatrix getCovariance(double minSingularValue) throws IllegalArgumentException; 112 113 /** 114 * Returns the L<sub>2</sub> norm of the matrix. 115 * <p>The L<sub>2</sub> norm is max(|A × u|<sub>2</sub> / 116 * |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm 117 * (i.e. the traditional euclidian norm).</p> 118 * @return norm 119 */ 120 double getNorm(); 121 122 /** 123 * Return the condition number of the matrix. 124 * @return condition number of the matrix 125 */ 126 double getConditionNumber(); 127 128 /** 129 * Return the effective numerical matrix rank. 130 * <p>The effective numerical rank is the number of non-negligible 131 * singular values. The threshold used to identify non-negligible 132 * terms is max(m,n) × ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>) 133 * is the least significant bit of the largest singular value.</p> 134 * @return effective numerical matrix rank 135 */ 136 int getRank(); 137 138 /** 139 * Get a solver for finding the A × X = B solution in least square sense. 140 * @return a solver 141 */ 142 DecompositionSolver getSolver(); 143 144 }