1 /* 2 * Licensed to the Apache Software Foundation (ASF) under one or more 3 * contributor license agreements. See the NOTICE file distributed with 4 * this work for additional information regarding copyright ownership. 5 * The ASF licenses this file to You under the Apache License, Version 2.0 6 * (the "License"); you may not use this file except in compliance with 7 * the License. You may obtain a copy of the License at 8 * 9 * http://www.apache.org/licenses/LICENSE-2.0 10 * 11 * Unless required by applicable law or agreed to in writing, software 12 * distributed under the License is distributed on an "AS IS" BASIS, 13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14 * See the License for the specific language governing permissions and 15 * limitations under the License. 16 */ 17 18 package org.apache.commons.math.linear; 19 20 21 22 /** 23 * An interface to classes that implement an algorithm to calculate the 24 * Singular Value Decomposition of a real matrix. 25 * <p> 26 * The Singular Value Decomposition of matrix A is a set of three matrices: U, 27 * Σ and V such that A = U × Σ × V<sup>T</sup>. Let A be 28 * a m × n matrix, then U is a m × p orthogonal matrix, Σ is a 29 * p × p diagonal matrix with positive or null elements, V is a p × 30 * n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where 31 * p=min(m,n). 32 * </p> 33 * <p>This interface is similar to the class with similar name from the 34 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the 35 * following changes:</p> 36 * <ul> 37 * <li>the <code>norm2</code> method which has been renamed as {@link #getNorm() 38 * getNorm},</li> 39 * <li>the <code>cond</code> method which has been renamed as {@link 40 * #getConditionNumber() getConditionNumber},</li> 41 * <li>the <code>rank</code> method which has been renamed as {@link #getRank() 42 * getRank},</li> 43 * <li>a {@link #getUT() getUT} method has been added,</li> 44 * <li>a {@link #getVT() getVT} method has been added,</li> 45 * <li>a {@link #getSolver() getSolver} method has been added,</li> 46 * <li>a {@link #getCovariance(double) getCovariance} method has been added.</li> 47 * </ul> 48 * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a> 49 * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a> 50 * @version $Revision: 928081 $ $Date: 2010-03-26 23:36:38 +0100 (ven. 26 mars 2010) $ 51 * @since 2.0 52 */ 53 public interface SingularValueDecomposition { 54 55 /** 56 * Returns the matrix U of the decomposition. 57 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 58 * @return the U matrix 59 * @see #getUT() 60 */ 61 RealMatrix getU(); 62 63 /** 64 * Returns the transpose of the matrix U of the decomposition. 65 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 66 * @return the U matrix (or null if decomposed matrix is singular) 67 * @see #getU() 68 */ 69 RealMatrix getUT(); 70 71 /** 72 * Returns the diagonal matrix Σ of the decomposition. 73 * <p>Σ is a diagonal matrix. The singular values are provided in 74 * non-increasing order, for compatibility with Jama.</p> 75 * @return the Σ matrix 76 */ 77 RealMatrix getS(); 78 79 /** 80 * Returns the diagonal elements of the matrix Σ of the decomposition. 81 * <p>The singular values are provided in non-increasing order, for 82 * compatibility with Jama.</p> 83 * @return the diagonal elements of the Σ matrix 84 */ 85 double[] getSingularValues(); 86 87 /** 88 * Returns the matrix V of the decomposition. 89 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 90 * @return the V matrix (or null if decomposed matrix is singular) 91 * @see #getVT() 92 */ 93 RealMatrix getV(); 94 95 /** 96 * Returns the transpose of the matrix V of the decomposition. 97 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 98 * @return the V matrix (or null if decomposed matrix is singular) 99 * @see #getV() 100 */ 101 RealMatrix getVT(); 102 103 /** 104 * Returns the n × n covariance matrix. 105 * <p>The covariance matrix is V × J × V<sup>T</sup> 106 * where J is the diagonal matrix of the inverse of the squares of 107 * the singular values.</p> 108 * @param minSingularValue value below which singular values are ignored 109 * (a 0 or negative value implies all singular value will be used) 110 * @return covariance matrix 111 * @exception IllegalArgumentException if minSingularValue is larger than 112 * the largest singular value, meaning all singular values are ignored 113 */ 114 RealMatrix getCovariance(double minSingularValue) throws IllegalArgumentException; 115 116 /** 117 * Returns the L<sub>2</sub> norm of the matrix. 118 * <p>The L<sub>2</sub> norm is max(|A × u|<sub>2</sub> / 119 * |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm 120 * (i.e. the traditional euclidian norm).</p> 121 * @return norm 122 */ 123 double getNorm(); 124 125 /** 126 * Return the condition number of the matrix. 127 * @return condition number of the matrix 128 */ 129 double getConditionNumber(); 130 131 /** 132 * Return the effective numerical matrix rank. 133 * <p>The effective numerical rank is the number of non-negligible 134 * singular values. The threshold used to identify non-negligible 135 * terms is max(m,n) × ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>) 136 * is the least significant bit of the largest singular value.</p> 137 * @return effective numerical matrix rank 138 */ 139 int getRank(); 140 141 /** 142 * Get a solver for finding the A × X = B solution in least square sense. 143 * @return a solver 144 */ 145 DecompositionSolver getSolver(); 146 147 } 148
