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>
026 * The Singular Value Decomposition of matrix A is a set of three matrices: U,
027 * Σ and V such that A = U × Σ × V<sup>T</sup>. Let A be
028 * a m × n matrix, then U is a m × p orthogonal matrix, Σ is a
029 * p × p diagonal matrix with positive or null elements, V is a p ×
030 * n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
031 * p=min(m,n).
032 * </p>
033 * <p>This interface is similar to the class with similar name from the
034 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
035 * following changes:</p>
036 * <ul>
037 * <li>the <code>norm2</code> method which has been renamed as {@link #getNorm()
038 * getNorm},</li>
039 * <li>the <code>cond</code> method which has been renamed as {@link
040 * #getConditionNumber() getConditionNumber},</li>
041 * <li>the <code>rank</code> method which has been renamed as {@link #getRank()
042 * getRank},</li>
043 * <li>a {@link #getUT() getUT} method has been added,</li>
044 * <li>a {@link #getVT() getVT} method has been added,</li>
045 * <li>a {@link #getSolver() getSolver} method has been added,</li>
046 * <li>a {@link #getCovariance(double) getCovariance} method has been added.</li>
047 * </ul>
048 * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>
049 * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a>
050 * @version $Revision: 928081 $ $Date: 2010-03-26 18:36:38 -0400 (Fri, 26 Mar 2010) $
051 * @since 2.0
052 */
053 public interface SingularValueDecomposition {
054
055 /**
056 * Returns the matrix U of the decomposition.
057 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
058 * @return the U matrix
059 * @see #getUT()
060 */
061 RealMatrix getU();
062
063 /**
064 * Returns the transpose of the matrix U of the decomposition.
065 * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
066 * @return the U matrix (or null if decomposed matrix is singular)
067 * @see #getU()
068 */
069 RealMatrix getUT();
070
071 /**
072 * Returns the diagonal matrix Σ of the decomposition.
073 * <p>Σ is a diagonal matrix. The singular values are provided in
074 * non-increasing order, for compatibility with Jama.</p>
075 * @return the Σ matrix
076 */
077 RealMatrix getS();
078
079 /**
080 * Returns the diagonal elements of the matrix Σ of the decomposition.
081 * <p>The singular values are provided in non-increasing order, for
082 * compatibility with Jama.</p>
083 * @return the diagonal elements of the Σ matrix
084 */
085 double[] getSingularValues();
086
087 /**
088 * Returns the matrix V of the decomposition.
089 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
090 * @return the V matrix (or null if decomposed matrix is singular)
091 * @see #getVT()
092 */
093 RealMatrix getV();
094
095 /**
096 * Returns the transpose of the matrix V of the decomposition.
097 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
098 * @return the V matrix (or null if decomposed matrix is singular)
099 * @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 }