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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>The Singular Value Decomposition of matrix A is a set of three matrices:
26   * U, &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>.
27   * Let A be an m &times; n matrix, then U is an m &times; m orthogonal matrix,
28   * &Sigma; is a m &times; n diagonal matrix with positive diagonal elements,
29   * and V is an n &times; n orthogonal matrix.</p>
30   * <p>This interface is similar to the class with similar name from the now defunct
31   * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
32   * following changes:</p>
33   * <ul>
34   *   <li>the <code>norm2</code> method which has been renamed as {@link #getNorm()
35   *   getNorm},</li>
36   *   <li>the <code>cond</code> method which has been renamed as {@link
37   *   #getConditionNumber() getConditionNumber},</li>
38   *   <li>the <code>rank</code> method which has been renamed as {@link #getRank()
39   *   getRank},</li>
40   *   <li>a {@link #getUT() getUT} method has been added,</li>
41   *   <li>a {@link #getVT() getVT} method has been added,</li>
42   *   <li>a {@link #getSolver() getSolver} method has been added,</li>
43   *   <li>a {@link #getCovariance(double) getCovariance} method has been added.</li>
44   * </ul>
45   * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>
46   * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a>
47   * @version $Revision: 799857 $ $Date: 2009-08-01 09:07:12 -0400 (Sat, 01 Aug 2009) $
48   * @since 2.0
49   */
50  public interface SingularValueDecomposition {
51  
52      /**
53       * Returns the matrix U of the decomposition. 
54       * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
55       * @return the U matrix
56       * @see #getUT()
57       */
58      RealMatrix getU();
59  
60      /**
61       * Returns the transpose of the matrix U of the decomposition. 
62       * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
63       * @return the U matrix (or null if decomposed matrix is singular)
64       * @see #getU()
65       */
66      RealMatrix getUT();
67  
68      /**
69       * Returns the diagonal matrix &Sigma; of the decomposition. 
70       * <p>&Sigma; is a diagonal matrix. The singular values are provided in
71       * non-increasing order, for compatibility with Jama.</p>
72       * @return the &Sigma; matrix
73       */
74      RealMatrix getS();
75  
76      /**
77       * Returns the diagonal elements of the matrix &Sigma; of the decomposition.
78       * <p>The singular values are provided in non-increasing order, for
79       * compatibility with Jama.</p>
80       * @return the diagonal elements of the &Sigma; matrix
81       */
82      double[] getSingularValues();
83  
84      /**
85       * Returns the matrix V of the decomposition. 
86       * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
87       * @return the V matrix (or null if decomposed matrix is singular)
88       * @see #getVT()
89       */
90      RealMatrix getV();
91  
92      /**
93       * Returns the transpose of the matrix V of the decomposition. 
94       * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
95       * @return the V matrix (or null if decomposed matrix is singular)
96       * @see #getV()
97       */
98      RealMatrix getVT();
99  
100     /**
101      * Returns the n &times; n covariance matrix.
102      * <p>The covariance matrix is V &times; J &times; 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 &times; 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) &times; 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 &times; X = B solution in least square sense.
140      * @return a solver
141      */
142     DecompositionSolver getSolver();
143 
144 }