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, &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>.
027     * Let A be an m &times; n matrix, then U is an m &times; m orthogonal matrix,
028     * &Sigma; is a m &times; n diagonal matrix with positive diagonal elements,
029     * and V is an n &times; 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 &Sigma; of the decomposition. 
070         * <p>&Sigma; is a diagonal matrix. The singular values are provided in
071         * non-increasing order, for compatibility with Jama.</p>
072         * @return the &Sigma; matrix
073         */
074        RealMatrix getS();
075    
076        /**
077         * Returns the diagonal elements of the matrix &Sigma; 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 &Sigma; 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 &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    }