Java tutorial
/* * Licensed to the Apache Software Foundation (ASF) under one or more * contributor license agreements. See the NOTICE file distributed with * this work for additional information regarding copyright ownership. * The ASF licenses this file to You under the Apache License, Version 2.0 * (the "License"); you may not use this file except in compliance with * the License. You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ package org.apache.commons.math.linear; /** * An interface to classes that implement an algorithm to calculate the * Singular Value Decomposition of a real matrix. * <p> * The Singular Value Decomposition of matrix A is a set of three matrices: U, * Σ and V such that A = U × Σ × V<sup>T</sup>. Let A be * a m × n matrix, then U is a m × p orthogonal matrix, Σ is a * p × p diagonal matrix with positive or null elements, V is a p × * n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where * p=min(m,n). * </p> * <p>This interface is similar to the class with similar name from the * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the * following changes:</p> * <ul> * <li>the <code>norm2</code> method which has been renamed as {@link #getNorm() * getNorm},</li> * <li>the <code>cond</code> method which has been renamed as {@link * #getConditionNumber() getConditionNumber},</li> * <li>the <code>rank</code> method which has been renamed as {@link #getRank() * getRank},</li> * <li>a {@link #getUT() getUT} method has been added,</li> * <li>a {@link #getVT() getVT} method has been added,</li> * <li>a {@link #getSolver() getSolver} method has been added,</li> * <li>a {@link #getCovariance(double) getCovariance} method has been added.</li> * </ul> * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a> * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a> * @version $Revision: 928081 $ $Date: 2010-03-26 23:36:38 +0100 (ven. 26 mars 2010) $ * @since 2.0 */ public interface SingularValueDecomposition { /** * Returns the matrix U of the decomposition. * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> * @return the U matrix * @see #getUT() */ RealMatrix getU(); /** * Returns the transpose of the matrix U of the decomposition. * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p> * @return the U matrix (or null if decomposed matrix is singular) * @see #getU() */ RealMatrix getUT(); /** * Returns the diagonal matrix Σ of the decomposition. * <p>Σ is a diagonal matrix. The singular values are provided in * non-increasing order, for compatibility with Jama.</p> * @return the Σ matrix */ RealMatrix getS(); /** * Returns the diagonal elements of the matrix Σ of the decomposition. * <p>The singular values are provided in non-increasing order, for * compatibility with Jama.</p> * @return the diagonal elements of the Σ matrix */ double[] getSingularValues(); /** * Returns the matrix V of the decomposition. * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> * @return the V matrix (or null if decomposed matrix is singular) * @see #getVT() */ RealMatrix getV(); /** * Returns the transpose of the matrix V of the decomposition. * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> * @return the V matrix (or null if decomposed matrix is singular) * @see #getV() */ RealMatrix getVT(); /** * Returns the n × n covariance matrix. * <p>The covariance matrix is V × J × V<sup>T</sup> * where J is the diagonal matrix of the inverse of the squares of * the singular values.</p> * @param minSingularValue value below which singular values are ignored * (a 0 or negative value implies all singular value will be used) * @return covariance matrix * @exception IllegalArgumentException if minSingularValue is larger than * the largest singular value, meaning all singular values are ignored */ RealMatrix getCovariance(double minSingularValue) throws IllegalArgumentException; /** * Returns the L<sub>2</sub> norm of the matrix. * <p>The L<sub>2</sub> norm is max(|A × u|<sub>2</sub> / * |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm * (i.e. the traditional euclidian norm).</p> * @return norm */ double getNorm(); /** * Return the condition number of the matrix. * @return condition number of the matrix */ double getConditionNumber(); /** * Return the effective numerical matrix rank. * <p>The effective numerical rank is the number of non-negligible * singular values. The threshold used to identify non-negligible * terms is max(m,n) × ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>) * is the least significant bit of the largest singular value.</p> * @return effective numerical matrix rank */ int getRank(); /** * Get a solver for finding the A × X = B solution in least square sense. * @return a solver */ DecompositionSolver getSolver(); }