1 /** 2 * The linear algebra package provides classes and computational methods for 3 * operations on matrices and vectors. 4 * <p> 5 * Some content of this package is adapted from the Jama package. 6 * <p> 7 * Five fundamental matrix decompositions, which consist of pairs or triples 8 * of matrices, permutation vectors, and the like, produce results in five 9 * decomposition classes. These decompositions are accessed by the Matrix 10 * class to compute solutions of simultaneous linear equations, determinants, 11 * inverses and other matrix functions. The five decompositions are: 12 * <ul> 13 * <li>Cholesky Decomposition of symmetric, positive definite matrices.</li> 14 * <li>LU Decomposition of rectangular matrices.</li> 15 * <li>QR Decomposition of rectangular matrices.</li> 16 * <li>Singular Value Decomposition of rectangular matrices.</li> 17 * <li>Eigenvalue Decomposition of both symmetric and nonsymmetric square 18 * matrices.</li> 19 * </ul> 20 * <b>Example of use:</b> 21 * <p> 22 * Solve a linear system \(Ax=b\) and compute the residual norm, \(||b-Ax||\). 23 * <p> 24 * {@code double[][] matrix = { {1.,2.,3}, {4.,5.,6.}, {7.,8.,10.} };}<br> 25 * {@code double[] b = MathUtil.randomDoubleArray(3, new Random());}<br> 26 * {@code double[] x = VMath.solve(matrix, b);}<br> 27 * {@code double[] r = VMath.minusEquals(VMath.times(matrix, x), b);}<br> 28 * {@code double norm = VMath.euclideanLength(r);} 29 * <p> 30 * The original Jama-package has been developed by 31 * the <a href="http://www.mathworks.com/">MathWorks</a> and 32 * <a href="http://www.nist.gov/">NIST</a> 33 * and can be found at 34 * <a href="http://math.nist.gov/javanumerics/jama/">math.nist.gov</a>. 35 * <p> 36 * Here, for the adaption some classes and methods convenient for data mining 37 * applications within ELKI were added. 38 * Furthermore some erroneous comments were corrected and the coding-style was 39 * subtly changed to a more Java-typical style. 40 * 41 * @opt hide ^java\.(io|lang)\..* 42 */ 43 /* 44 * This file is part of ELKI: 45 * Environment for Developing KDD-Applications Supported by Index-Structures 46 * 47 * Copyright (C) 2018 48 * ELKI Development Team 49 * 50 * This program is free software: you can redistribute it and/or modify 51 * it under the terms of the GNU Affero General Public License as published by 52 * the Free Software Foundation, either version 3 of the License, or 53 * (at your option) any later version. 54 * 55 * This program is distributed in the hope that it will be useful, 56 * but WITHOUT ANY WARRANTY; without even the implied warranty of 57 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 58 * GNU Affero General Public License for more details. 59 * 60 * You should have received a copy of the GNU Affero General Public License 61 * along with this program. If not, see <http://www.gnu.org/licenses/>. 62 */ 63 package de.lmu.ifi.dbs.elki.math.linearalgebra; 64