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 018package org.apache.commons.math3.linear; 019 020/** 021 * Interface handling decomposition algorithms that can solve A × X = B. 022 * <p> 023 * Decomposition algorithms decompose an A matrix has a product of several specific 024 * matrices from which they can solve A × X = B in least squares sense: they find X 025 * such that ||A × X - B|| is minimal. 026 * <p> 027 * Some solvers like {@link LUDecomposition} can only find the solution for 028 * square matrices and when the solution is an exact linear solution, i.e. when 029 * ||A × X - B|| is exactly 0. Other solvers can also find solutions 030 * with non-square matrix A and with non-null minimal norm. If an exact linear 031 * solution exists it is also the minimal norm solution. 032 * 033 * @since 2.0 034 */ 035public interface DecompositionSolver { 036 037 /** 038 * Solve the linear equation A × X = B for matrices A. 039 * <p> 040 * The A matrix is implicit, it is provided by the underlying 041 * decomposition algorithm. 042 * 043 * @param b right-hand side of the equation A × X = B 044 * @return a vector X that minimizes the two norm of A × X - B 045 * @throws org.apache.commons.math3.exception.DimensionMismatchException 046 * if the matrices dimensions do not match. 047 * @throws SingularMatrixException if the decomposed matrix is singular. 048 */ 049 RealVector solve(final RealVector b) throws SingularMatrixException; 050 051 /** 052 * Solve the linear equation A × X = B for matrices A. 053 * <p> 054 * The A matrix is implicit, it is provided by the underlying 055 * decomposition algorithm. 056 * 057 * @param b right-hand side of the equation A × X = B 058 * @return a matrix X that minimizes the two norm of A × X - B 059 * @throws org.apache.commons.math3.exception.DimensionMismatchException 060 * if the matrices dimensions do not match. 061 * @throws SingularMatrixException if the decomposed matrix is singular. 062 */ 063 RealMatrix solve(final RealMatrix b) throws SingularMatrixException; 064 065 /** 066 * Check if the decomposed matrix is non-singular. 067 * @return true if the decomposed matrix is non-singular. 068 */ 069 boolean isNonSingular(); 070 071 /** 072 * Get the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a> 073 * of the decomposed matrix. 074 * <p> 075 * <em>This is equal to the inverse of the decomposed matrix, if such an inverse exists.</em> 076 * <p> 077 * If no such inverse exists, then the result has properties that resemble that of an inverse. 078 * <p> 079 * In particular, in this case, if the decomposed matrix is A, then the system of equations 080 * \( A x = b \) may have no solutions, or many. If it has no solutions, then the pseudo-inverse 081 * \( A^+ \) gives the "closest" solution \( z = A^+ b \), meaning \( \left \| A z - b \right \|_2 \) 082 * is minimized. If there are many solutions, then \( z = A^+ b \) is the smallest solution, 083 * meaning \( \left \| z \right \|_2 \) is minimized. 084 * <p> 085 * Note however that some decompositions cannot compute a pseudo-inverse for all matrices. 086 * For example, the {@link LUDecomposition} is not defined for non-square matrices to begin 087 * with. The {@link QRDecomposition} can operate on non-square matrices, but will throw 088 * {@link SingularMatrixException} if the decomposed matrix is singular. Refer to the javadoc 089 * of specific decomposition implementations for more details. 090 * 091 * @return pseudo-inverse matrix (which is the inverse, if it exists), 092 * if the decomposition can pseudo-invert the decomposed matrix 093 * @throws SingularMatrixException if the decomposed matrix is singular and the decomposition 094 * can not compute a pseudo-inverse 095 */ 096 RealMatrix getInverse() throws SingularMatrixException; 097}