| 1 | /*! |
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| 2 | * \file |
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| 3 | * \brief Definitions of functions for solving linear equation systems |
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| 4 | * \author Tony Ottosson |
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| 5 | * |
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| 6 | * ------------------------------------------------------------------------- |
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| 7 | * |
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| 8 | * IT++ - C++ library of mathematical, signal processing, speech processing, |
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| 9 | * and communications classes and functions |
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| 10 | * |
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| 11 | * Copyright (C) 1995-2007 (see AUTHORS file for a list of contributors) |
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| 12 | * |
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| 13 | * This program is free software; you can redistribute it and/or modify |
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| 14 | * it under the terms of the GNU General Public License as published by |
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| 15 | * the Free Software Foundation; either version 2 of the License, or |
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| 16 | * (at your option) any later version. |
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| 17 | * |
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| 18 | * This program is distributed in the hope that it will be useful, |
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| 19 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 20 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 21 | * GNU General Public License for more details. |
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| 22 | * |
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| 23 | * You should have received a copy of the GNU General Public License |
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| 24 | * along with this program; if not, write to the Free Software |
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| 25 | * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA |
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| 26 | * |
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| 27 | * ------------------------------------------------------------------------- |
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| 28 | */ |
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| 29 | |
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| 30 | #ifndef LS_SOLVE_H |
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| 31 | #define LS_SOLVE_H |
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| 32 | |
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| 33 | #include <itpp/base/mat.h> |
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| 34 | |
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| 35 | |
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| 36 | namespace itpp { |
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| 37 | |
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| 38 | |
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| 39 | /*! \addtogroup linearequations |
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| 40 | */ |
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| 41 | //!@{ |
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| 42 | |
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| 43 | /*! \brief Solve linear equation system by LU factorisation. |
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| 44 | |
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| 45 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix. |
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| 46 | Uses the LAPACK routine DGESV. |
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| 47 | */ |
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| 48 | bool ls_solve(const mat &A, const vec &b, vec &x); |
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| 49 | |
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| 50 | /*! \brief Solve linear equation system by LU factorisation. |
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| 51 | |
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| 52 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix. |
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| 53 | Uses the LAPACK routine DGESV. |
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| 54 | */ |
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| 55 | vec ls_solve(const mat &A, const vec &b); |
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| 56 | |
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| 57 | /*! \brief Solve multiple linear equations by LU factorisation. |
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| 58 | |
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| 59 | Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix. |
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| 60 | Uses the LAPACK routine DGESV. |
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| 61 | */ |
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| 62 | bool ls_solve(const mat &A, const mat &B, mat &X); |
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| 63 | |
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| 64 | /*! \brief Solve multiple linear equations by LU factorisation. |
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| 65 | |
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| 66 | Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix. |
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| 67 | Uses the LAPACK routine DGESV. |
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| 68 | */ |
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| 69 | mat ls_solve(const mat &A, const mat &B); |
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| 70 | |
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| 71 | |
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| 72 | /*! \brief Solve linear equation system by LU factorisation. |
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| 73 | |
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| 74 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix. |
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| 75 | Uses the LAPACK routine ZGESV. |
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| 76 | */ |
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| 77 | bool ls_solve(const cmat &A, const cvec &b, cvec &x); |
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| 78 | |
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| 79 | /*! \brief Solve linear equation system by LU factorisation. |
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| 80 | |
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| 81 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix. |
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| 82 | Uses the LAPACK routine ZGESV. |
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| 83 | */ |
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| 84 | cvec ls_solve(const cmat &A, const cvec &b); |
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| 85 | |
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| 86 | /*! \brief Solve multiple linear equations by LU factorisation. |
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| 87 | |
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| 88 | Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix. |
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| 89 | Uses the LAPACK routine ZGESV. |
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| 90 | */ |
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| 91 | bool ls_solve(const cmat &A, const cmat &B, cmat &X); |
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| 92 | |
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| 93 | /*! \brief Solve multiple linear equations by LU factorisation. |
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| 94 | |
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| 95 | Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix. |
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| 96 | Uses the LAPACK routine ZGESV. |
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| 97 | */ |
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| 98 | cmat ls_solve(const cmat &A, const cmat &B); |
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| 99 | |
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| 100 | |
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| 101 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 102 | |
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| 103 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric postive definite \f$n \times n\f$ matrix. |
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| 104 | Uses the LAPACK routine DPOSV. |
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| 105 | */ |
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| 106 | bool ls_solve_chol(const mat &A, const vec &b, vec &x); |
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| 107 | |
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| 108 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 109 | |
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| 110 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric postive definite \f$n \times n\f$ matrix. |
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| 111 | Uses the LAPACK routine DPOSV. |
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| 112 | */ |
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| 113 | vec ls_solve_chol(const mat &A, const vec &b); |
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| 114 | |
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| 115 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 116 | |
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| 117 | Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric postive definite \f$n \times n\f$ matrix. |
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| 118 | Uses the LAPACK routine DPOSV. |
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| 119 | */ |
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| 120 | bool ls_solve_chol(const mat &A, const mat &B, mat &X); |
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| 121 | |
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| 122 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 123 | |
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| 124 | Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric postive definite \f$n \times n\f$ matrix. |
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| 125 | Uses the LAPACK routine DPOSV. |
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| 126 | */ |
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| 127 | mat ls_solve_chol(const mat &A, const mat &B); |
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| 128 | |
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| 129 | |
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| 130 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 131 | |
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| 132 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian postive definite \f$n \times n\f$ matrix. |
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| 133 | Uses the LAPACK routine ZPOSV. |
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| 134 | */ |
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| 135 | bool ls_solve_chol(const cmat &A, const cvec &b, cvec &x); |
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| 136 | |
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| 137 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 138 | |
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| 139 | Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian postive definite \f$n \times n\f$ matrix. |
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| 140 | Uses the LAPACK routine ZPOSV. |
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| 141 | */ |
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| 142 | cvec ls_solve_chol(const cmat &A, const cvec &b); |
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| 143 | |
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| 144 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 145 | |
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| 146 | Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian postive definite \f$n \times n\f$ matrix. |
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| 147 | Uses the LAPACK routine ZPOSV. |
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| 148 | */ |
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| 149 | bool ls_solve_chol(const cmat &A, const cmat &B, cmat &X); |
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| 150 | |
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| 151 | /*! \brief Solve linear equation system by Cholesky factorisation. |
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| 152 | |
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| 153 | Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian postive definite \f$n \times n\f$ matrix. |
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| 154 | Uses the LAPACK routine ZPOSV. |
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| 155 | */ |
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| 156 | cmat ls_solve_chol(const cmat &A, const cmat &B); |
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| 157 | |
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| 158 | |
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| 159 | |
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| 160 | /*! \brief Solves overdetermined linear equation systems. |
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| 161 | |
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| 162 | Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 163 | Uses QR-factorization and is built upon the LAPACK routine DGELS. |
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| 164 | */ |
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| 165 | bool ls_solve_od(const mat &A, const vec &b, vec &x); |
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| 166 | |
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| 167 | /*! \brief Solves overdetermined linear equation systems. |
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| 168 | |
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| 169 | Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 170 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 171 | */ |
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| 172 | vec ls_solve_od(const mat &A, const vec &b); |
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| 173 | |
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| 174 | /*! \brief Solves overdetermined linear equation systems. |
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| 175 | |
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| 176 | Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 177 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 178 | */ |
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| 179 | bool ls_solve_od(const mat &A, const mat &B, mat &X); |
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| 180 | |
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| 181 | /*! \brief Solves overdetermined linear equation systems. |
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| 182 | |
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| 183 | Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 184 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 185 | */ |
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| 186 | mat ls_solve_od(const mat &A, const mat &B); |
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| 187 | |
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| 188 | |
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| 189 | /*! \brief Solves overdetermined linear equation systems. |
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| 190 | |
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| 191 | Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 192 | Uses QR-factorization and is built upon the LAPACK routine ZGELS. |
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| 193 | */ |
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| 194 | bool ls_solve_od(const cmat &A, const cvec &b, cvec &x); |
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| 195 | |
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| 196 | /*! \brief Solves overdetermined linear equation systems. |
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| 197 | |
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| 198 | Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 199 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 200 | */ |
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| 201 | cvec ls_solve_od(const cmat &A, const cvec &b); |
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| 202 | |
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| 203 | /*! \brief Solves overdetermined linear equation systems. |
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| 204 | |
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| 205 | Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 206 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 207 | */ |
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| 208 | bool ls_solve_od(const cmat &A, const cmat &B, cmat &X); |
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| 209 | |
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| 210 | /*! \brief Solves overdetermined linear equation systems. |
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| 211 | |
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| 212 | Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$. |
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| 213 | Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 214 | */ |
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| 215 | cmat ls_solve_od(const cmat &A, const cmat &B); |
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| 216 | |
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| 217 | |
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| 218 | |
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| 219 | /*! \brief Solves underdetermined linear equation systems. |
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| 220 | |
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| 221 | Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 222 | Uses LQ-factorization and is built upon the LAPACK routine DGELS. |
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| 223 | */ |
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| 224 | bool ls_solve_ud(const mat &A, const vec &b, vec &x); |
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| 225 | |
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| 226 | /*! \brief Solves overdetermined linear equation systems. |
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| 227 | |
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| 228 | Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 229 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 230 | */ |
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| 231 | vec ls_solve_ud(const mat &A, const vec &b); |
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| 232 | |
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| 233 | /*! \brief Solves underdetermined linear equation systems. |
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| 234 | |
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| 235 | Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 236 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 237 | */ |
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| 238 | bool ls_solve_ud(const mat &A, const mat &B, mat &X); |
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| 239 | |
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| 240 | /*! \brief Solves underdetermined linear equation systems. |
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| 241 | |
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| 242 | Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 243 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS. |
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| 244 | */ |
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| 245 | mat ls_solve_ud(const mat &A, const mat &B); |
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| 246 | |
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| 247 | |
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| 248 | /*! \brief Solves underdetermined linear equation systems. |
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| 249 | |
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| 250 | Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 251 | Uses LQ-factorization and is built upon the LAPACK routine ZGELS. |
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| 252 | */ |
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| 253 | bool ls_solve_ud(const cmat &A, const cvec &b, cvec &x); |
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| 254 | |
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| 255 | /*! \brief Solves overdetermined linear equation systems. |
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| 256 | |
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| 257 | Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 258 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 259 | */ |
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| 260 | cvec ls_solve_ud(const cmat &A, const cvec &b); |
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| 261 | |
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| 262 | /*! \brief Solves underdetermined linear equation systems. |
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| 263 | |
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| 264 | Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 265 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 266 | */ |
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| 267 | bool ls_solve_ud(const cmat &A, const cmat &B, cmat &X); |
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| 268 | |
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| 269 | /*! \brief Solves underdetermined linear equation systems. |
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| 270 | |
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| 271 | Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$. |
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| 272 | Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS. |
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| 273 | */ |
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| 274 | cmat ls_solve_ud(const cmat &A, const cmat &B); |
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| 275 | |
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| 276 | |
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| 277 | /*! \brief A general linear equation system solver. |
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| 278 | |
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| 279 | Tries to emulate the backslash operator in Matlab by calling |
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| 280 | ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x) |
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| 281 | */ |
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| 282 | bool backslash(const mat &A, const vec &b, vec &x); |
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| 283 | |
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| 284 | /*! \brief A general linear equation system solver. |
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| 285 | |
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| 286 | Tries to emulate the backslash operator in Matlab by calling |
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| 287 | ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b) |
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| 288 | */ |
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| 289 | vec backslash(const mat &A, const vec &b); |
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| 290 | |
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| 291 | /*! \brief A general linear equation system solver. |
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| 292 | |
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| 293 | Tries to emulate the backslash operator in Matlab by calling |
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| 294 | ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X). |
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| 295 | */ |
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| 296 | bool backslash(const mat &A, const mat &B, mat &X); |
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| 297 | |
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| 298 | /*! \brief A general linear equation system solver. |
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| 299 | |
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| 300 | Tries to emulate the backslash operator in Matlab by calling |
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| 301 | ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B). |
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| 302 | */ |
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| 303 | mat backslash(const mat &A, const mat &B); |
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| 304 | |
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| 305 | |
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| 306 | /*! \brief A general linear equation system solver. |
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| 307 | |
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| 308 | Tries to emulate the backslash operator in Matlab by calling |
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| 309 | ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x) |
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| 310 | */ |
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| 311 | bool backslash(const cmat &A, const cvec &b, cvec &x); |
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| 312 | |
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| 313 | /*! \brief A general linear equation system solver. |
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| 314 | |
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| 315 | Tries to emulate the backslash operator in Matlab by calling |
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| 316 | ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b) |
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| 317 | */ |
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| 318 | cvec backslash(const cmat &A, const cvec &b); |
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| 319 | |
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| 320 | /*! \brief A general linear equation system solver. |
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| 321 | |
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| 322 | Tries to emulate the backslash operator in Matlab by calling |
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| 323 | ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X). |
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| 324 | */ |
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| 325 | bool backslash(const cmat &A, const cmat &B, cmat &X); |
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| 326 | |
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| 327 | /*! \brief A general linear equation system solver. |
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| 328 | |
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| 329 | Tries to emulate the backslash operator in Matlab by calling |
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| 330 | ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B). |
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| 331 | */ |
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| 332 | cmat backslash(const cmat &A, const cmat &B); |
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| 333 | |
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| 334 | |
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| 335 | |
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| 336 | /*! \brief Forward substitution of square matrix. |
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| 337 | |
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| 338 | Solves Lx=b, where L is a lower triangular n by n matrix. |
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| 339 | Assumes that L is nonsingular. Requires n^2 flops. |
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| 340 | Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89. |
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| 341 | */ |
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| 342 | vec forward_substitution(const mat &L, const vec &b); |
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| 343 | |
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| 344 | /*! \brief Forward substitution of square matrix. |
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| 345 | |
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| 346 | Solves Lx=b, where L is a lower triangular n by n matrix. |
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| 347 | Assumes that L is nonsingular. Requires n^2 flops. |
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| 348 | Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89. |
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| 349 | */ |
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| 350 | void forward_substitution(const mat &L, const vec &b, vec &x); |
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| 351 | |
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| 352 | /*! \brief Forward substitution of band matricies. |
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| 353 | |
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| 354 | Solves Lx=b, where L is a lower triangular n by n band-matrix with lower |
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| 355 | bandwidth p. |
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| 356 | Assumes that L is nonsingular. Requires about 2np flops (if n >> p). |
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| 357 | Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153. |
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| 358 | */ |
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| 359 | vec forward_substitution(const mat &L, int p, const vec &b); |
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| 360 | |
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| 361 | /*! \brief Forward substitution of band matricies. |
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| 362 | |
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| 363 | Solves Lx=b, where L is a lower triangular n by n band-matrix with |
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| 364 | lower bandwidth p. |
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| 365 | Assumes that L is nonsingular. Requires about 2np flops (if n >> p). |
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| 366 | Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153. |
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| 367 | */ |
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| 368 | void forward_substitution(const mat &L, int p, const vec &b, vec &x); |
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| 369 | |
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| 370 | /*! \brief Backward substitution of square matrix. |
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| 371 | |
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| 372 | Solves Ux=b, where U is a upper triangular n by n matrix. |
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| 373 | Assumes that U is nonsingular. Requires n^2 flops. |
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| 374 | Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89. |
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| 375 | */ |
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| 376 | vec backward_substitution(const mat &U, const vec &b); |
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| 377 | |
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| 378 | /*! \brief Backward substitution of square matrix. |
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| 379 | |
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| 380 | Solves Ux=b, where U is a upper triangular n by n matrix. |
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| 381 | Assumes that U is nonsingular. Requires n^2 flops. |
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| 382 | Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89. |
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| 383 | */ |
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| 384 | void backward_substitution(const mat &U, const vec &b, vec &x); |
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| 385 | |
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| 386 | /*! \brief Backward substitution of band matrix. |
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| 387 | |
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| 388 | Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with |
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| 389 | upper bandwidth q. |
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| 390 | Assumes that U is nonsingular. Requires about 2nq flops (if n >> q). |
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| 391 | Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153. |
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| 392 | */ |
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| 393 | vec backward_substitution(const mat &U, int q, const vec &b); |
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| 394 | |
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| 395 | /*! \brief Backward substitution of band matrix. |
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| 396 | |
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| 397 | Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with |
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| 398 | upper bandwidth q. |
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| 399 | Assumes that U is nonsingular. Requires about 2nq flops (if n >> q). |
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| 400 | Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153. |
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| 401 | */ |
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| 402 | void backward_substitution(const mat &U, int q, const vec &b, vec &x); |
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| 403 | |
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| 404 | //!@} |
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| 405 | |
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| 406 | } //namespace itpp |
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| 407 | |
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| 408 | #endif // #ifndef LS_SOLVE_H |
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| 409 | |
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| 410 | |
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