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* \file
* \brief Definitions of functions for solving linear equation systems
* \author Tony Ottosson
*
* -------------------------------------------------------------------------
*
* Copyright (C) 1995-2010 (see AUTHORS file for a list of contributors)
*
* This file is part of IT++ - a C++ library of mathematical, signal
* processing, speech processing, and communications classes and functions.
*
* IT++ is free software: you can redistribute it and/or modify it under the
* terms of the GNU General Public License as published by the Free Software
* Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* IT++ is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
* FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
* details.
*
* You should have received a copy of the GNU General Public License along
* with IT++. If not, see <http://www.gnu.org/licenses/>.
*
* -------------------------------------------------------------------------
*/
#ifndef LS_SOLVE_H
#define LS_SOLVE_H
#include <itpp/base/mat.h>
#include <itpp/itexports.h>
namespace itpp
{
/*! \addtogroup linearequations
*/
//!@{
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
ITPP_EXPORT bool ls_solve(const mat &A, const vec &b, vec &x);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
ITPP_EXPORT vec ls_solve(const mat &A, const vec &b);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
bool ls_solve(const mat &A, const mat &B, mat &X);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
ITPP_EXPORT mat ls_solve(const mat &A, const mat &B);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
ITPP_EXPORT bool ls_solve(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
ITPP_EXPORT cvec ls_solve(const cmat &A, const cvec &b);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
ITPP_EXPORT bool ls_solve(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
ITPP_EXPORT cmat ls_solve(const cmat &A, const cmat &B);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
ITPP_EXPORT bool ls_solve_chol(const mat &A, const vec &b, vec &x);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
ITPP_EXPORT vec ls_solve_chol(const mat &A, const vec &b);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
ITPP_EXPORT bool ls_solve_chol(const mat &A, const mat &B, mat &X);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
ITPP_EXPORT mat ls_solve_chol(const mat &A, const mat &B);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
ITPP_EXPORT bool ls_solve_chol(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
ITPP_EXPORT cvec ls_solve_chol(const cmat &A, const cvec &b);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
ITPP_EXPORT bool ls_solve_chol(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
ITPP_EXPORT cmat ls_solve_chol(const cmat &A, const cmat &B);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and is built upon the LAPACK routine DGELS.
*/
ITPP_EXPORT bool ls_solve_od(const mat &A, const vec &b, vec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT vec ls_solve_od(const mat &A, const vec &b);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT bool ls_solve_od(const mat &A, const mat &B, mat &X);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT mat ls_solve_od(const mat &A, const mat &B);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and is built upon the LAPACK routine ZGELS.
*/
ITPP_EXPORT bool ls_solve_od(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT cvec ls_solve_od(const cmat &A, const cvec &b);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT bool ls_solve_od(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT cmat ls_solve_od(const cmat &A, const cmat &B);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and is built upon the LAPACK routine DGELS.
*/
ITPP_EXPORT bool ls_solve_ud(const mat &A, const vec &b, vec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT vec ls_solve_ud(const mat &A, const vec &b);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT bool ls_solve_ud(const mat &A, const mat &B, mat &X);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
ITPP_EXPORT mat ls_solve_ud(const mat &A, const mat &B);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and is built upon the LAPACK routine ZGELS.
*/
ITPP_EXPORT bool ls_solve_ud(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT cvec ls_solve_ud(const cmat &A, const cvec &b);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT bool ls_solve_ud(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
ITPP_EXPORT cmat ls_solve_ud(const cmat &A, const cmat &B);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)
*/
ITPP_EXPORT bool backslash(const mat &A, const vec &b, vec &x);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)
*/
ITPP_EXPORT vec backslash(const mat &A, const vec &b);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).
*/
ITPP_EXPORT bool backslash(const mat &A, const mat &B, mat &X);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).
*/
ITPP_EXPORT mat backslash(const mat &A, const mat &B);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)
*/
ITPP_EXPORT bool backslash(const cmat &A, const cvec &b, cvec &x);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)
*/
ITPP_EXPORT cvec backslash(const cmat &A, const cvec &b);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).
*/
ITPP_EXPORT bool backslash(const cmat &A, const cmat &B, cmat &X);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).
*/
ITPP_EXPORT cmat backslash(const cmat &A, const cmat &B);
/*! \brief Forward substitution of square matrix.
Solves Lx=b, where L is a lower triangular n by n matrix.
Assumes that L is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
ITPP_EXPORT vec forward_substitution(const mat &L, const vec &b);
/*! \brief Forward substitution of square matrix.
Solves Lx=b, where L is a lower triangular n by n matrix.
Assumes that L is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
ITPP_EXPORT void forward_substitution(const mat &L, const vec &b, vec &x);
/*! \brief Forward substitution of band matrices.
Solves Lx=b, where L is a lower triangular n by n band-matrix with lower
bandwidth p.
Assumes that L is nonsingular. Requires about 2np flops (if n >> p).
Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
ITPP_EXPORT vec forward_substitution(const mat &L, int p, const vec &b);
/*! \brief Forward substitution of band matrices.
Solves Lx=b, where L is a lower triangular n by n band-matrix with
lower bandwidth p.
Assumes that L is nonsingular. Requires about 2np flops (if n >> p).
Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
ITPP_EXPORT void forward_substitution(const mat &L, int p, const vec &b, vec &x);
/*! \brief Backward substitution of square matrix.
Solves Ux=b, where U is a upper triangular n by n matrix.
Assumes that U is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
ITPP_EXPORT vec backward_substitution(const mat &U, const vec &b);
/*! \brief Backward substitution of square matrix.
Solves Ux=b, where U is a upper triangular n by n matrix.
Assumes that U is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
ITPP_EXPORT void backward_substitution(const mat &U, const vec &b, vec &x);
/*! \brief Backward substitution of band matrix.
Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with
upper bandwidth q.
Assumes that U is nonsingular. Requires about 2nq flops (if n >> q).
Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
ITPP_EXPORT vec backward_substitution(const mat &U, int q, const vec &b);
/*! \brief Backward substitution of band matrix.
Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with
upper bandwidth q.
Assumes that U is nonsingular. Requires about 2nq flops (if n >> q).
Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
ITPP_EXPORT void backward_substitution(const mat &U, int q, const vec &b, vec &x);
//!@}
} //namespace itpp
#endif // #ifndef LS_SOLVE_H
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