/*!
 * \file
 * \brief Definitions of functions for solving linear equation systems
 * \author Tony Ottosson
 *
 * -------------------------------------------------------------------------
 *
 * IT++ - C++ library of mathematical, signal processing, speech processing,
 *        and communications classes and functions
 *
 * Copyright (C) 1995-2007  (see AUTHORS file for a list of contributors)
 *
 * This program 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 2 of the License, or
 * (at your option) any later version.
 *
 * This program 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 this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
 *
 * -------------------------------------------------------------------------
 */

#ifndef LS_SOLVE_H
#define LS_SOLVE_H

#include <itpp/base/mat.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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine DPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine DPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine DPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine DPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine ZPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine ZPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine ZPOSV.
  */
  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 postive definite \f$n \times n\f$ matrix.
  Uses the LAPACK routine ZPOSV.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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)
  */
  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)
  */
  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).
  */
  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).
  */
  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)
  */
  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)
  */
  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).
  */
  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).
  */
  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.
  */
  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.
  */
  void forward_substitution(const mat &L, const vec &b, vec &x);

  /*! \brief Forward substitution of band matricies.

  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.
  */
  vec forward_substitution(const mat &L, int p, const vec &b);

  /*! \brief Forward substitution of band matricies.

  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.
  */
  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.
  */
  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.
  */
  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.
  */
  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.
  */
  void backward_substitution(const mat &U, int q, const vec &b, vec &x);

  //!@}

} //namespace itpp

#endif // #ifndef LS_SOLVE_H