ls_solve.h

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  */
  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 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.
  */
  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.
  */
  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