NAME

cg -- conjugate gradient algorithm.

SYNOPSIS

  template <class Matrix, class Vector, class Preconditioner, class Real>


  int cg (const Matrix &A, Vector &x, const Vector &b,


    const solver_option& sopt = solver_option())

EXAMPLE

The simplest call to cg has the folling form:

    solver_option sopt;


    sopt.max_iter = 100;


    sopt.tol = 1e-7;


    int status = cg(a, x, b, eye(), sopt);

DESCRIPTION

cg solves the symmetric positive definite linear system Ax=b using the conjugate gradient method. The return value indicates convergence within max_iter (input) iterations (0), or no convergence within max_iter iterations (1). Upon successful return, output arguments have the following values:

x

approximate solution to Ax = b

sopt.n_iter

the number of iterations performed before the tolerance was reached

sopt.residue

the residual after the final iteration See also the solver_option(2).

NOTE

cg is an iterative template routine.

cg follows the algorithm described on p. 15 in

Templates for the solution of linear systems: building blocks for iterative methods, 2nd Edition, R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst, SIAM, 1994, ftp.netlib.org/templates/templates.ps.

The present implementation is inspired from IML++ 1.2 iterative method library, http://math.nist.gov/iml++.

IMPLEMENTATION

template <class Matrix, class Vector, class Vector2, class Preconditioner>
int cg (const Matrix &A, Vector &x, const Vector2 &Mb, const Preconditioner &M,


        const solver_option& sopt = solver_option())
{


  typedef typename Vector::size_type  Size;


  typedef typename Vector::float_type Real;


  std::string label = (sopt.label != "" ? sopt.label : "cg");


  Vector b = M.solve(Mb);


  Real norm2_b = dot(Mb,b);


  if (sopt.absolute_stopping || norm2_b == Real(0)) norm2_b = 1;


  Vector Mr = Mb - A*x;


  Real  norm2_r = 0;


  if (sopt.p_err) (*sopt.p_err) << "[" << label << "] #iteration residue" << std::endl;


  Vector p;


  for (sopt.n_iter = 0; sopt.n_iter <= sopt.max_iter; sopt.n_iter++) {


    Vector r = M.solve(Mr);


    Real prev_norm2_r = norm2_r;


    norm2_r = dot(Mr, r);


    sopt.residue = sqrt(norm2_r/norm2_b);


    if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;


    if (sopt.residue <= sopt.tol) return 0;


    if (sopt.n_iter == 0) {


      p = r;


    } else {


      Real beta = norm2_r/prev_norm2_r;


      p = r + beta*p;


    }


    Vector Mq = A*p;


    Real alpha = norm2_r/dot(Mq, p);


    x  += alpha*p;


    Mr -= alpha*Mq;


  }


  return 1;
}

SEE ALSO

solver_option(2)

COPYRIGHT

Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>. This is free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.