Loading...
Searching...
No Matches
cg

conjugate gradient algorithm

# Synopsis

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())

# Example

        solver_option sopt;
        sopt.max_iter = 100;
        sopt.tol = 1e-7;
        int status = cg (A, x, b, eye(), sopt);

# Description

This function solves the symmetric positive definite linear system A*x=b with the conjugate gradient method. The cg function follows the algorithm described on p. 15 in

        R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato,
        J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst,
        Templates for the solution of linear systems: building blocks for iterative methods,
        SIAM, 1994.

The fourth argument of cg is a preconditionner: here, the eye one is a do-nothing preconditionner, for simplicity. Finally, the solver_option variable sopt transmits the stopping criterion with sopt.tol and sopt.max_iter.

On return, the sopt.residue and sopt.n_iter indicate the reached residue and the number of iterations effectively performed. The return status is zero when the prescribed tolerance tol has been obtained, and non-zero otherwise. Also, the x variable contains the approximate solution. See also the solver_option for more controls upon the stopping criterion.

# Implementation

This documentation has been generated from file linalg/lib/cg.h

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

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;
}