pcg(4rheolef) rheolef-6.1 pcg(4rheolef)
NAME
pcg -- conjugate gradient algorithm.
SYNOPSIS
template <class Matrix, class Vector, class Preconditioner, class Real>
int pcg (const Matrix &A, Vector &x, const Vector &b,
const Preconditioner &M, int &max_iter, Real &tol, odiststream *p_derr=0);
EXAMPLE
The simplest call to 'pcg' has the folling form:
size_t max_iter = 100;
double tol = 1e-7;
int status = pcg(a, x, b, EYE, max_iter, tol, &derr);
DESCRIPTION
pcg 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 suc-
cessful return, output arguments have the following values:
x approximate solution to Ax = b
max_iter
the number of iterations performed before the tolerance was reached
tol the residual after the final iteration
NOTE
pcg is an iterative template routine.
pcg follows the algorithm described on p. 15 in
@quotation 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/tem-
plates.ps. @end quotation
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, class Real, class Size>
int pcg(const Matrix &A, Vector &x, const Vector2 &Mb, const Preconditioner &M,
Size &max_iter, Real &tol, odiststream *p_derr = 0, std::string label = "cg")
{
Vector b = M.solve(Mb);
Real norm2_b = dot(Mb,b);
if (norm2_b == Real(0)) norm2_b = 1;
Vector Mr = Mb - A*x;
Real norm2_r = 0;
if (p_derr) (*p_derr) << "[" << label << "] #iteration residue" << std::endl;
Vector p;
for (Size n = 0; n <= max_iter; n++) {
Vector r = M.solve(Mr);
Real prev_norm2_r = norm2_r;
norm2_r = dot(Mr, r);
if (p_derr) (*p_derr) << "[" << label << "] " << n << " " << ::sqrt(norm2_r/norm2_b) << std::endl;
if (norm2_r <= sqr(tol)*norm2_b) {
tol = ::sqrt(norm2_r/norm2_b);
max_iter = n;
return 0;
}
if (n == 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;
}
tol = ::sqrt(norm2_r/norm2_b);
return 1;
}
rheolef-6.1 rheolef-6.1 pcg(4rheolef)