pcg(4) conjugate gradient algorithm.


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);


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);


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 successful return, output arguments have the following values:

approximate solution to Ax = b

the number of iterations performed before the tolerance was reached

the residual after the final iteration


pcg is an iterative template routine.

pcg 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++.


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;