SYNOPSIS

template <class Matrix, class Vector, class Preconditioner, class Real>
int pminres (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 'pminres' has the folling form:

    size_t max_iter = 100;
    double tol = 1e-7;
    int status = pminres(a, x, b, EYE, max_iter, tol, &derr);

DESCRIPTION

pminres 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

max_iter

the number of iterations performed before the tolerance was reached

tol

the residual after the final iteration

NOTE

pminres follows the algorithm described in "Solution of sparse indefinite systems of linear equations", C. C. Paige and M. A. Saunders, SIAM J. Numer. Anal., 12(4), 1975. For more, see http://www.stanford.edu/group/SOL/software.html and also the PhD "Iterative methods for singular linear equations and least-squares problems", S.-C. T. Choi, Stanford University, 2006, http://www.stanford.edu/group/SOL/dissertations/sou-cheng-choi-thesis.pdf at page 60. The present implementation style is inspired from IML++ 1.2 iterative method library, http://math.nist.gov/iml++.

IMPLEMENTATION

template <class Matrix, class Vector, class Preconditioner, class Real, class Size>
int pminres(const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
  Size &max_iter, Real &tol, odiststream *p_derr = 0, std::string label = "minres")
{
  Vector b = M.solve(Mb);
  Real norm_b = sqrt(fabs(dot(Mb,b)));
  if (norm_b == Real(0.)) norm_b = 1;
  Vector Mr = Mb - A*x;
  Vector z = M.solve(Mr);
  Real beta2 = dot(Mr, z);
  Real norm_r = sqrt(fabs(beta2));
  if (p_derr) (*p_derr) << "[" << label << "] # norm_b=" <<norm_b<< std::endl;
  if (p_derr) (*p_derr) << "[" << label << "] #iteration residue/norm_b" << std::endl;
  if (p_derr) (*p_derr) << "[" << label << "] 0 " << norm_r/norm_b << std::endl;
  if (beta2 < 0 || norm_r <= tol*norm_b) {
    tol = norm_r/norm_b;
    max_iter = 0;
    dis_warning_macro ("beta2 = " << beta2 << " < 0: stop");
    return 0;
  }
  Real beta = sqrt(beta2);
  Real eta = beta;
  Vector Mv = Mr/beta;
  Vector  u =  z/beta;
  Real c_old = 1.;
  Real s_old = 0.;
  Real c = 1.;
  Real s = 0.;
  Vector u_old  (x.ownership(), 0.);
  Vector Mv_old (x.ownership(), 0.);
  Vector w      (x.ownership(), 0.);
  Vector w_old  (x.ownership(), 0.);
  Vector w_old2 (x.ownership(), 0.);
  for (Size n = 1; n <= max_iter; n++) {
    // Lanczos
    Mr = A*u;
    z = M.solve(Mr);
    Real alpha = dot(Mr, u);
    Mr = Mr - alpha*Mv - beta*Mv_old;
    z  =  z - alpha*u  - beta*u_old;
    beta2 = dot(Mr, z);
    if (beta2 < 0) {
        dis_warning_macro ("pminres: machine precision problem");
        tol = norm_r/norm_b;
        max_iter = n;
        return 2;
    }
    Real beta_old = beta;
    beta = sqrt(beta2);
    // QR factorisation
    Real c_old2 = c_old;
    Real s_old2 = s_old;
    c_old = c;
    s_old = s;
    Real rho0 = c_old*alpha - c_old2*s_old*beta_old;
    Real rho2 = s_old*alpha + c_old2*c_old*beta_old;
    Real rho1 = sqrt(sqr(rho0) + sqr(beta));
    Real rho3 = s_old2 * beta_old;
    // Givens rotation
    c = rho0 / rho1;
    s = beta / rho1;
    // update
    w_old2 = w_old;
    w_old  = w;
    w = (u - rho2*w_old - rho3*w_old2)/rho1;
    x += c*eta*w;
    eta = -s*eta;
    Mv_old = Mv;
     u_old = u;
    Mv = Mr/beta;
     u =  z/beta;
    // check residue
    norm_r *= s;
    if (p_derr) (*p_derr) << "[" << label << "] " << n << " " << norm_r/norm_b << std::endl;
    if (norm_r <= tol*max(1.0,norm_b)) {
      tol = norm_r/norm_b;
      max_iter = n;
      return 0;
    }
  }
  tol = norm_r/norm_b;
  return 1;
}