NAME¶

pgmres -- generalized minimum residual method

SYNOPSIS¶

   template <class Matrix, class Vector, class Preconditioner,
     class SmallMatrix, class SmallVector, class Real, class Size>
   int pgmres (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
     SmallMatrix &H, const SmallVector& dummy,
     Size m, Size &max_iter, Real &tol);
 

EXAMPLE¶

The simplest call to pgmres has the folling form:

        double tol = 1e-7;
        size_t max_iter = 100;
        size_t m = 6;
        boost::numeric::ublas::matrix<double> H(m+1,m+1);
        vec<double,sequential> dummy;
        int status = pgmres (a, x, b, ic0(a), H, dummy, m, max_iter, tol);
 

DESCRIPTION¶

pgmres solves the unsymmetric linear system Ax = b using the generalized minimum residual 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 In addition, M specifies a preconditioner, H specifies a matrix to hold the coefficients of the upper Hessenberg matrix constructed by the pgmres iterations, m specifies the number of iterations for each restart.

pgmres requires two matrices as input, A and H. The matrix A, which will typically be a sparse matrix) corresponds to the matrix in the linear system Ax=b. The matrix H, which will be typically a dense matrix, corresponds to the upper Hessenberg matrix H that is constructed during the pgmres iterations. Within pgmres, H is used in a different way than A, so its class must supply different functionality. That is, A is only accessed though its matrix-vector and transpose-matrix-vector multiplication functions. On the other hand, pgmres solves a dense upper triangular linear system of equations on H. Therefore, the class to which H belongs must provide H(i,j) operator for element acess.

NOTE¶

It is important to remember that we use the convention that indices are 0-based. That is H(0,0) is the first component of the matrix H. Also, the type of the matrix must be compatible with the type of single vector entry. That is, operations such as H(i,j)*x(j) must be able to be carried out. pgmres is an iterative template routine. pgmres follows the algorithm described on p. 20 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/templates.ps. @end quotation The present implementation is inspired from IML++ 1.2 iterative method library, http://math.nist.gov/iml++.

IMPLEMENTATION¶

template <class SmallMatrix, class Vector, class SmallVector, class Size>
void
Update(Vector &x, Size k, SmallMatrix &h, SmallVector &s, Vector v[])
{
  SmallVector y = s;
  // Back solve:
  for (int i = k; i >= 0; i--) {
    y(i) /= h(i,i);
    for (int j = i - 1; j >= 0; j--)
      y(j) -= h(j,i) * y(i);
  }
  for (Size j = 0; j <= k; j++) {
    x += v[j] * y(j);
  }
}
template<class Real>
void GeneratePlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn)
{
  if (dy == Real(0)) {
    cs = 1.0;
    sn = 0.0;
  } else if (abs(dy) > abs(dx)) {
    Real temp = dx / dy;
    sn = 1.0 / ::sqrt( 1.0 + temp*temp );
    cs = temp * sn;
  } else {
    Real temp = dy / dx;
    cs = 1.0 / ::sqrt( 1.0 + temp*temp );
    sn = temp * cs;
  }
}
template<class Real>
void ApplyPlaneRotation(Real &dx, Real &dy, Real &cs, Real &sn)
{
  Real temp  =  cs * dx + sn * dy;
  dy = -sn * dx + cs * dy;
  dx = temp;
}
template <class Matrix, class Vector, class Preconditioner,
          class SmallMatrix, class SmallVector, class Real, class Size>
int
pgmres (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
      SmallMatrix &H, const SmallVector&, const Size &m, Size &max_iter, Real &tol)
{
  odiststream* p_derr = &derr;
  std::string label = "pgmres";
  if (p_derr) (*p_derr) << "[" << label << "] #iteration residue" << std::endl;
  Vector w;
  SmallVector s(m+1), cs(m+1), sn(m+1);
  Size i;
  Size j = 1;
  Size k;
  Real resid;
  Real normb = norm(M.solve(b));
  Vector r = M.solve(b - A * x);
  Real beta = norm(r);
  if (normb == Real(0)) {
    normb = 1;
  }
  resid = norm(r) / normb;
  if (resid  <= tol) {
    tol = resid;
    max_iter = 0;
    return 0;
  }
  Vector *v = new Vector[m+1];
  while (j <= max_iter) {
    v[0] = r * (1.0 / beta);    // ??? r / beta
    s = 0.0;
    s(0) = beta;
    for (i = 0; i < m && j <= max_iter; i++, j++) {
      w = M.solve(A * v[i]);
      for (k = 0; k <= i; k++) {
        H(k, i) = dot(w, v[k]);
        w -= H(k, i) * v[k];
      }
      H(i+1, i) = norm(w);
      v[i+1] = w * (1.0 / H(i+1, i)); // ??? w / H(i+1, i)
      for (k = 0; k < i; k++) {
        ApplyPlaneRotation(H(k,i), H(k+1,i), cs(k), sn(k));
      }
      GeneratePlaneRotation(H(i,i), H(i+1,i), cs(i), sn(i));
      ApplyPlaneRotation(H(i,i), H(i+1,i), cs(i), sn(i));
      ApplyPlaneRotation(s(i), s(i+1), cs(i), sn(i));
      resid = abs(s(i+1)) / normb;
      if (p_derr) (*p_derr) << "[" << label << "] " << j << " " << resid << std::endl;
      if (resid  < tol) {
        Update(x, i, H, s, v);
        tol = resid;
        max_iter = j;
        delete [] v;
        return 0;
      }
    }
    Update(x, m - 1, H, s, v);
    r = M.solve(b - A * x);
    beta = norm(r);
    resid = beta / normb;
    if (p_derr) (*p_derr) << "[" << label << "] " << j << " " << resid << std::endl;
    if (resid < tol) {
      tol = resid;
      max_iter = j;
      delete [] v;
      return 0;
    }
  }
  tol = resid;
  delete [] v;
  return 1;
}