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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
   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

   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++.

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;
   }



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