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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, std::ostream *p_cerr=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, &cerr);

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

       @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 Matrix, class Vector, class Preconditioner, class Real, class Size>
       int pcg(const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
               Size &max_iter, Real &tol, std::ostream *p_cerr = 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_cerr) (*p_cerr) << "[" << 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_cerr) (*p_cerr) << "[" << 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;
       }