NAME

       puzawa -- Uzawa algorithm.

SYNOPSIS

           template <class Matrix, class Vector, class Preconditioner, class Real>
           int puzawa (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
             int &max_iter, Real &tol, const Real& rho, odiststream *p_derr=0);

EXAMPLE

       The simplest call to 'puzawa' has the folling form:

           size_t max_iter = 100;
           double tol = 1e-7;
           int status = puzawa(A, x, b, EYE, max_iter, tol, 1.0, &derr);

DESCRIPTION

       puzawa  solves  the  linear  system  A*x=b  using  the Uzawa method. The Uzawa method is a
       descent method in the direction opposite to the  gradient, with  a  constant  step  length
       'rho'. The convergence is assured when the step length 'rho' is small enough.  If matrix A
       is symmetric positive definite, please uses 'pcg' that computes automatically the  optimal
       descdnt step length at each iteration.

       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

IMPLEMENTATION

       template < class Matrix, class Vector, class Preconditioner, class Real, class Size>
       int puzawa(const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
           Size &max_iter, Real &tol, const Real& rho,
           odiststream *p_derr, std::string label)
       {
           Vector b = M.solve(Mb);
           Real norm2_b = dot(Mb,b);
           Real norm2_r = norm2_b;
           if (norm2_b == Real(0)) norm2_b = 1;
           if (p_derr) (*p_derr) << "[" << label << "] #iteration residue" << std::endl;
           for (Size n = 0; n <= max_iter; n++) {
               Vector Mr = A*x - Mb;
               Vector r = M.solve(Mr);
               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;
               }
               x  -= rho*r;
           }
           tol = sqrt(norm2_r/norm2_b);
           return 1;
       }