NAME

       newton -- Newton nonlinear algorithm

DESCRIPTION

       Nonlinear Newton algorithm for the resolution of the following problem:

              F(u) = 0

       A simple call to the algorithm writes:

           my_problem P;
           field uh (Vh);
           newton (P, uh, tol, max_iter);

       The  my_problem  class  may contains methods for the evaluation of F (aka residue) and its
       derivative:

           class my_problem {
           public:
             my_problem();
             field residue          (const field& uh) const;
             Float dual_space_norm  (const field& mrh) const;
             void update_derivative (const field& uh) const;
             field derivative_solve (const field& mrh) const;
           };

       The dual_space_norm returns a scalar from the weighted residual field term mrh returned by
       the  residue  function:  this  scalar is used as stopping criteria for the algorithm.  The
       update_derivative and derivative_solver members are called at  each  step  of  the  Newton
       algorithm.  See the example p_laplacian.h in the user's documentation for more.

IMPLEMENTATION

       template <class Problem, class Field>
       int newton (Problem P, Field& uh, Float& tol, size_t& max_iter, odiststream *p_derr = 0) {
           if (p_derr) *p_derr << "# Newton:" << std::endl << "# n r" << std::endl << std::flush;
           for (size_t n = 0; true; n++) {
             Field rh = P.residue(uh);
             Float r = P.dual_space_norm(rh);
             if (p_derr) *p_derr << n << " " << r << std::endl << std::flush;
             if (r <= tol) { tol = r; max_iter = n; return 0; }
             if (n == max_iter) { tol = r; return 1; }
             P.update_derivative (uh);
             Field delta_uh = P.derivative_solve (-rh);
             uh += delta_uh;
           }
       }