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:


      typedef value_type;


      value_type residue          (const value_type& uh)  const;


      Float dual_space_norm  (const value_type& mrh) const;


      void update_derivative      (const value_type& uh)  const;


      value_type derivative_solve (const value_type& 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 (const 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;


    }
}

COPYRIGHT

Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>. This is free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.

rheolef-7.0