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 [AT] 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.