Provided by: librheolef-dev_6.6-1build2_amd64 #### 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;
}
}
```