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# newton -- Newton nonlinear algorithm

 newton(4rheolef) rheolef-6.7 newton(4rheolef)

## 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;
}
}

```
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