newton(4rheolef)						    rheolef-6.1 						  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;
	     void update_derivative (const field& uh) const;
	     field derivative_solve (const field& mrh) const;
	     Float norm (const field& uh) const;
	     Float dual_norm (const field& Muh) const;
	   };

       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: n r" << std::endl;
	   for (size_t n = 0; true; n++) {
	     Field rh = P.residue(uh);
	     Float r = P.dual_norm(rh);
	     if (p_derr) *p_derr << n << " " << r << std::endl;
	     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;
	   }
       }

rheolef-6.1							    rheolef-6.1 						  newton(4rheolef)