.TH "newton" 3rheolef "Sat Mar 13 2021" "Version 7.1" "rheolef" \" -*- nroff -*-
.ad l
.nh
.SH NAME
newton \- nonlinear solver (rheolef-7\&.1)
.PP
 
.SH "SYNOPSIS"
.PP
.PP
.nf
template <class Problem, class Field>
int newton (const Problem& P, Field& uh, Float& tol, size_t& max_iter, odiststream *p_derr = 0)
.fi
.PP
 
.SH "DESCRIPTION"
.PP
This function implements a generic Newton method for the resolution of the following problem: 
.PP
.nf
    F(u) = 0

.fi
.PP
 A simple call to the algorithm writes: 
.PP
.nf
    my_problem P;
    field uh (Xh);
    newton (P, uh, tol, max_iter);

.fi
.PP
 The \fCmy_problem\fP class should contain some methods for the evaluation of F, i\&.e\&. the \fCresidue\fP of the problem, and its derivative\&. The minimal requirements are: 
.PP
.nf
    class my_problem {
    public:
      typedef value_type;
      value_type residue           (const value_type& uh)  const;
      void update_derivative       (const value_type& uh)  const;
      value_type  derivative_solve (const value_type& mrh) const;
      Float dual_space_norm        (const value_type& mrh) const;
    };

.fi
.PP
 The \fCvalue_type\fP could be a \fBfield(2)\fP\&. The Newton method could also be applied when \fCvalue_type\fP is a simple \fCFloat\fP scalar\&. Conversely, it supports multi-field extensions\&.
.PP
The \fCupdate_derivative\fP and \fCderivative_solver\fP members are called at each step of the Newton algorithm\&.
.PP
The \fCdual_space_norm\fP member function returns a scalar from the weighted residual field term \fCmrh\fP returned by the \fCresidue\fP function: this scalar is used as the stopping criterion of the algorithm\&.
.SH "EXAMPLE"
.PP
See the \fBp_laplacian_newton\&.cc\fP example and the \fBusersguide\fP for more\&.
.SH "IMPLEMENTATION"
.PP
This documentation has been generated from file main/lib/newton\&.h 
.SH AUTHOR
Pierre  Saramito  <Pierre.Saramito@imag.fr>
.SH 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.