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.TH newton 4rheolef "rheolef-6.7" "rheolef-6.7" "rheolef-6.7"
.\" label: /*Class:newton
.SH NAME
\fBnewton\fP -- Newton nonlinear algorithm
.\" skip: @findex newton
.\" skip: @cindex nonlinear problem
.\" skip: @cindex Newton method
.SH DESCRIPTION

Nonlinear Newton algorithm for the resolution of the following problem:
.\" begin_example
.Sp
.nf
       F(u) = 0
.Sp
.fi
.\" end_example
A simple call to the algorithm writes:
.\" begin_example
.Sp
.nf
    my_problem P;
    field uh (Vh);
    newton (P, uh, tol, max_iter);
.Sp
.fi
.\" end_example
The \fBmy_problem\fP class may contains methods for the evaluation
of F (aka residue) and its derivative:
.\" begin_example
.Sp
.nf
    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;
    };
.Sp
.fi
.\" end_example
The \fBdual_space_norm\fP returns a scalar from the weighted residual field term
\fBmrh\fP returned by the \fBresidue\fP function:
this scalar is used as stopping criteria for the algorithm.
The \fBupdate_derivative\fP and \fBderivative_solver\fP
members are called at each step of the Newton algorithm.
See the example \fBp_laplacian.h\fP in the user's documentation for more.
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.\" skip start:DATE:
.\" skip start:METHODS:
.\" END
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
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;
    }
}
.Sp
.fi
.\" end_example