.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH continuation 4rheolef "rheolef-7.0" "rheolef-7.0" "rheolef-7.0"
.\" label: /*Class:continuation
.SH NAME
\fBcontinuation\fP -- continuation algorithm for nonlinear problems
.\" skip: @findex continuation
.\" skip: @cindex nonlinear problem
.\" skip: @cindex Newton method
.\" skip: @cindex continuation method
.SH DESCRIPTION

This function applies a continuation algorithm for the resolution of the following problem:
.\" begin_example
.Sp
.nf
       F(lambda,u) = 0
.Sp
.fi
.\" end_example
where \fBlambda\fP is a parameter and \fBu\fP is the corresponding solution.
The main idea is to follow a branch of solution
\fBu(lambda)\fP when the parameter \fBlambda\fP varies.
A simple call to the algorithm writes:
.\" begin_example
.Sp
.nf
    my_problem P;
    field uh (Vh,0);
    continuation (P, uh, &dout, &derr);
.Sp
.fi
.\" end_example
The continuation algorithm bases on the
\fBdamped_newton\fP (see damped_newton(4))
and thus requirement for the \fBmy_problem\fP class
may contains methods for the evaluation
of F (aka the residue) and its derivative:
.\" begin_example
.Sp
.nf
    class my_problem {
    public:
      typedef value_type;
      typedef float_type;
      string parameter_name() const;
      float_type parameter() const;
      void set_parameter (float_type lambda);
      value_type residue          (const value_type& uh) const;
      void update_derivative      (const value_type& uh) const;
      csr<float_type> derivative  (const value_type& uh) const;
      value_type derivative_versus_parameter (const field& uh) const;
      value_type derivative_solve      (const value_type& mrh) const;
      value_type derivative_trans_mult (const value_type& mrh) const;
      bool stop (const value_type& xh) const;
      idiststream& get (idiststream& is,       value_type& uh);
      odiststream& put (odiststream& os, const value_type& uh) const;
      float_type space_norm (const value_type& uh) const;
      float_type dual_space_norm (const value_type& mrh) const;
      float_type space_dot (const value_type& xh, const value_type& yh) const;
      float_type dual_space_dot (const value_type& mrh, const value_type& msh) const;
      value_type massify   (const value_type& uh) const;
      value_type unmassify (const value_type& mrh) const;
    };
.Sp
.fi
.\" end_example
See the example \fBcombustion.h\fP in the user's documentation for more.
The optional argument of type
class \fBcontinuation_option\fP (see continuation_option(2))
allows one to control some features of the algorithm.
.PP
.\" skip start:AUTHOR:

.\" skip start:DATE:
.\" END
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
template<class Problem>
void
continuation (
  Problem&                        F,
  typename Problem::value_type&   uh,
  odiststream*                    p_out,
  odiststream*                    p_err,
  const continuation_option& opts = continuation_option())
.Sp
.fi
.\" end_example

.\" LENGTH = 2
.SH SEE ALSO
damped_newton(4), continuation_option(2)
.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.