.. index::
   single: solving nonlinear systems of equations
   single: nonlinear systems of equations, solution of
   single: systems of equations, nonlinear

*****************************
Multidimensional Root-Finding
*****************************

.. include:: include.rst

This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with :math:`n` equations in :math:`n`
unknowns).  The library provides low level components for a variety of
iterative solvers and convergence tests.  These can be combined by the
user to achieve the desired solution, with full access to the
intermediate steps of the iteration.  Each class of methods uses the
same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.  The solvers are based on the original Fortran
library |minpack|.

The header file :file:`gsl_multiroots.h` contains prototypes for the
multidimensional root finding functions and related declarations.

.. index::
   single: multidimensional root finding, overview

Overview
========

The problem of multidimensional root finding requires the simultaneous
solution of :math:`n` equations, :math:`f_i`, in :math:`n` variables,
:math:`x_i`,

.. only:: not texinfo

   .. math:: f_i (x_1, \dots, x_n) = 0 \qquad\hbox{for}~i = 1 \dots n.

.. only:: texinfo

   ::

      f_i (x_1, ..., x_n) = 0    for i = 1 ... n.

In general there are no bracketing methods available for :math:`n`
dimensional systems, and no way of knowing whether any solutions
exist.  All algorithms proceed from an initial guess using a variant of
the Newton iteration,

.. only:: not texinfo

   .. math:: x \to x' = x - J^{-1} f(x)

.. only:: texinfo

   ::

      x -> x' = x - J^{-1} f(x)

where :math:`x`, :math:`f` are vector quantities and :math:`J` is the
Jacobian matrix :math:`J_{ij} = \partial f_i / \partial x_j`.
Additional strategies can be used to enlarge the region of
convergence.  These include requiring a decrease in the norm :math:`|f|` on
each step proposed by Newton's method, or taking steepest-descent steps in
the direction of the negative gradient of :math:`|f|`.

Several root-finding algorithms are available within a single framework.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,

* initialize solver state, :data:`s`, for algorithm :data:`T`
* update :data:`s` using the iteration :data:`T`
* test :data:`s` for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
:math:`n^2` terms of the matrix becomes too expensive.  For these reasons
the algorithms provided by the library are divided into two classes according
to whether the derivatives are available or not.

.. index::
   single: Jacobian matrix, root finding

The state for solvers with an analytic Jacobian matrix is held in a
:type:`gsl_multiroot_fdfsolver` struct.  The updating procedure requires
both the function and its derivatives to be supplied by the user.

The state for solvers which do not use an analytic Jacobian matrix is
held in a :type:`gsl_multiroot_fsolver` struct.  The updating procedure
uses only function evaluations (not derivatives).  The algorithms
estimate the matrix :math:`J` or :math:`J^{-1}`
by approximate methods.

Initializing the Solver
=======================

The following functions initialize a multidimensional solver, either
with or without derivatives.  The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.

.. type:: gsl_multiroot_fsolver

   This is a workspace for multidimensional root-finding without derivatives.

.. type:: gsl_multiroot_fdfsolver

   This is a workspace for multidimensional root-finding with derivatives.

.. function:: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type * T, size_t n)

   This function returns a pointer to a newly allocated instance of a
   solver of type :data:`T` for a system of :data:`n` dimensions.
   For example, the following code creates an instance of a hybrid solver, 
   to solve a 3-dimensional system of equations::

      const gsl_multiroot_fsolver_type * T = gsl_multiroot_fsolver_hybrid;
      gsl_multiroot_fsolver * s = gsl_multiroot_fsolver_alloc (T, 3);

   If there is insufficient memory to create the solver then the function
   returns a null pointer and the error handler is invoked with an error
   code of :macro:`GSL_ENOMEM`.

.. function:: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc (const gsl_multiroot_fdfsolver_type * T, size_t n)

   This function returns a pointer to a newly allocated instance of a
   derivative solver of type :data:`T` for a system of :data:`n` dimensions.
   For example, the following code creates an instance of a Newton-Raphson solver,
   for a 2-dimensional system of equations::

      const gsl_multiroot_fdfsolver_type * T = gsl_multiroot_fdfsolver_newton;
      gsl_multiroot_fdfsolver * s = gsl_multiroot_fdfsolver_alloc (T, 2);

   If there is insufficient memory to create the solver then the function
   returns a null pointer and the error handler is invoked with an error
   code of :macro:`GSL_ENOMEM`.

.. function:: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * s, gsl_multiroot_function * f, const gsl_vector * x)
              int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver * s, gsl_multiroot_function_fdf * fdf, const gsl_vector * x)

   These functions set, or reset, an existing solver :data:`s` to use the
   function :data:`f` or function and derivative :data:`fdf`, and the initial
   guess :data:`x`.  Note that the initial position is copied from :data:`x`, this
   argument is not modified by subsequent iterations.

.. function:: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * s)
              void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver * s)

   These functions free all the memory associated with the solver :data:`s`.

.. function:: const char * gsl_multiroot_fsolver_name (const gsl_multiroot_fsolver * s)
              const char * gsl_multiroot_fdfsolver_name (const gsl_multiroot_fdfsolver * s)

   These functions return a pointer to the name of the solver.  For example::

      printf ("s is a '%s' solver\n", gsl_multiroot_fdfsolver_name (s));

   would print something like :code:`s is a 'newton' solver`.

.. index::
   single: multidimensional root finding, providing a function to solve

Providing the function to solve
===============================

You must provide :math:`n` functions of :math:`n` variables for the root
finders to operate on.  In order to allow for general parameters the
functions are defined by the following data types:

.. type:: gsl_multiroot_function 

   This data type defines a general system of functions with parameters.

   :code:`int (* f) (const gsl_vector * x, void * params, gsl_vector * f)`

      this function should store the vector result
      :math:`f(x,params)` in :data:`f` for argument :data:`x` and parameters :data:`params`,
      returning an appropriate error code if the function cannot be computed.

   :code:`size_t n`

      the dimension of the system, i.e. the number of components of the
      vectors :data:`x` and :data:`f`.

   :code:`void * params`

      a pointer to the parameters of the function.

Here is an example using Powell's test function,

.. only:: not texinfo

   .. math::

      f_1(x) &= A x_0 x_1 - 1 \\
      f_2(x) &= \exp(-x_0) + \exp(-x_1) - (1 + 1/A)

.. only:: texinfo

   ::

      f_1(x) = A x_0 x_1 - 1,
      f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)

with :math:`A = 10^4`.  The following code defines a
:type:`gsl_multiroot_function` system :code:`F` which you could pass to a
solver::

  struct powell_params { double A; };

  int
  powell (gsl_vector * x, void * p, gsl_vector * f) {
     struct powell_params * params 
       = (struct powell_params *)p;
     const double A = (params->A);
     const double x0 = gsl_vector_get(x,0);
     const double x1 = gsl_vector_get(x,1);

     gsl_vector_set (f, 0, A * x0 * x1 - 1);
     gsl_vector_set (f, 1, (exp(-x0) + exp(-x1) 
                            - (1.0 + 1.0/A)));
     return GSL_SUCCESS
  }

  gsl_multiroot_function F;
  struct powell_params params = { 10000.0 };

  F.f = &powell;
  F.n = 2;
  F.params = &params;

.. type:: gsl_multiroot_function_fdf

   This data type defines a general system of functions with parameters and
   the corresponding Jacobian matrix of derivatives,

   :code:`int (* f) (const gsl_vector * x, void * params, gsl_vector * f)`

      this function should store the vector result
      :math:`f(x,params)` in :data:`f` for argument :data:`x` and parameters :data:`params`,
      returning an appropriate error code if the function cannot be computed.

   :code:`int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)`

      this function should store the :data:`n`-by-:data:`n` matrix result

      .. only:: not texinfo

         .. math:: J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j

      .. only:: texinfo

         ::

            J_ij = d f_i(x,params) / d x_j
            
      in :data:`J` for argument :data:`x` 
      and parameters :data:`params`, returning an appropriate error code if the
      function cannot be computed.

   :code:`int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f, gsl_matrix * J)`

      This function should set the values of the :data:`f` and :data:`J` as above,
      for arguments :data:`x` and parameters :data:`params`.  This function
      provides an optimization of the separate functions for :math:`f(x)` and
      :math:`J(x)`---it is always faster to compute the function and its
      derivative at the same time.

   :code:`size_t n`

      the dimension of the system, i.e. the number of components of the
      vectors :data:`x` and :data:`f`.

   :code:`void * params`

      a pointer to the parameters of the function.

The example of Powell's test function defined above can be extended to
include analytic derivatives using the following code::

  int
  powell_df (gsl_vector * x, void * p, gsl_matrix * J) 
  {
     struct powell_params * params 
       = (struct powell_params *)p;
     const double A = (params->A);
     const double x0 = gsl_vector_get(x,0);
     const double x1 = gsl_vector_get(x,1);
     gsl_matrix_set (J, 0, 0, A * x1);
     gsl_matrix_set (J, 0, 1, A * x0);
     gsl_matrix_set (J, 1, 0, -exp(-x0));
     gsl_matrix_set (J, 1, 1, -exp(-x1));
     return GSL_SUCCESS
  }

  int
  powell_fdf (gsl_vector * x, void * p, 
              gsl_matrix * f, gsl_matrix * J) {
     struct powell_params * params 
       = (struct powell_params *)p;
     const double A = (params->A);
     const double x0 = gsl_vector_get(x,0);
     const double x1 = gsl_vector_get(x,1);

     const double u0 = exp(-x0);
     const double u1 = exp(-x1);

     gsl_vector_set (f, 0, A * x0 * x1 - 1);
     gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));

     gsl_matrix_set (J, 0, 0, A * x1);
     gsl_matrix_set (J, 0, 1, A * x0);
     gsl_matrix_set (J, 1, 0, -u0);
     gsl_matrix_set (J, 1, 1, -u1);
     return GSL_SUCCESS
  }

  gsl_multiroot_function_fdf FDF;

  FDF.f = &powell_f;
  FDF.df = &powell_df;
  FDF.fdf = &powell_fdf;
  FDF.n = 2;
  FDF.params = 0;

Note that the function :code:`powell_fdf` is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.

Iteration
=========

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

.. function:: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver * s)
              int gsl_multiroot_fdfsolver_iterate (gsl_multiroot_fdfsolver * s)

   These functions perform a single iteration of the solver :data:`s`.  If the
   iteration encounters an unexpected problem then an error code will be
   returned,

   :macro:`GSL_EBADFUNC`

      the iteration encountered a singular point where the function or its
      derivative evaluated to :code:`Inf` or :code:`NaN`.

   :macro:`GSL_ENOPROG`

      the iteration is not making any progress, preventing the algorithm from
      continuing.

The solver maintains a current best estimate of the root :code:`s->x`
and its function value :code:`s->f` at all times.  This information can
be accessed with the following auxiliary functions,

.. function:: gsl_vector * gsl_multiroot_fsolver_root (const gsl_multiroot_fsolver * s)
              gsl_vector * gsl_multiroot_fdfsolver_root (const gsl_multiroot_fdfsolver * s)

   These functions return the current estimate of the root for the solver :data:`s`, given by :code:`s->x`.

.. function:: gsl_vector * gsl_multiroot_fsolver_f (const gsl_multiroot_fsolver * s)
              gsl_vector * gsl_multiroot_fdfsolver_f (const gsl_multiroot_fdfsolver * s)

   These functions return the function value :math:`f(x)` at the current
   estimate of the root for the solver :data:`s`, given by :code:`s->f`.

.. function:: gsl_vector * gsl_multiroot_fsolver_dx (const gsl_multiroot_fsolver * s)
              gsl_vector * gsl_multiroot_fdfsolver_dx (const gsl_multiroot_fdfsolver * s)

   These functions return the last step :math:`dx` taken by the solver
   :data:`s`, given by :code:`s->dx`.

.. index::
   single: root finding, stopping parameters

Search Stopping Parameters
==========================

A root finding procedure should stop when one of the following conditions is
true:

* A multidimensional root has been found to within the user-specified precision.
* A user-specified maximum number of iterations has been reached.
* An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

.. function:: int gsl_multiroot_test_delta (const gsl_vector * dx, const gsl_vector * x, double epsabs, double epsrel)

   This function tests for the convergence of the sequence by comparing the
   last step :data:`dx` with the absolute error :data:`epsabs` and relative
   error :data:`epsrel` to the current position :data:`x`.  The test returns
   :macro:`GSL_SUCCESS` if the following condition is achieved,

   .. only:: not texinfo

      .. math:: |dx_i| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_i|

   .. only:: texinfo

      ::

         |dx_i| < epsabs + epsrel |x_i|

   for each component of :data:`x` and returns :macro:`GSL_CONTINUE` otherwise.

.. index::
   single: residual, in nonlinear systems of equations

.. function:: int gsl_multiroot_test_residual (const gsl_vector * f, double epsabs)

   This function tests the residual value :data:`f` against the absolute
   error bound :data:`epsabs`.  The test returns :macro:`GSL_SUCCESS` if the
   following condition is achieved,

   .. only:: not texinfo

      .. math:: \sum_i |f_i| < \hbox{\it epsabs}

   .. only:: texinfo

      ::

         \sum_i |f_i| < epsabs

   and returns :macro:`GSL_CONTINUE` otherwise.  This criterion is suitable
   for situations where the precise location of the root, :math:`x`, is
   unimportant provided a value can be found where the residual is small
   enough.

Algorithms using Derivatives
============================

The root finding algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the root, but there is no absolute guarantee of
convergence---the function must be suitable for this technique and the
initial guess must be sufficiently close to the root for it to work.
When the conditions are satisfied then convergence is quadratic.

.. type:: gsl_multiroot_fdfsolver_type

   The following are available algorithms for minimizing functions using
   derivatives.

   .. index:: HYBRID algorithms for nonlinear systems

   .. index::
      single: HYBRIDSJ algorithm
      single: MINPACK, minimization algorithms

   .. var:: gsl_multiroot_fdfsolver_type * gsl_multiroot_fdfsolver_hybridsj

      This is a modified version of Powell's Hybrid method as implemented in
      the HYBRJ algorithm in |minpack|.  Minpack was written by Jorge
      J. |More|, Burton S. Garbow and Kenneth E. Hillstrom.  The Hybrid
      algorithm retains the fast convergence of Newton's method but will also
      reduce the residual when Newton's method is unreliable. 

      The algorithm uses a generalized trust region to keep each step under
      control.  In order to be accepted a proposed new position :math:`x'` must
      satisfy the condition :math:`|D (x' - x)| < \delta`, where :math:`D` is a
      diagonal scaling matrix and :math:`\delta` is the size of the trust
      region.  The components of :math:`D` are computed internally, using the
      column norms of the Jacobian to estimate the sensitivity of the residual
      to each component of :math:`x`.  This improves the behavior of the
      algorithm for badly scaled functions.

      On each iteration the algorithm first determines the standard Newton
      step by solving the system :math:`J dx = - f`.  If this step falls inside
      the trust region it is used as a trial step in the next stage.  If not,
      the algorithm uses the linear combination of the Newton and gradient
      directions which is predicted to minimize the norm of the function while
      staying inside the trust region,

      .. math:: dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2

      This combination of Newton and gradient directions is referred to as a
      *dogleg step*.

      The proposed step is now tested by evaluating the function at the
      resulting point, :math:`x'`.  If the step reduces the norm of the function
      sufficiently then it is accepted and size of the trust region is
      increased.  If the proposed step fails to improve the solution then the
      size of the trust region is decreased and another trial step is
      computed.

      The speed of the algorithm is increased by computing the changes to the
      Jacobian approximately, using a rank-1 update.  If two successive
      attempts fail to reduce the residual then the full Jacobian is
      recomputed.  The algorithm also monitors the progress of the solution
      and returns an error if several steps fail to make any improvement,

      :macro:`GSL_ENOPROG`

         the iteration is not making any progress, preventing the algorithm from
         continuing.

      :macro:`GSL_ENOPROGJ`

         re-evaluations of the Jacobian indicate that the iteration is not
         making any progress, preventing the algorithm from continuing.

   .. index:: HYBRIDJ algorithm

   .. var:: gsl_multiroot_fdfsolver_type * gsl_multiroot_fdfsolver_hybridj

      This algorithm is an unscaled version of HYBRIDSJ.  The steps are
      controlled by a spherical trust region :math:`|x' - x| < \delta`, instead
      of a generalized region.  This can be useful if the generalized region
      estimated by HYBRIDSJ is inappropriate.

   .. index:: Newton's method for systems of nonlinear equations

   .. var:: gsl_multiroot_fdfsolver_type * gsl_multiroot_fdfsolver_newton

      Newton's Method is the standard root-polishing algorithm.  The algorithm
      begins with an initial guess for the location of the solution.  On each
      iteration a linear approximation to the function :math:`F` is used to
      estimate the step which will zero all the components of the residual.
      The iteration is defined by the following sequence,

      .. only:: not texinfo

         .. math:: x \to x' = x - J^{-1} f(x)

      .. only:: texinfo

         ::

            x -> x' = x - J^{-1} f(x)

      where the Jacobian matrix :math:`J` is computed from the derivative
      functions provided by :data:`f`.  The step :math:`dx` is obtained by solving
      the linear system,

      .. math:: J dx = - f(x)

      using LU decomposition.  If the Jacobian matrix is singular, an error
      code of :macro:`GSL_EDOM` is returned.

   .. index::
      single: Modified Newton's method for nonlinear systems
      single: Newton algorithm, globally convergent

   .. var:: gsl_multiroot_fdfsolver_type * gsl_multiroot_fdfsolver_gnewton

      This is a modified version of Newton's method which attempts to improve
      global convergence by requiring every step to reduce the Euclidean norm
      of the residual, :math:`|f(x)|`.  If the Newton step leads to an increase
      in the norm then a reduced step of relative size,

      .. math:: t = (\sqrt{1 + 6 r} - 1) / (3 r)

      is proposed, with :math:`r` being the ratio of norms
      :math:`|f(x')|^2/|f(x)|^2`.  This procedure is repeated until a suitable step
      size is found. 

Algorithms without Derivatives
==============================

The algorithms described in this section do not require any derivative
information to be supplied by the user.  Any derivatives needed are
approximated by finite differences.  Note that if the
finite-differencing step size chosen by these routines is inappropriate,
an explicit user-supplied numerical derivative can always be used with
the algorithms described in the previous section.

.. type:: gsl_multiroot_fsolver_type

   The following are available algorithms for minimizing functions without
   derivatives.

   .. index::
      single: HYBRIDS algorithm, scaled without derivatives

   .. var:: gsl_multiroot_fsolver_type * gsl_multiroot_fsolver_hybrids

      This is a version of the Hybrid algorithm which replaces calls to the
      Jacobian function by its finite difference approximation.  The finite
      difference approximation is computed using :func:`gsl_multiroots_fdjac`
      with a relative step size of :macro:`GSL_SQRT_DBL_EPSILON`.  Note that
      this step size will not be suitable for all problems.

   .. index::
      single: HYBRID algorithm, unscaled without derivatives

   .. var:: gsl_multiroot_fsolver_type * gsl_multiroot_fsolver_hybrid

      This is a finite difference version of the Hybrid algorithm without
      internal scaling.

   .. index::
      single: Discrete Newton algorithm for multidimensional roots
      single: Newton algorithm, discrete

   .. var:: gsl_multiroot_fsolver_type * gsl_multiroot_fsolver_dnewton

      The *discrete Newton algorithm* is the simplest method of solving a
      multidimensional system.  It uses the Newton iteration

      .. only:: not texinfo

         .. math:: x \to x - J^{-1} f(x)

      .. only:: texinfo

         ::

            x -> x - J^{-1} f(x)

      where the Jacobian matrix :math:`J` is approximated by taking finite
      differences of the function :data:`f`.  The approximation scheme used by
      this implementation is,

      .. math:: J_{ij} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j

      where :math:`\delta_j` is a step of size :math:`\sqrt\epsilon |x_j|` with
      :math:`\epsilon` being the machine precision 
      (:math:`\epsilon \approx 2.22 \times 10^{-16}`).
      The order of convergence of Newton's algorithm is quadratic, but the
      finite differences require :math:`n^2` function evaluations on each
      iteration.  The algorithm may become unstable if the finite differences
      are not a good approximation to the true derivatives.

   .. index::
      single: Broyden algorithm for multidimensional roots
      single: multidimensional root finding, Broyden algorithm

   .. var:: gsl_multiroot_fsolver_type * gsl_multiroot_fsolver_broyden

      The *Broyden algorithm* is a version of the discrete Newton
      algorithm which attempts to avoids the expensive update of the Jacobian
      matrix on each iteration.  The changes to the Jacobian are also
      approximated, using a rank-1 update,

      .. math:: J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df

      where the vectors :math:`dx` and :math:`df` are the changes in :math:`x`
      and :math:`f`.  On the first iteration the inverse Jacobian is estimated
      using finite differences, as in the discrete Newton algorithm.
 
      This approximation gives a fast update but is unreliable if the changes
      are not small, and the estimate of the inverse Jacobian becomes worse as
      time passes.  The algorithm has a tendency to become unstable unless it
      starts close to the root.  The Jacobian is refreshed if this instability
      is detected (consult the source for details).

      This algorithm is included only for demonstration purposes, and is not
      recommended for serious use.

Examples
========

The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms.  This first example
demonstrates the HYBRIDS scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of equations,

.. only:: not texinfo

   .. math::

      f_1 (x, y) &= a (1 - x) \\
      f_2 (x, y) &= b (y - x^2)

.. only:: texinfo

   ::

      f_1 (x, y) = a (1 - x)
      f_2 (x, y) = b (y - x^2)

with :math:`a = 1, b = 10`. The solution of this system lies at
:math:`(x,y) = (1,1)` in a narrow valley.

The first stage of the program is to define the system of equations::

  #include <stdlib.h>
  #include <stdio.h>
  #include <gsl/gsl_vector.h>
  #include <gsl/gsl_multiroots.h>

  struct rparams
    {
      double a;
      double b;
    };

  int
  rosenbrock_f (const gsl_vector * x, void *params, 
                gsl_vector * f)
  {
    double a = ((struct rparams *) params)->a;
    double b = ((struct rparams *) params)->b;

    const double x0 = gsl_vector_get (x, 0);
    const double x1 = gsl_vector_get (x, 1);

    const double y0 = a * (1 - x0);
    const double y1 = b * (x1 - x0 * x0);

    gsl_vector_set (f, 0, y0);
    gsl_vector_set (f, 1, y1);

    return GSL_SUCCESS;
  }

The main program begins by creating the function object :code:`f`, with
the arguments :code:`(x,y)` and parameters :code:`(a,b)`. The solver
:code:`s` is initialized to use this function, with the :data:`gsl_multiroot_fsolver_hybrids`
method::

  int
  main (void)
  {
    const gsl_multiroot_fsolver_type *T;
    gsl_multiroot_fsolver *s;

    int status;
    size_t i, iter = 0;

    const size_t n = 2;
    struct rparams p = {1.0, 10.0};
    gsl_multiroot_function f = {&rosenbrock_f, n, &p};

    double x_init[2] = {-10.0, -5.0};
    gsl_vector *x = gsl_vector_alloc (n);

    gsl_vector_set (x, 0, x_init[0]);
    gsl_vector_set (x, 1, x_init[1]);

    T = gsl_multiroot_fsolver_hybrids;
    s = gsl_multiroot_fsolver_alloc (T, 2);
    gsl_multiroot_fsolver_set (s, &f, x);

    print_state (iter, s);

    do
      {
        iter++;
        status = gsl_multiroot_fsolver_iterate (s);

        print_state (iter, s);

        if (status)   /* check if solver is stuck */
          break;

        status = 
          gsl_multiroot_test_residual (s->f, 1e-7);
      }
    while (status == GSL_CONTINUE && iter < 1000);

    printf ("status = %s\n", gsl_strerror (status));

    gsl_multiroot_fsolver_free (s);
    gsl_vector_free (x);
    return 0;
  }

Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck.  If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.

The intermediate state of the solution is displayed by the following
function.  The solver state contains the vector :code:`s->x` which is the
current position, and the vector :code:`s->f` with corresponding function
values::

  int
  print_state (size_t iter, gsl_multiroot_fsolver * s)
  {
    printf ("iter = %3u x = % .3f % .3f "
            "f(x) = % .3e % .3e\n",
            iter,
            gsl_vector_get (s->x, 0), 
            gsl_vector_get (s->x, 1),
            gsl_vector_get (s->f, 0), 
            gsl_vector_get (s->f, 1));
  }

Here are the results of running the program. The algorithm is started at
:math:`(-10,-5)` far from the solution.  Since the solution is hidden in
a narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual. Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid::

  iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
  iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
  iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
  iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
  iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
  iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
  iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
  iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
  iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
  iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
  iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
  iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
  status = success

Note that the algorithm does not update the location on every
iteration. Some iterations are used to adjust the trust-region
parameter, after trying a step which was found to be divergent, or to
recompute the Jacobian, when poor convergence behavior is detected.

The next example program adds derivative information, in order to
accelerate the solution. There are two derivative functions
:code:`rosenbrock_df` and :code:`rosenbrock_fdf`. The latter computes both
the function and its derivative simultaneously. This allows the
optimization of any common terms.  For simplicity we substitute calls to
the separate :code:`f` and :code:`df` functions at this point in the code
below::

  int
  rosenbrock_df (const gsl_vector * x, void *params, 
                 gsl_matrix * J)
  {
    const double a = ((struct rparams *) params)->a;
    const double b = ((struct rparams *) params)->b;

    const double x0 = gsl_vector_get (x, 0);

    const double df00 = -a;
    const double df01 = 0;
    const double df10 = -2 * b  * x0;
    const double df11 = b;

    gsl_matrix_set (J, 0, 0, df00);
    gsl_matrix_set (J, 0, 1, df01);
    gsl_matrix_set (J, 1, 0, df10);
    gsl_matrix_set (J, 1, 1, df11);

    return GSL_SUCCESS;
  }

  int
  rosenbrock_fdf (const gsl_vector * x, void *params,
                  gsl_vector * f, gsl_matrix * J)
  {
    rosenbrock_f (x, params, f);
    rosenbrock_df (x, params, J);

    return GSL_SUCCESS;
  }

The main program now makes calls to the corresponding :code:`fdfsolver`
versions of the functions::

  int
  main (void)
  {
    const gsl_multiroot_fdfsolver_type *T;
    gsl_multiroot_fdfsolver *s;

    int status;
    size_t i, iter = 0;

    const size_t n = 2;
    struct rparams p = {1.0, 10.0};
    gsl_multiroot_function_fdf f = {&rosenbrock_f, 
                                    &rosenbrock_df, 
                                    &rosenbrock_fdf, 
                                    n, &p};

    double x_init[2] = {-10.0, -5.0};
    gsl_vector *x = gsl_vector_alloc (n);

    gsl_vector_set (x, 0, x_init[0]);
    gsl_vector_set (x, 1, x_init[1]);

    T = gsl_multiroot_fdfsolver_gnewton;
    s = gsl_multiroot_fdfsolver_alloc (T, n);
    gsl_multiroot_fdfsolver_set (s, &f, x);

    print_state (iter, s);

    do
      {
        iter++;

        status = gsl_multiroot_fdfsolver_iterate (s);

        print_state (iter, s);

        if (status)
          break;

        status = gsl_multiroot_test_residual (s->f, 1e-7);
      }
    while (status == GSL_CONTINUE && iter < 1000);

    printf ("status = %s\n", gsl_strerror (status));

    gsl_multiroot_fdfsolver_free (s);
    gsl_vector_free (x);
    return 0;
  }

The addition of derivative information to the :data:`gsl_multiroot_fsolver_hybrids` solver does
not make any significant difference to its behavior, since it able to
approximate the Jacobian numerically with sufficient accuracy.  To
illustrate the behavior of a different derivative solver we switch to
:data:`gsl_multiroot_fdfsolver_gnewton`. This is a traditional Newton solver with the constraint
that it scales back its step if the full step would lead "uphill". Here
is the output for the :data:`gsl_multiroot_fdfsolver_gnewton` algorithm::

  iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
  iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
  iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
  iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
  status = success

The convergence is much more rapid, but takes a wide excursion out to
the point :math:`(-4.23,-65.3)`. This could cause the algorithm to go
astray in a realistic application.  The hybrid algorithm follows the
downhill path to the solution more reliably.

References and Further Reading
==============================

The original version of the Hybrid method is described in the following
articles by Powell,

* M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p
  87--114) and "A Fortran Subroutine for Solving systems of Nonlinear
  Algebraic Equations" (Chap 7, p 115--161), in *Numerical Methods for
  Nonlinear Algebraic Equations*, P. Rabinowitz, editor.  Gordon and
  Breach, 1970.

The following papers are also relevant to the algorithms described in
this section,

* J.J. |More|, M.Y. Cosnard, "Numerical Solution of Nonlinear Equations",
  *ACM Transactions on Mathematical Software*, Vol 5, No 1, (1979), p 64--85

* C.G. Broyden, "A Class of Methods for Solving Nonlinear
  Simultaneous Equations", *Mathematics of Computation*, Vol 19 (1965),
  p 577--593

* J.J. |More|, B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
  Optimization Software", ACM Transactions on Mathematical Software, Vol
  7, No 1 (1981), p 17--41