@cindex solving nonlinear systems of equations
@cindex nonlinear systems of equations, solution of
@cindex systems of equations, nonlinear

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 @sc{minpack}.

The header file @file{gsl_multiroots.h} contains prototypes for the
multidimensional root finding functions and related declarations.

@menu
* Overview of Multidimensional Root Finding::  
* Initializing the Multidimensional Solver::  
* Providing the multidimensional system of equations to solve::  
* Iteration of the multidimensional solver::  
* Search Stopping Parameters for the multidimensional solver::  
* Algorithms using Derivatives::  
* Algorithms without Derivatives::  
* Example programs for Multidimensional Root finding::  
* References and Further Reading for Multidimensional Root Finding::  
@end menu

@node Overview of Multidimensional Root Finding
@section Overview
@cindex multidimensional root finding, 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},
@tex
\beforedisplay
$$
f_i (x_1, \dots, x_n) = 0 \qquad\hbox{for}~i = 1 \dots n.
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
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,
@tex
\beforedisplay
$$
x \to x' = x - J^{-1} f(x)
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
where @math{x}, @math{f} are vector quantities and @math{J} is the
Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$} 
@math{J_@{ij@} = d f_i / d 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,

@itemize @bullet
@item
initialize solver state, @var{s}, for algorithm @var{T}

@item
update @var{s} using the iteration @var{T}

@item
test @var{s} for convergence, and repeat iteration if necessary
@end itemize

@noindent
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.

@cindex Jacobian matrix, root finding
The state for solvers with an analytic Jacobian matrix is held in a
@code{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 @code{gsl_multiroot_fsolver} struct.  The updating procedure
uses only function evaluations (not derivatives).  The algorithms
estimate the matrix @math{J} or @c{$J^{-1}$} 
@math{J^@{-1@}} by approximate methods.

@node Initializing the Multidimensional Solver
@section 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.

@deftypefun {gsl_multiroot_fsolver *} gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type * @var{T}, size_t @var{n})
@tpindex gsl_multiroot_fsolver
@tpindex gsl_multiroot_fsolver_type
This function returns a pointer to a newly allocated instance of a
solver of type @var{T} for a system of @var{n} dimensions.
For example, the following code creates an instance of a hybrid solver, 
to solve a 3-dimensional system of equations.

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

@noindent
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 @code{GSL_ENOMEM}.
@end deftypefun

@deftypefun {gsl_multiroot_fdfsolver *} gsl_multiroot_fdfsolver_alloc (const gsl_multiroot_fdfsolver_type * @var{T}, size_t @var{n})
@tpindex gsl_multiroot_fdfsolver
@tpindex gsl_multiroot_fdfsolver_type
This function returns a pointer to a newly allocated instance of a
derivative solver of type @var{T} for a system of @var{n} dimensions.
For example, the following code creates an instance of a Newton-Raphson solver,
for a 2-dimensional system of equations.

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

@noindent
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 @code{GSL_ENOMEM}.
@end deftypefun

@deftypefun int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * @var{s}, gsl_multiroot_function * @var{f}, const gsl_vector * @var{x})
@deftypefunx int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver * @var{s}, gsl_multiroot_function_fdf * @var{fdf}, const gsl_vector * @var{x})
These functions set, or reset, an existing solver @var{s} to use the
function @var{f} or function and derivative @var{fdf}, and the initial
guess @var{x}.  Note that the initial position is copied from @var{x}, this
argument is not modified by subsequent iterations.
@end deftypefun

@deftypefun void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * @var{s})
@deftypefunx void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver * @var{s})
These functions free all the memory associated with the solver @var{s}.
@end deftypefun

@deftypefun {const char *} gsl_multiroot_fsolver_name (const gsl_multiroot_fsolver * @var{s})
@deftypefunx {const char *} gsl_multiroot_fdfsolver_name (const gsl_multiroot_fdfsolver * @var{s})
These functions return a pointer to the name of the solver.  For example,

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

@noindent
would print something like @code{s is a 'newton' solver}.
@end deftypefun

@node Providing the multidimensional system of equations to solve
@section Providing the function to solve
@cindex multidimensional root finding, providing a 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:

@deftp {Data Type} gsl_multiroot_function 
This data type defines a general system of functions with parameters.

@table @code
@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
this function should store the vector result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} in @var{f} for argument @var{x} and parameters @var{params},
returning an appropriate error code if the function cannot be computed.

@item size_t n
the dimension of the system, i.e. the number of components of the
vectors @var{x} and @var{f}.

@item void * params
a pointer to the parameters of the function.
@end table
@end deftp

@noindent
Here is an example using Powell's test function,
@tex
\beforedisplay
$$
f_1(x) = A x_0 x_1 - 1,
f_2(x) = \exp(-x_0) + \exp(-x_1) - (1 + 1/A)
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
with @math{A = 10^4}.  The following code defines a
@code{gsl_multiroot_function} system @code{F} which you could pass to a
solver:

@example
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;
@end example

@deftp {Data Type} gsl_multiroot_function_fdf
This data type defines a general system of functions with parameters and
the corresponding Jacobian matrix of derivatives,

@table @code
@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
this function should store the vector result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} in @var{f} for argument @var{x} and parameters @var{params},
returning an appropriate error code if the function cannot be computed.

@item int (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_matrix * @var{J})
this function should store the @var{n}-by-@var{n} matrix result
@c{$J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j$}
@math{J_ij = d f_i(x,params) / d x_j} in @var{J} for argument @var{x} 
and parameters @var{params}, returning an appropriate error code if the
function cannot be computed.

@item int (* fdf) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f}, gsl_matrix * @var{J})
This function should set the values of the @var{f} and @var{J} as above,
for arguments @var{x} and parameters @var{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.

@item size_t n
the dimension of the system, i.e. the number of components of the
vectors @var{x} and @var{f}.

@item void * params
a pointer to the parameters of the function.
@end table
@end deftp

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

@example
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;
@end example

@noindent
Note that the function @code{powell_fdf} is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.

@node Iteration of the multidimensional solver
@section 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.

@deftypefun int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver * @var{s})
@deftypefunx int gsl_multiroot_fdfsolver_iterate (gsl_multiroot_fdfsolver * @var{s})
These functions perform a single iteration of the solver @var{s}.  If the
iteration encounters an unexpected problem then an error code will be
returned,

@table @code
@item GSL_EBADFUNC
the iteration encountered a singular point where the function or its
derivative evaluated to @code{Inf} or @code{NaN}.

@item GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from
continuing.
@end table
@end deftypefun

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,

@deftypefun {gsl_vector *} gsl_multiroot_fsolver_root (const gsl_multiroot_fsolver * @var{s})
@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_root (const gsl_multiroot_fdfsolver * @var{s})
These functions return the current estimate of the root for the solver @var{s}, given by @code{s->x}.
@end deftypefun

@deftypefun {gsl_vector *} gsl_multiroot_fsolver_f (const gsl_multiroot_fsolver * @var{s})
@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_f (const gsl_multiroot_fdfsolver * @var{s})
These functions return the function value @math{f(x)} at the current
estimate of the root for the solver @var{s}, given by @code{s->f}.
@end deftypefun

@deftypefun {gsl_vector *} gsl_multiroot_fsolver_dx (const gsl_multiroot_fsolver * @var{s})
@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_dx (const gsl_multiroot_fdfsolver * @var{s})
These functions return the last step @math{dx} taken by the solver
@var{s}, given by @code{s->dx}.
@end deftypefun

@node Search Stopping Parameters for the multidimensional solver
@section Search Stopping Parameters
@cindex root finding, stopping parameters

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

@itemize @bullet
@item
A multidimensional root has been found to within the user-specified precision.

@item
A user-specified maximum number of iterations has been reached.

@item
An error has occurred.
@end itemize

@noindent
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.

@deftypefun int gsl_multiroot_test_delta (const gsl_vector * @var{dx}, const gsl_vector * @var{x}, double @var{epsabs}, double @var{epsrel})

This function tests for the convergence of the sequence by comparing the
last step @var{dx} with the absolute error @var{epsabs} and relative
error @var{epsrel} to the current position @var{x}.  The test returns
@code{GSL_SUCCESS} if the following condition is achieved,
@tex
\beforedisplay
$$
|dx_i| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_i|
$$
\afterdisplay
@end tex
@ifinfo

@example
|dx_i| < epsabs + epsrel |x_i|
@end example

@end ifinfo
@noindent
for each component of @var{x} and returns @code{GSL_CONTINUE} otherwise.
@end deftypefun

@cindex residual, in nonlinear systems of equations
@deftypefun int gsl_multiroot_test_residual (const gsl_vector * @var{f}, double @var{epsabs})
This function tests the residual value @var{f} against the absolute
error bound @var{epsabs}.  The test returns @code{GSL_SUCCESS} if the
following condition is achieved,
@tex
\beforedisplay
$$
\sum_i |f_i| < \hbox{\it epsabs}
$$
\afterdisplay
@end tex
@ifinfo

@example
\sum_i |f_i| < epsabs
@end example

@end ifinfo
@noindent
and returns @code{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.
@end deftypefun

@comment ============================================================

@node Algorithms using Derivatives
@section 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.


@comment ============================================================
@cindex HYBRID algorithms for nonlinear systems
@deffn {Derivative Solver} gsl_multiroot_fdfsolver_hybridsj
@cindex HYBRIDSJ algorithm
@cindex MINPACK, minimization algorithms
This is a modified version of Powell's Hybrid method as implemented in
the @sc{hybrj} algorithm in @sc{minpack}.  Minpack was written by Jorge
J. Mor@'e, 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,
@tex
\beforedisplay
$$
dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.
$$
\afterdisplay
@end tex
@ifinfo

@example
dx = - \alpha J^@{-1@} f(x) - \beta \nabla |f(x)|^2.
@end example

@end ifinfo
@noindent
This combination of Newton and gradient directions is referred to as a
@dfn{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,

@table @code
@item GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from
continuing.

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

@end deffn

@deffn {Derivative Solver} gsl_multiroot_fdfsolver_hybridj
@cindex HYBRIDJ algorithm
This algorithm is an unscaled version of @code{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 @code{hybridsj} is inappropriate.
@end deffn


@deffn {Derivative Solver} gsl_multiroot_fdfsolver_newton
@cindex Newton's method for systems of nonlinear equations

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,
@tex
\beforedisplay
$$
x \to x' = x - J^{-1} f(x)
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
where the Jacobian matrix @math{J} is computed from the derivative
functions provided by @var{f}.  The step @math{dx} is obtained by solving
the linear system,
@tex
\beforedisplay
$$
J \,dx = - f(x)
$$
\afterdisplay
@end tex
@ifinfo

@example
J dx = - f(x)
@end example

@end ifinfo
@noindent
using LU decomposition.
@end deffn

@comment ============================================================

@deffn {Derivative Solver} gsl_multiroot_fdfsolver_gnewton
@cindex Modified Newton's method for nonlinear systems
@cindex Newton algorithm, globally convergent
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,
@tex
\beforedisplay
$$
t = (\sqrt{1 + 6 r} - 1) / (3 r)
$$
\afterdisplay
@end tex
@ifinfo

@example
t = (\sqrt(1 + 6 r) - 1) / (3 r)
@end example

@end ifinfo
@noindent
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. 
@end deffn

@comment ============================================================

@node Algorithms without Derivatives
@section 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.

@deffn {Solver} gsl_multiroot_fsolver_hybrids
@cindex HYBRIDS algorithm, scaled without derivatives
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 @code{gsl_multiroots_fdjac}
with a relative step size of @code{GSL_SQRT_DBL_EPSILON}.  Note that
this step size will not be suitable for all problems.
@end deffn

@deffn {Solver} gsl_multiroot_fsolver_hybrid
@cindex HYBRID algorithm, unscaled without derivatives
This is a finite difference version of the Hybrid algorithm without
internal scaling.
@end deffn

@comment ============================================================

@deffn {Solver} gsl_multiroot_fsolver_dnewton

@cindex Discrete Newton algorithm for multidimensional roots
@cindex Newton algorithm, discrete

The @dfn{discrete Newton algorithm} is the simplest method of solving a
multidimensional system.  It uses the Newton iteration
@tex
\beforedisplay
$$
x \to x - J^{-1} f(x)
$$
\afterdisplay
@end tex
@ifinfo

@example
x -> x - J^@{-1@} f(x)
@end example

@end ifinfo
@noindent
where the Jacobian matrix @math{J} is approximated by taking finite
differences of the function @var{f}.  The approximation scheme used by
this implementation is,
@tex
\beforedisplay
$$
J_{ij} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j
$$
\afterdisplay
@end tex
@ifinfo

@example
J_@{ij@} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j
@end example

@end ifinfo
@noindent
where @math{\delta_j} is a step of size @math{\sqrt\epsilon |x_j|} with
@math{\epsilon} being the machine precision 
(@c{$\epsilon \approx 2.22 \times 10^{-16}$}
@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.
@end deffn

@comment ============================================================

@deffn {Solver} gsl_multiroot_fsolver_broyden
@cindex Broyden algorithm for multidimensional roots
@cindex multidimensional root finding, Broyden algorithm

The @dfn{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,
@tex
\beforedisplay
$$
J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
$$
\afterdisplay
@end tex
@ifinfo

@example
J^@{-1@} \to J^@{-1@} - (J^@{-1@} df - dx) dx^T J^@{-1@} / dx^T J^@{-1@} df
@end example

@end ifinfo
@noindent
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.
@end deffn

@comment ============================================================


@node Example programs for Multidimensional Root finding
@section Examples

The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms.  This first example
demonstrates the @code{hybrids} scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of equations,
@tex
\beforedisplay
$$
f_1 (x, y) = a (1 - x),~
f_2 (x, y) = b (y - x^2)
$$
\afterdisplay
@end tex
@ifinfo

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

@end ifinfo
@noindent
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,

@example
#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;
@}
@end example

@noindent
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 @code{hybrids}
method.

@example
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;
@}
@end example

@noindent
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.

@example
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));
@}
@end example

@noindent
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.

@smallexample
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
@end smallexample

@noindent
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.

@example
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;
@}
@end example

@noindent
The main program now makes calls to the corresponding @code{fdfsolver}
versions of the functions,

@example
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;
@}
@end example

@noindent
The addition of derivative information to the @code{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
@code{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 @code{gnewton} algorithm,

@smallexample
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
@end smallexample

@noindent
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.

@node References and Further Reading for Multidimensional Root Finding
@section References and Further Reading

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

@itemize @asis
@item
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 @cite{Numerical Methods for
Nonlinear Algebraic Equations}, P. Rabinowitz, editor.  Gordon and
Breach, 1970.
@end itemize

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

@itemize @asis
@item
J.J. Mor@'e, M.Y. Cosnard, ``Numerical Solution of Nonlinear Equations'',
@cite{ACM Transactions on Mathematical Software}, Vol 5, No 1, (1979), p 64--85

@item
C.G. Broyden, ``A Class of Methods for Solving Nonlinear
Simultaneous Equations'', @cite{Mathematics of Computation}, Vol 19 (1965),
p 577--593

@item 
J.J. Mor@'e, B.S. Garbow, K.E. Hillstrom, ``Testing Unconstrained
Optimization Software'', ACM Transactions on Mathematical Software, Vol
7, No 1 (1981), p 17--41
@end itemize

