@cindex minimization, multidimensional

This chapter describes routines for finding minima of arbitrary
multidimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs. The minimization
algorithms can be used to maximize a function by inverting its sign.

The header file @file{gsl_multimin.h} contains prototypes for the
minimization functions and related declarations.  

@menu
* Multimin Overview::       
* Multimin Caveats::        
* Initializing the Multidimensional Minimizer::  
* Providing a function to minimize::  
* Multimin Iteration::      
* Multimin Stopping Criteria::  
* Multimin Algorithms::     
* Multimin Examples::       
* Multimin References and Further Reading::  
@end menu

@node Multimin Overview
@section Overview

The problem of multidimensional minimization requires finding a point
@math{x} such that the scalar function,
@tex
\beforedisplay
$$
f(x_1, \dots, x_n)
$$
\afterdisplay
@end tex
@ifinfo

@example
f(x_1, @dots{}, x_n)
@end example

@end ifinfo
@noindent
takes a value which is lower than at any neighboring point. For smooth
functions the gradient @math{g = \nabla f} vanishes at the minimum. In
general there are no bracketing methods available for the
minimization of @math{n}-dimensional functions.  The algorithms
proceed from an initial guess using a search algorithm which attempts
to move in a downhill direction. 

Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance.  The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true @math{n}-dimensional
minimum is found.

The Nelder-Mead Simplex algorithm applies a different strategy.  It
maintains @math{n+1} trial parameter vectors as the vertices of a
@math{n}-dimensional simplex.  In each iteration step it tries to
improve the worst vertex by a simple geometrical transformation until
the size of the simplex falls below a given tolerance.

Both types of algorithms use a standard 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 minimizer 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
Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a
@code{gsl_multimin_fdfminimizer} struct or a
@code{gsl_multimin_fminimizer} struct.

@node Multimin Caveats
@section Caveats
@cindex Multimin, caveats

Note that the minimization algorithms can only search for one local
minimum at a time.  When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be.  In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.

It is also important to note that the minimization algorithms find local
minima; there is no way to determine whether a minimum is a global
minimum of the function in question.

@node Initializing the Multidimensional Minimizer
@section Initializing the Multidimensional Minimizer

The following function initializes a multidimensional minimizer.  The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.

@deftypefun {gsl_multimin_fdfminimizer *} gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type * @var{T}, size_t @var{n})
@deftypefunx {gsl_multimin_fminimizer *} gsl_multimin_fminimizer_alloc (const gsl_multimin_fminimizer_type * @var{T}, size_t @var{n})
This function returns a pointer to a newly allocated instance of a
minimizer of type @var{T} for an @var{n}-dimension function.  If there
is insufficient memory to create the minimizer 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_multimin_fdfminimizer_set (gsl_multimin_fdfminimizer * @var{s}, gsl_multimin_function_fdf * @var{fdf}, const gsl_vector * @var{x}, double @var{step_size}, double @var{tol})
This function initializes the minimizer @var{s} to minimize the function
@var{fdf} starting from the initial point @var{x}.  The size of the
first trial step is given by @var{step_size}.  The accuracy of the line
minimization is specified by @var{tol}.  The precise meaning of this
parameter depends on the method used.  Typically the line minimization
is considered successful if the gradient of the function @math{g} is
orthogonal to the current search direction @math{p} to a relative
accuracy of @var{tol}, where @c{$p\cdot g < tol |p| |g|$} 
@math{dot(p,g) < tol |p| |g|}.  A @var{tol} value of 0.1 is 
suitable for most purposes, since line minimization only needs to
be carried out approximately.    Note that setting @var{tol} to zero will
force the use of ``exact'' line-searches, which are extremely expensive.

@deftypefunx int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer * @var{s}, gsl_multimin_function * @var{f}, const gsl_vector * @var{x}, const gsl_vector * @var{step_size})
This function initializes the minimizer @var{s} to minimize the function
@var{f}, starting from the initial point
@var{x}. The size of the initial trial steps is given in vector
@var{step_size}. The precise meaning of this parameter depends on the
method used. 
@end deftypefun

@deftypefun void gsl_multimin_fdfminimizer_free (gsl_multimin_fdfminimizer * @var{s})
@deftypefunx void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer * @var{s})
This function frees all the memory associated with the minimizer
@var{s}.
@end deftypefun

@deftypefun {const char *} gsl_multimin_fdfminimizer_name (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx {const char *} gsl_multimin_fminimizer_name (const gsl_multimin_fminimizer * @var{s})
This function returns a pointer to the name of the minimizer.  For example,

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

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

@node Providing a function to minimize
@section Providing a function to minimize

You must provide a parametric function of @math{n} variables for the
minimizers to operate on.  You may also need to provide a routine which
calculates the gradient of the function and a third routine which
calculates both the function value and the gradient together.  In order
to allow for general parameters the functions are defined by the
following data types:

@deftp {Data Type} gsl_multimin_function_fdf
This data type defines a general function of @math{n} variables with
parameters and the corresponding gradient vector of derivatives,

@table @code
@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
this function should return the result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} for argument @var{x} and parameters @var{params}.

@item void (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{g})
this function should store the @var{n}-dimensional gradient
@c{$g_i = \partial f(x,\hbox{\it params}) / \partial x_i$}
@math{g_i = d f(x,params) / d x_i} in the vector @var{g} for argument @var{x} 
and parameters @var{params}, returning an appropriate error code if the
function cannot be computed.

@item void (* fdf) (const gsl_vector * @var{x}, void * @var{params}, double * f, gsl_vector * @var{g})
This function should set the values of the @var{f} and @var{g} 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{g(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}.

@item void * params
a pointer to the parameters of the function.
@end table
@end deftp
@deftp {Data Type} gsl_multimin_function
This data type defines a general function of @math{n} variables with
parameters,

@table @code
@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
this function should return the result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} for argument @var{x} and parameters @var{params}.

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

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

@noindent
The following example function defines a simple paraboloid with two
parameters,

@example
/* Paraboloid centered on (dp[0],dp[1]) */

double
my_f (const gsl_vector *v, void *params)
@{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  return 10.0 * (x - dp[0]) * (x - dp[0]) +
           20.0 * (y - dp[1]) * (y - dp[1]) + 30.0; 
@}

/* The gradient of f, df = (df/dx, df/dy). */
void 
my_df (const gsl_vector *v, void *params, 
       gsl_vector *df)
@{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
  gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
@}

/* Compute both f and df together. */
void 
my_fdf (const gsl_vector *x, void *params, 
        double *f, gsl_vector *df) 
@{
  *f = my_f(x, params); 
  my_df(x, params, df);
@}
@end example

@noindent
The function can be initialized using the following code,

@example
gsl_multimin_function_fdf my_func;

double p[2] = @{ 1.0, 2.0 @}; /* center at (1,2) */

my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = (void *)p;
@end example

@node Multimin Iteration
@section Iteration

The following function drives the iteration of each algorithm.  The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.

@deftypefun int gsl_multimin_fdfminimizer_iterate (gsl_multimin_fdfminimizer * @var{s})
@deftypefunx int gsl_multimin_fminimizer_iterate (gsl_multimin_fminimizer * @var{s})
These functions perform a single iteration of the minimizer @var{s}.  If
the iteration encounters an unexpected problem then an error code will
be returned.
@end deftypefun

@noindent
The minimizer maintains a current best estimate of the minimum at all
times.  This information can be accessed with the following auxiliary
functions,

@deftypefun {gsl_vector *} gsl_multimin_fdfminimizer_x (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx {gsl_vector *} gsl_multimin_fminimizer_x (const gsl_multimin_fminimizer * @var{s})
@deftypefunx double gsl_multimin_fdfminimizer_minimum (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx double gsl_multimin_fminimizer_minimum (const gsl_multimin_fminimizer * @var{s})
@deftypefunx {gsl_vector *} gsl_multimin_fdfminimizer_gradient (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx double gsl_multimin_fminimizer_size (const gsl_multimin_fminimizer * @var{s})
These functions return the current best estimate of the location of the
minimum, the value of the function at that point, its gradient, 
and minimizer specific characteristic size for the minimizer @var{s}.
@end deftypefun

@deftypefun int gsl_multimin_fdfminimizer_restart (gsl_multimin_fdfminimizer * @var{s})
This function resets the minimizer @var{s} to use the current point as a
new starting point.
@end deftypefun

@node Multimin Stopping Criteria
@section Stopping Criteria

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

@itemize @bullet
@item
A minimum 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.

@deftypefun int gsl_multimin_test_gradient (const gsl_vector * @var{g}, double @var{epsabs})
This function tests the norm of the gradient @var{g} against the
absolute tolerance @var{epsabs}. The gradient of a multidimensional
function goes to zero at a minimum. The test returns @code{GSL_SUCCESS}
if the following condition is achieved,
@tex
\beforedisplay
$$
|g| < \hbox{\it epsabs}
$$
\afterdisplay
@end tex
@ifinfo

@example
|g| < epsabs
@end example

@end ifinfo
@noindent
and returns @code{GSL_CONTINUE} otherwise.  A suitable choice of
@var{epsabs} can be made from the desired accuracy in the function for
small variations in @math{x}.  The relationship between these quantities
is given by @c{$\delta{f} = g\,\delta{x}$}
@math{\delta f = g \delta x}.
@end deftypefun

@deftypefun int gsl_multimin_test_size (const double @var{size}, double @var{epsabs})
This function tests the minimizer specific characteristic
size (if applicable to the used minimizer) against absolute tolerance @var{epsabs}. 
The test returns @code{GSL_SUCCESS} if the size is smaller than tolerance,
otherwise @code{GSL_CONTINUE} is returned.
@end deftypefun

@node Multimin Algorithms
@section Algorithms

There are several minimization methods available. The best choice of
algorithm depends on the problem.  All of the algorithms use the value
of the function and its gradient at each evaluation point, except for
the Simplex algorithm which uses function values only.

@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_fr
@cindex Fletcher-Reeves conjugate gradient algorithm, minimization
@cindex Conjugate gradient algorithm, minimization
@cindex minimization, conjugate gradient algorithm
This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate
gradient algorithm proceeds as a succession of line minimizations. The
sequence of search directions is used to build up an approximation to the
curvature of the function in the neighborhood of the minimum.  

An initial search direction @var{p} is chosen using the gradient, and line
minimization is carried out in that direction.  The accuracy of the line
minimization is specified by the parameter @var{tol}.  The minimum
along this line occurs when the function gradient @var{g} and the search direction
@var{p} are orthogonal.  The line minimization terminates when
@c{$p\cdot g < tol |p| |g|$} 
@math{dot(p,g) < tol |p| |g|}.  The
search direction is updated  using the Fletcher-Reeves formula
@math{p' = g' - \beta g} where @math{\beta=-|g'|^2/|g|^2}, and
the line minimization is then repeated for the new search
direction.
@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_pr
@cindex Polak-Ribiere algorithm, minimization
@cindex minimization, Polak-Ribiere algorithm
This is the Polak-Ribiere conjugate gradient algorithm.  It is similar
to the Fletcher-Reeves method, differing only in the choice of the
coefficient @math{\beta}. Both methods work well when the evaluation
point is close enough to the minimum of the objective function that it
is well approximated by a quadratic hypersurface.
@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs2
@deffnx {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs
@cindex BFGS algorithm, minimization
@cindex minimization, BFGS algorithm
These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS)
algorithm.  This is a quasi-Newton method which builds up an approximation
to the second derivatives of the function @math{f} using the difference
between successive gradient vectors.  By combining the first and second
derivatives the algorithm is able to take Newton-type steps towards the
function minimum, assuming quadratic behavior in that region.

The @code{bfgs2} version of this minimizer is the most efficient
version available, and is a faithful implementation of the line
minimization scheme described in Fletcher's @cite{Practical Methods of
Optimization}, Algorithms 2.6.2 and 2.6.4.  It supercedes the original
@code{bfgs} routine and requires substantially fewer function and
gradient evaluations.  The user-supplied tolerance @var{tol}
corresponds to the parameter @math{\sigma} used by Fletcher.  A value
of 0.1 is recommended for typical use (larger values correspond to
less accurate line searches).

@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_steepest_descent
@cindex steepest descent algorithm, minimization
@cindex minimization, steepest descent algorithm
The steepest descent algorithm follows the downhill gradient of the
function at each step. When a downhill step is successful the step-size
is increased by a factor of two.  If the downhill step leads to a higher
function value then the algorithm backtracks and the step size is
decreased using the parameter @var{tol}.  A suitable value of @var{tol}
for most applications is 0.1.  The steepest descent method is
inefficient and is included only for demonstration purposes.
@end deffn

@deffn {Minimizer} gsl_multimin_fminimizer_nmsimplex
@cindex Nelder-Mead simplex algorithm for minimization
@cindex simplex algorithm, minimization
@cindex minimization, simplex algorithm
This is the Simplex algorithm of Nelder and Mead. It constructs 
@math{n} vectors @math{p_i} from the
starting vector @var{x} and the vector @var{step_size} as follows:
@tex
\beforedisplay
$$
\eqalign{
p_0 & = (x_0, x_1, \cdots , x_n) \cr
p_1 & = (x_0 + step\_size_0, x_1, \cdots , x_n) \cr
p_2 & = (x_0, x_1 + step\_size_1, \cdots , x_n) \cr
\dots &= \dots \cr
p_n & = (x_0, x_1, \cdots , x_n+step\_size_n) \cr
}
$$
\afterdisplay
@end tex
@ifinfo

@example
p_0 = (x_0, x_1, ... , x_n) 
p_1 = (x_0 + step_size_0, x_1, ... , x_n) 
p_2 = (x_0, x_1 + step_size_1, ... , x_n) 
... = ...
p_n = (x_0, x_1, ... , x_n+step_size_n)
@end example

@end ifinfo
@noindent
These vectors form the @math{n+1} vertices of a simplex in @math{n}
dimensions.  On each iteration the algorithm tries to improve
the parameter vector @math{p_i} corresponding to the highest
function value by simple geometrical transformations.  These
are reflection, reflection followed by expansion, contraction and multiple
contraction. Using these transformations the simplex moves through 
the parameter space towards the minimum, where it contracts itself.  

After each iteration, the best vertex is returned.  Note, that due to
the nature of the algorithm not every step improves the current
best parameter vector.  Usually several iterations are required.

The routine calculates the minimizer specific characteristic size as the
average distance from the geometrical center of the simplex to all its
vertices.  This size can be used as a stopping criteria, as the simplex
contracts itself near the minimum. The size is returned by the function
@code{gsl_multimin_fminimizer_size}.
@end deffn

@node Multimin Examples
@section Examples

This example program finds the minimum of the paraboloid function
defined earlier.  The location of the minimum is offset from the origin
in @math{x} and @math{y}, and the function value at the minimum is
non-zero. The main program is given below, it requires the example
function given earlier in this chapter.

@smallexample
int
main (void)
@{
  size_t iter = 0;
  int status;

  const gsl_multimin_fdfminimizer_type *T;
  gsl_multimin_fdfminimizer *s;

  /* Position of the minimum (1,2). */
  double par[2] = @{ 1.0, 2.0 @};

  gsl_vector *x;
  gsl_multimin_function_fdf my_func;

  my_func.f = &my_f;
  my_func.df = &my_df;
  my_func.fdf = &my_fdf;
  my_func.n = 2;
  my_func.params = &par;

  /* Starting point, x = (5,7) */
  x = gsl_vector_alloc (2);
  gsl_vector_set (x, 0, 5.0);
  gsl_vector_set (x, 1, 7.0);

  T = gsl_multimin_fdfminimizer_conjugate_fr;
  s = gsl_multimin_fdfminimizer_alloc (T, 2);

  gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

  do
    @{
      iter++;
      status = gsl_multimin_fdfminimizer_iterate (s);

      if (status)
        break;

      status = gsl_multimin_test_gradient (s->gradient, 1e-3);

      if (status == GSL_SUCCESS)
        printf ("Minimum found at:\n");

      printf ("%5d %.5f %.5f %10.5f\n", iter,
              gsl_vector_get (s->x, 0), 
              gsl_vector_get (s->x, 1), 
              s->f);

    @}
  while (status == GSL_CONTINUE && iter < 100);

  gsl_multimin_fdfminimizer_free (s);
  gsl_vector_free (x);

  return 0;
@}
@end smallexample  

@noindent
The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001.  The program
terminates when the norm of the gradient has been reduced below
0.001. The output of the program is shown below,

@example
         x       y         f
    1 4.99629 6.99072  687.84780
    2 4.98886 6.97215  683.55456
    3 4.97400 6.93501  675.01278
    4 4.94429 6.86073  658.10798
    5 4.88487 6.71217  625.01340
    6 4.76602 6.41506  561.68440
    7 4.52833 5.82083  446.46694
    8 4.05295 4.63238  261.79422
    9 3.10219 2.25548   75.49762
   10 2.85185 1.62963   67.03704
   11 2.19088 1.76182   45.31640
   12 0.86892 2.02622   30.18555
Minimum found at:
   13 1.00000 2.00000   30.00000
@end example

@noindent
Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points.

@iftex
@sp 1
@center @image{multimin,3.4in}
@end iftex

@noindent
The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic. Additional
iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.

@smallexample
int 
main(void)
@{
  size_t np = 2;
  double par[2] = @{1.0, 2.0@};

  const gsl_multimin_fminimizer_type *T = 
    gsl_multimin_fminimizer_nmsimplex;
  gsl_multimin_fminimizer *s = NULL;
  gsl_vector *ss, *x;
  gsl_multimin_function minex_func;

  size_t iter = 0, i;
  int status;
  double size;

  /* Initial vertex size vector */
  ss = gsl_vector_alloc (np);

  /* Set all step sizes to 1 */
  gsl_vector_set_all (ss, 1.0);

  /* Starting point */
  x = gsl_vector_alloc (np);

  gsl_vector_set (x, 0, 5.0);
  gsl_vector_set (x, 1, 7.0);

  /* Initialize method and iterate */
  minex_func.f = &my_f;
  minex_func.n = np;
  minex_func.params = (void *)&par;

  s = gsl_multimin_fminimizer_alloc (T, np);
  gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

  do
    @{
      iter++;
      status = gsl_multimin_fminimizer_iterate(s);
      
      if (status) 
        break;

      size = gsl_multimin_fminimizer_size (s);
      status = gsl_multimin_test_size (size, 1e-2);

      if (status == GSL_SUCCESS)
        @{
          printf ("converged to minimum at\n");
        @}

      printf ("%5d ", iter);
      for (i = 0; i < np; i++)
        @{
          printf ("%10.3e ", gsl_vector_get (s->x, i));
        @}
      printf ("f() = %7.3f size = %.3f\n", s->fval, size);
    @}
  while (status == GSL_CONTINUE && iter < 100);
  
  gsl_vector_free(x);
  gsl_vector_free(ss);
  gsl_multimin_fminimizer_free (s);

  return status;
@}
@end smallexample

@noindent
The minimum search stops when the Simplex size drops to 0.01. The output is
shown below.

@example
    1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
    2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
    3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
    4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
    5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
    6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526
    7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
    8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
    9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
   10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
   11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
   12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
   13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
   14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
   15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
   16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
   17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
   18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
   19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
   20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
   21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
   22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
   23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
   24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
converged to minimum at                  
   25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010
@end example

@noindent
The simplex size first increases, while the simplex moves towards the
minimum. After a while the size begins to decrease as the simplex
contracts around the minimum.

@node Multimin References and Further Reading
@section References and Further Reading

The conjugate gradient and BFGS methods are described in detail in the
following book,

@itemize @asis
@item R. Fletcher,
@cite{Practical Methods of Optimization (Second Edition)} Wiley
(1987), ISBN 0471915475.
@end itemize

A brief description of multidimensional minimization algorithms and
more recent references can be found in,

@itemize @asis
@item C.W. Ueberhuber,
@cite{Numerical Computation (Volume 2)}, Chapter 14, Section 4.4
``Minimization Methods'', p.@: 325--335, Springer (1997), ISBN
3-540-62057-5.
@end itemize

@noindent
The simplex algorithm is described in the following paper, 

@itemize @asis
@item J.A. Nelder and R. Mead,
@cite{A simplex method for function minimization}, Computer Journal
vol.@: 7 (1965), 308--315.
@end itemize

@noindent