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   This file documents the GNU Scientific Library.

   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 The GSL Team.

   Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.1 or
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Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.  A
copy of the license is included in the section entitled "GNU Free
Documentation License".


File: gsl-ref.info,  Node: Minimization Stopping Parameters,  Next: Minimization Algorithms,  Prev: Minimization Iteration,  Up: One dimensional Minimization

Stopping Parameters
===================

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

   * A minimum 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 function
below allows the user to test the precision of the current result.

 - Function: int gsl_min_test_interval (double X_LOWER, double X_UPPER,
          double EPSABS, double EPSREL)
     This function tests for the convergence of the interval [X_LOWER,
     X_UPPER] with absolute error EPSABS and relative error EPSREL.
     The test returns `GSL_SUCCESS' if the following condition is
     achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by
     zero (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for minima
     close to the origin.

     This condition on the interval also implies that any estimate of
     the minimum x_m in the interval satisfies the same condition with
     respect to the true minimum x_m^*,

          |x_m - x_m^*| < epsabs + epsrel x_m^*

     assuming that the true minimum x_m^* is contained within the
     interval.


File: gsl-ref.info,  Node: Minimization Algorithms,  Next: Minimization Examples,  Prev: Minimization Stopping Parameters,  Up: One dimensional Minimization

Minimization Algorithms
=======================

   The minimization algorithms described in this section require an
initial interval which is guaranteed to contain a minimum -- if a and b
are the endpoints of the interval and x is an estimate of the minimum
then f(a) > f(x) < f(b).  This ensures that the function has at least
one minimum somewhere in the interval.  If a valid initial interval is
used then these algorithm cannot fail, provided the function is
well-behaved.

 - Minimizer: gsl_min_fminimizer_goldensection
     The "golden section algorithm" is the simplest method of bracketing
     the minimum of a function.  It is the slowest algorithm provided
     by the library, with linear convergence.

     On each iteration, the algorithm first compares the subintervals
     from the endpoints to the current minimum.  The larger subinterval
     is divided in a golden section (using the famous ratio (3-\sqrt
     5)/2 = 0.3189660...) and the value of the function at this new
     point is calculated.  The new value is used with the constraint
     f(a') > f(x') < f(b') to a select new interval containing the
     minimum, by discarding the least useful point.  This procedure can
     be continued indefinitely until the interval is sufficiently
     small.  Choosing the golden section as the bisection ratio can be
     shown to provide the fastest convergence for this type of
     algorithm.


 - Minimizer: gsl_min_fminimizer_brent
     The "Brent minimization algorithm" combines a parabolic
     interpolation with the golden section algorithm.  This produces a
     fast algorithm which is still robust.

     The outline of the algorithm can be summarized as follows: on each
     iteration Brent's method approximates the function using an
     interpolating parabola through three existing points.  The minimum
     of the parabola is taken as a guess for the minimum.  If it lies
     within the bounds of the current interval then the interpolating
     point is accepted, and used to generate a smaller interval.  If
     the interpolating point is not accepted then the algorithm falls
     back to an ordinary golden section step.  The full details of
     Brent's method include some additional checks to improve
     convergence.


File: gsl-ref.info,  Node: Minimization Examples,  Next: Minimization References and Further Reading,  Prev: Minimization Algorithms,  Up: One dimensional Minimization

Examples
========

   The following program uses the Brent algorithm to find the minimum of
the function f(x) = \cos(x) + 1, which occurs at x = \pi.  The starting
interval is (0,6), with an initial guess for the minimum of 2.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_min.h>
     
     double fn1 (double x, void * params)
     {
       return cos(x) + 1.0;
     }
     
     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_min_fminimizer_type *T;
       gsl_min_fminimizer *s;
       double m = 2.0, m_expected = M_PI;
       double a = 0.0, b = 6.0;
       gsl_function F;
     
       F.function = &fn1;
       F.params = 0;
     
       T = gsl_min_fminimizer_brent;
       s = gsl_min_fminimizer_alloc (T);
       gsl_min_fminimizer_set (s, &F, m, a, b);
     
       printf ("using %s method\n",
               gsl_min_fminimizer_name (s));
     
       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "min",
               "err", "err(est)");
     
       printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
               iter, a, b,
               m, m - m_expected, b - a);
     
       do
         {
           iter++;
           status = gsl_min_fminimizer_iterate (s);
     
           m = gsl_min_fminimizer_minimum (s);
           a = gsl_min_fminimizer_x_lower (s);
           b = gsl_min_fminimizer_x_upper (s);
     
           status
             = gsl_min_test_interval (a, b, 0.001, 0.0);
     
           if (status == GSL_SUCCESS)
             printf ("Converged:\n");
     
           printf ("%5d [%.7f, %.7f] "
                   "%.7f %.7f %+.7f %.7f\n",
                   iter, a, b,
                   m, m_expected, m - m_expected, b - a);
         }
       while (status == GSL_CONTINUE && iter < max_iter);
     
       return status;
     }

Here are the results of the minimization procedure.

     bash$ ./a.out
         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
         1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
         2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
         3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
         4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
         5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
         6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
         7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
         8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
         9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
        10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
     Converged:
        11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175


File: gsl-ref.info,  Node: Minimization References and Further Reading,  Prev: Minimization Examples,  Up: One dimensional Minimization

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

Further information on Brent's algorithm is available in the following
book,

     Richard Brent, `Algorithms for minimization without derivatives',
     Prentice-Hall (1973), republished by Dover in paperback (2002),
     ISBN 0-486-41998-3.


File: gsl-ref.info,  Node: Multidimensional Root-Finding,  Next: Multidimensional Minimization,  Prev: One dimensional Minimization,  Up: Top

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

   This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in 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 `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::


File: gsl-ref.info,  Node: Overview of Multidimensional Root Finding,  Next: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding

Overview
========

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

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

In general there are no bracketing methods available for 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,

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

where x, f are vector quantities and J is the Jacobian matrix 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 |f| on
each step proposed by Newton's method, or taking steepest-descent steps
in the direction of the negative gradient of |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, S, for algorithm T

   * update S using the iteration T

   * test 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
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.

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


File: gsl-ref.info,  Node: Initializing the Multidimensional Solver,  Next: Providing the multidimensional system of equations to solve,  Prev: Overview of Multidimensional Root Finding,  Up: Multidimensional Root-Finding

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.

 - Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const
          gsl_multiroot_fsolver_type * T, size_t N)
     This function returns a pointer to a a newly allocated instance of
     a solver of type T for a system of 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 `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 a newly allocated instance of
     a derivative solver of type T for a system of 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 `GSL_ENOMEM'.

 - Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * S,
          gsl_multiroot_function * F, gsl_vector * X)
     This function sets, or resets, an existing solver S to use the
     function F and the initial guess X.

 - Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver *
          S, gsl_function_fdf * FDF, gsl_vector * X)
     This function sets, or resets, an existing solver S to use the
     function and derivative FDF and the initial guess X.

 - Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * S)
 - Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver
          * S)
     These functions free all the memory associated with the solver S.

 - Function: const char * gsl_multiroot_fsolver_name (const
          gsl_multiroot_fsolver * S)
 - Function: 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 `s is a 'newton' solver'.


File: gsl-ref.info,  Node: Providing the multidimensional system of equations to solve,  Next: Iteration of the multidimensional solver,  Prev: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding

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

   You must provide n functions of 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:

 - Data Type: gsl_multiroot_function
     This data type defines a general system of functions with
     parameters.

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `size_t N'
          the dimension of the system, i.e. the number of components of
          the vectors X and F.

    `void * PARAMS'
          a pointer to the parameters of the function.

Here is an example using Powell's test function,

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

with A = 10^4.  The following code defines a `gsl_multiroot_function'
system `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;
        double A = (params->A);
        double x0 = gsl_vector_get(x,0);
        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.function = &powell;
     F.n = 2;
     F.params = &params;

 - Data Type: gsl_multiroot_function_fdf
     This data type defines a general system of functions with
     parameters and the corresponding Jacobian matrix of derivatives,

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
          this function should store the N-by-N matrix result J_ij = d
          f_i(x,params) / d x_j in J for argument X and parameters
          PARAMS, returning an appropriate error code if the function
          cannot be computed.

    `int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
          This function should set the values of the F and J as above,
          for arguments X and parameters PARAMS.  This function provides
          an optimization of the separate functions for f(x) and J(x) -
          it is always faster to compute the function and its
          derivative at the same time.

    `size_t N'
          the dimension of the system, i.e. the number of components of
          the vectors X and F.

    `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;
        double A = (params->A);
        double x0 = gsl_vector_get(x,0);
        double x1 = gsl_vector_get(x,1);
        gsl_vector_set (J, 0, 0, A * x1)
        gsl_vector_set (J, 0, 1, A * x0)
        gsl_vector_set (J, 1, 0, -exp(-x0))
        gsl_vector_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;
        double A = (params->A);
        double x0 = gsl_vector_get(x,0);
        double x1 = gsl_vector_get(x,1);
     
        double u0 = exp(-x0);
        double u1 = exp(-x1);
     
        gsl_vector_set (f, 0, A * x0 * x1 - 1)
        gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A))
     
        gsl_vector_set (J, 0, 0, A * x1)
        gsl_vector_set (J, 0, 1, A * x0)
        gsl_vector_set (J, 1, 0, -u0)
        gsl_vector_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 `powell_fdf' is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.


File: gsl-ref.info,  Node: Iteration of the multidimensional solver,  Next: Search Stopping Parameters for the multidimensional solver,  Prev: Providing the multidimensional system of equations to solve,  Up: Multidimensional Root-Finding

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)
 - Function: int gsl_multiroot_fdfsolver_iterate
          (gsl_multiroot_fdfsolver * S)
     These functions perform a single iteration of the solver S.  If the
     iteration encounters an unexpected problem then an error code will
     be returned,

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

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

   The solver maintains a current best estimate of the root 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)
 - Function: gsl_vector * gsl_multiroot_fdfsolver_root (const
          gsl_multiroot_fdfsolver * S)
     These functions return the current estimate of the root for the
     solver S.


File: gsl-ref.info,  Node: Search Stopping Parameters for the multidimensional solver,  Next: Algorithms using Derivatives,  Prev: Iteration of the multidimensional solver,  Up: Multidimensional Root-Finding

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 DX with the absolute error EPSABS and
     relative error EPSREL to the current position X.  The test returns
     `GSL_SUCCESS' if the following condition is achieved,

          |dx_i| < epsabs + epsrel |x_i|

     for each component of X and returns `GSL_CONTINUE' otherwise.

 - Function: int gsl_multiroot_test_residual (const gsl_vector * F,
          double EPSABS)
     This function tests the residual value F against the absolute
     error bound EPSABS.  The test returns `GSL_SUCCESS' if the
     following condition is achieved,

          \sum_i |f_i| < epsabs

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


File: gsl-ref.info,  Node: Algorithms using Derivatives,  Next: Algorithms without Derivatives,  Prev: Search Stopping Parameters for the multidimensional solver,  Up: Multidimensional Root-Finding

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.

 - Derivative Solver: 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 x'
     must satisfy the condition |D (x' - x)| < \delta, where D is a
     diagonal scaling matrix and \delta is the size of the trust
     region.  The components of D are computed internally, using the
     column norms of the Jacobian to estimate the sensitivity of the
     residual to each component of 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 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.

          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, 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,

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

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


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

 - Derivative Solver: 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 F is used to estimate the step which will zero all the
     components of the residual.  The iteration is defined by the
     following sequence,

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

     where the Jacobian matrix J is computed from the derivative
     functions provided by F.  The step dx is obtained by solving the
     linear system,

          J dx = - f(x)

     using LU decomposition.

 - Derivative Solver: 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, |f(x)|.  If the Newton step leads
     to an increase in the norm then a reduced step of relative size,

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

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


File: gsl-ref.info,  Node: Algorithms without Derivatives,  Next: Example programs for Multidimensional Root finding,  Prev: Algorithms using Derivatives,  Up: Multidimensional Root-Finding

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 from by finite difference.

 - Solver: 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
     `gsl_multiroots_fdjac' with a relative step size of
     `GSL_SQRT_DBL_EPSILON'.

 - Solver: gsl_multiroot_fsolver_hybrid
     This is a finite difference version of the Hybrid algorithm without
     internal scaling.

 - Solver: gsl_multiroot_fsolver_dnewton
     The "discrete Newton algorithm" is the simplest method of solving a
     multidimensional system.  It uses the Newton iteration

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

     where the Jacobian matrix J is approximated by taking finite
     differences of the function F.  The approximation scheme used by
     this implementation is,

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

     where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon
     being the machine precision (\epsilon \approx 2.22 \times 10^-16).
     The order of convergence of Newton's algorithm is quadratic, but
     the finite differences require 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.

 - Solver: 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,

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

     where the vectors dx and df are the changes in x and 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 not recommended and is included only for
     demonstration purposes.


File: gsl-ref.info,  Node: Example programs for Multidimensional Root finding,  Next: References and Further Reading for Multidimensional Root Finding,  Prev: Algorithms without Derivatives,  Up: Multidimensional Root-Finding

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,

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

with a = 1, b = 10. The solution of this system lies at (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;
     
       double x0 = gsl_vector_get (x, 0);
       double x1 = gsl_vector_get (x, 1);
     
       double y0 = a * (1 - x0);
       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 `f', with the
arguments `(x,y)' and parameters `(a,b)'. The solver `s' is initialized
to use this function, with the `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 `s->x' which is the
current position, and the vector `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
(-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
`rosenbrock_df' and `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 `f' and `df' functions at this point in the code below.

     int
     rosenbrock_df (const gsl_vector * x, void *params,
                    gsl_matrix * J)
     {
       double a = ((struct rparams *) params)->a;
       double b = ((struct rparams *) params)->b;
     
       double x0 = gsl_vector_get (x, 0);
     
       double df00 = -a;
       double df01 = 0;
       double df10 = -2 * b  * x0;
       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 `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 `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
`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 `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 (-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.


File: gsl-ref.info,  Node: References and Further Reading for Multidimensional Root Finding,  Prev: Example programs for Multidimensional Root finding,  Up: Multidimensional Root-Finding

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


File: gsl-ref.info,  Node: Multidimensional Minimization,  Next: Least-Squares Fitting,  Prev: Multidimensional Root-Finding,  Up: Top

Multidimensional Minimization
*****************************

   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 `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::


File: gsl-ref.info,  Node: Multimin Overview,  Next: Multimin Caveats,  Up: Multidimensional Minimization

Overview
========

   The problem of multidimensional minimization requires finding a point
x such that the scalar function,

     f(x_1, ..., x_n)

takes a value which is lower than at any neighboring point. For smooth
functions the gradient g = \nabla f vanishes at the minimum. In general
there are no bracketing methods available for the minimization of
n-dimensional functions.  All algorithms proceed from an initial guess
using a search algorithm which attempts to move in a downhill
direction. A one-dimensional line minimisation is performed 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 n-dimensional minimum is found.

   Several minimization 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 minimizer state, S, for algorithm T

   * update S using the iteration T

   * test S for convergence, and repeat iteration if necessary

Each iteration step consists either of an improvement to the
line-mimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a
`gsl_multimin_fdfminimizer' struct.


File: gsl-ref.info,  Node: Multimin Caveats,  Next: Initializing the Multidimensional Minimizer,  Prev: Multimin Overview,  Up: Multidimensional Minimization

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.


File: gsl-ref.info,  Node: Initializing the Multidimensional Minimizer,  Next: Providing a function to minimize,  Prev: Multimin Caveats,  Up: Multidimensional Minimization

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.

 - Function: gsl_multimin_fdfminimizer *
          gsl_multimin_fdfminimizer_alloc (const
          gsl_multimin_fdfminimizer_type *T, size_t N)
     This function returns a pointer to a a newly allocated instance of
     a minimizer of type T for an 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 `GSL_ENOMEM'.

 - Function: int gsl_multimin_fdfminimizer_set
          (gsl_multimin_fdfminimizer * S, gsl_multimin_function_fdf
          *FDF, const gsl_vector * X, double STEP_SIZE, double TOL)
     This function initializes the minimizer S to minimize the function
     FDF starting from the initial point X.  The size of the first
     trial step is given by STEP_SIZE.  The accuracy of the line
     minimization is specified by 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 g is orthogonal to the current search direction p to a
     relative accuracy of TOL, where dot(p,g) < tol |p| |g|.

 - Function: void gsl_multimin_fdfminimizer_free
          (gsl_multimin_fdfminimizer *S)
     This function frees all the memory associated with the minimizer S.

 - Function: const char * gsl_multimin_fdfminimizer_name (const
          gsl_multimin_fdfminimizer * S)
     This function returns a pointer to the name of the minimizer.  For
     example,

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

     would print something like `s is a 'conjugate_pr' minimizer'.


File: gsl-ref.info,  Node: Providing a function to minimize,  Next: Multimin Iteration,  Prev: Initializing the Multidimensional Minimizer,  Up: Multidimensional Minimization

Providing a function to minimize
================================

   You must provide a parametric function of n variables for the
minimizers to operate on.  You 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 type:

 - Data Type: gsl_multimin_function_fdf
     This data type defines a general function of n variables with
     parameters and the corresponding gradient vector of derivatives,

    `double (* f) (const gsl_vector * X, void * PARAMS)'
          this function should return the result f(x,params) for
          argument X and parameters PARAMS.

    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_vector * G)'
          this function should store the N-dimensional gradient g_i = d
          f(x,params) / d x_i in the vector G for argument X and
          parameters PARAMS, returning an appropriate error code if the
          function cannot be computed.

    `int (* fdf) (const gsl_vector * X, void * PARAMS, double * f, gsl_vector * G)'
          This function should set the values of the F and G as above,
          for arguments X and parameters PARAMS.  This function provides
          an optimization of the separate functions for f(x) and g(x) -
          it is always faster to compute the function and its
          derivative at the same time.

    `size_t N'
          the dimension of the system, i.e. the number of components of
          the vectors X.

    `void * PARAMS'
          a pointer to the parameters of the function.

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

     /* 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);
     }

The function can be initialized using the following code,

     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;

