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GNU Scientific Library Reference Manual - Third Edition (v1.12)
by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi
Paperback (6"x9"), 592 pages, 60 figures
ISBN 0954612078
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34.3 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;
   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;
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 Jacobian 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;
   const double A = (params->A);
   const double x0 = gsl_vector_get(x,0);
   const double x1 = gsl_vector_get(x,1);
   gsl_matrix_set (J, 0, 0, A * x1);
   gsl_matrix_set (J, 0, 1, A * x0);
   gsl_matrix_set (J, 1, 0, -exp(-x0));
   gsl_matrix_set (J, 1, 1, -exp(-x1));
   return GSL_SUCCESS
}

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

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

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

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

gsl_multiroot_function_fdf FDF;

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

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

ISBN 0954612078GNU Scientific Library Reference Manual - Third Edition (v1.12)See the print edition