GNU Scientific Library – Reference Manual: Providing the multidimensional system of equations to solve

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### 35.3 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, 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 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.

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