### 36.3 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 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.