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<title>Providing the multidimensional system of equations to solve - GNU Scientific Library -- Reference Manual</title>
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<hr>
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<h3 class="section">34.3 Providing the function to solve</h3>
<p><a name="index-multidimensional-root-finding_002c-providing-a-function-to-solve-2289"></a>
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:
<div class="defun">
— Data Type: <b>gsl_multiroot_function</b><var><a name="index-gsl_005fmultiroot_005ffunction-2290"></a></var><br>
<blockquote><p>This data type defines a general system of functions with parameters.
<dl>
<dt><code>int (* f) (const gsl_vector * </code><var>x</var><code>, void * </code><var>params</var><code>, gsl_vector * </code><var>f</var><code>)</code><dd>this function should store the vector result
<!-- {$f(x,\hbox{\it params})$} -->
f(x,params) in <var>f</var> for argument <var>x</var> and parameters <var>params</var>,
returning an appropriate error code if the function cannot be computed.
<br><dt><code>size_t n</code><dd>the dimension of the system, i.e. the number of components of the
vectors <var>x</var> and <var>f</var>.
<br><dt><code>void * params</code><dd>a pointer to the parameters of the function.
</dl>
</p></blockquote></div>
<p class="noindent">Here is an example using Powell's test function,
<pre class="example"> f_1(x) = A x_0 x_1 - 1,
f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)
</pre>
<p class="noindent">with A = 10^4. The following code defines a
<code>gsl_multiroot_function</code> system <code>F</code> which you could pass to a
solver:
<pre class="example"> 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;
</pre>
<div class="defun">
— Data Type: <b>gsl_multiroot_function_fdf</b><var><a name="index-gsl_005fmultiroot_005ffunction_005ffdf-2291"></a></var><br>
<blockquote><p>This data type defines a general system of functions with parameters and
the corresponding Jacobian matrix of derivatives,
<dl>
<dt><code>int (* f) (const gsl_vector * </code><var>x</var><code>, void * </code><var>params</var><code>, gsl_vector * </code><var>f</var><code>)</code><dd>this function should store the vector result
<!-- {$f(x,\hbox{\it params})$} -->
f(x,params) in <var>f</var> for argument <var>x</var> and parameters <var>params</var>,
returning an appropriate error code if the function cannot be computed.
<br><dt><code>int (* df) (const gsl_vector * </code><var>x</var><code>, void * </code><var>params</var><code>, gsl_matrix * </code><var>J</var><code>)</code><dd>this function should store the <var>n</var>-by-<var>n</var> matrix result
<!-- {$J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j$} -->
J_ij = d f_i(x,params) / d x_j in <var>J</var> for argument <var>x</var>
and parameters <var>params</var>, returning an appropriate error code if the
function cannot be computed.
<br><dt><code>int (* fdf) (const gsl_vector * </code><var>x</var><code>, void * </code><var>params</var><code>, gsl_vector * </code><var>f</var><code>, gsl_matrix * </code><var>J</var><code>)</code><dd>This function should set the values of the <var>f</var> and <var>J</var> as above,
for arguments <var>x</var> and parameters <var>params</var>. 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.
<br><dt><code>size_t n</code><dd>the dimension of the system, i.e. the number of components of the
vectors <var>x</var> and <var>f</var>.
<br><dt><code>void * params</code><dd>a pointer to the parameters of the function.
</dl>
</p></blockquote></div>
<p class="noindent">The example of Powell's test function defined above can be extended to
include analytic derivatives using the following code,
<pre class="example"> 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;
</pre>
<p class="noindent">Note that the function <code>powell_fdf</code> is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.
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