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<a name="Providing-the-multidimensional-system-of-equations-to-solve"></a>
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<p>
Next: <a href="Iteration-of-the-multidimensional-solver.html#Iteration-of-the-multidimensional-solver" accesskey="n" rel="next">Iteration of the multidimensional solver</a>, Previous: <a href="Initializing-the-Multidimensional-Solver.html#Initializing-the-Multidimensional-Solver" accesskey="p" rel="previous">Initializing the Multidimensional Solver</a>, Up: <a href="Multidimensional-Root_002dFinding.html#Multidimensional-Root_002dFinding" accesskey="u" rel="up">Multidimensional Root-Finding</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Providing-the-function-to-solve-2"></a>
<h3 class="section">36.3 Providing the function to solve</h3>
<a name="index-multidimensional-root-finding_002c-providing-a-function-to-solve"></a>

<p>You must provide <em>n</em> functions of <em>n</em> variables for the root
finders to operate on.  In order to allow for general parameters the
functions are defined by the following data types:
</p>
<dl>
<dt><a name="index-gsl_005fmultiroot_005ffunction"></a>Data Type: <strong>gsl_multiroot_function</strong></dt>
<dd><p>This data type defines a general system of functions with parameters.
</p>
<dl compact="compact">
<dt><code>int (* f) (const gsl_vector * <var>x</var>, void * <var>params</var>, gsl_vector * <var>f</var>)</code></dt>
<dd><p>this function should store the vector result
<em>f(x,params)</em> 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.
</p>
</dd>
<dt><code>size_t n</code></dt>
<dd><p>the dimension of the system, i.e. the number of components of the
vectors <var>x</var> and <var>f</var>.
</p>
</dd>
<dt><code>void * params</code></dt>
<dd><p>a pointer to the parameters of the function.
</p></dd>
</dl>
</dd></dl>

<p>Here is an example using Powell&rsquo;s test function,
</p>
<div class="example">
<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></div>

<p>with <em>A = 10^4</em>.  The following code defines a
<code>gsl_multiroot_function</code> system <code>F</code> which you could pass to a
solver:
</p>
<div class="example">
<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-&gt;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 = &amp;powell;
F.n = 2;
F.params = &amp;params;
</pre></div>

<dl>
<dt><a name="index-gsl_005fmultiroot_005ffunction_005ffdf"></a>Data Type: <strong>gsl_multiroot_function_fdf</strong></dt>
<dd><p>This data type defines a general system of functions with parameters and
the corresponding Jacobian matrix of derivatives,
</p>
<dl compact="compact">
<dt><code>int (* f) (const gsl_vector * <var>x</var>, void * <var>params</var>, gsl_vector * <var>f</var>)</code></dt>
<dd><p>this function should store the vector result
<em>f(x,params)</em> 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.
</p>
</dd>
<dt><code>int (* df) (const gsl_vector * <var>x</var>, void * <var>params</var>, gsl_matrix * <var>J</var>)</code></dt>
<dd><p>this function should store the <var>n</var>-by-<var>n</var> matrix result
<em>J_ij = d f_i(x,params) / d x_j</em> 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.
</p>
</dd>
<dt><code>int (* fdf) (const gsl_vector * <var>x</var>, void * <var>params</var>, gsl_vector * <var>f</var>, gsl_matrix * <var>J</var>)</code></dt>
<dd><p>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 <em>f(x)</em> and
<em>J(x)</em>&mdash;it is always faster to compute the function and its
derivative at the same time.
</p>
</dd>
<dt><code>size_t n</code></dt>
<dd><p>the dimension of the system, i.e. the number of components of the
vectors <var>x</var> and <var>f</var>.
</p>
</dd>
<dt><code>void * params</code></dt>
<dd><p>a pointer to the parameters of the function.
</p></dd>
</dl>
</dd></dl>

<p>The example of Powell&rsquo;s test function defined above can be extended to
include analytic derivatives using the following code,
</p>
<div class="example">
<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-&gt;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-&gt;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 = &amp;powell_f;
FDF.df = &amp;powell_df;
FDF.fdf = &amp;powell_fdf;
FDF.n = 2;
FDF.params = 0;
</pre></div>

<p>Note that the function <code>powell_fdf</code> is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Iteration-of-the-multidimensional-solver.html#Iteration-of-the-multidimensional-solver" accesskey="n" rel="next">Iteration of the multidimensional solver</a>, Previous: <a href="Initializing-the-Multidimensional-Solver.html#Initializing-the-Multidimensional-Solver" accesskey="p" rel="previous">Initializing the Multidimensional Solver</a>, Up: <a href="Multidimensional-Root_002dFinding.html#Multidimensional-Root_002dFinding" accesskey="u" rel="up">Multidimensional Root-Finding</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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