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<title>Providing the function to solve - GNU Scientific Library -- Reference Manual</title>
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<a name="Providing-the-function-to-solve"></a>
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<hr>
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<h3 class="section">33.4 Providing the function to solve</h3>
<p><a name="index-root-finding_002c-providing-a-function-to-solve-2360"></a>
You must provide a continuous function of one variable for the root
finders to operate on, and, sometimes, its first derivative. In order
to allow for general parameters the functions are defined by the
following data types:
<div class="defun">
— Data Type: <b>gsl_function</b><var><a name="index-gsl_005ffunction-2361"></a></var><br>
<blockquote><p>This data type defines a general function with parameters.
<dl>
<dt><code>double (* function) (double </code><var>x</var><code>, void * </code><var>params</var><code>)</code><dd>this function should return the value
<!-- {$f(x,\hbox{\it params})$} -->
f(x,params) for argument <var>x</var> and parameters <var>params</var>
<br><dt><code>void * params</code><dd>a pointer to the parameters of the function
</dl>
</p></blockquote></div>
<p>Here is an example for the general quadratic function,
<pre class="example"> f(x) = a x^2 + b x + c
</pre>
<p class="noindent">with a = 3, b = 2, c = 1. The following code
defines a <code>gsl_function</code> <code>F</code> which you could pass to a root
finder:
<pre class="example"> struct my_f_params { double a; double b; double c; };
double
my_f (double x, void * p) {
struct my_f_params * params
= (struct my_f_params *)p;
double a = (params->a);
double b = (params->b);
double c = (params->c);
return (a * x + b) * x + c;
}
gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };
F.function = &my_f;
F.params = &params;
</pre>
<p class="noindent">The function f(x) can be evaluated using the following macro,
<pre class="example"> #define GSL_FN_EVAL(F,x)
(*((F)->function))(x,(F)->params)
</pre>
<div class="defun">
— Data Type: <b>gsl_function_fdf</b><var><a name="index-gsl_005ffunction_005ffdf-2362"></a></var><br>
<blockquote><p>This data type defines a general function with parameters and its first
derivative.
<dl>
<dt><code>double (* f) (double </code><var>x</var><code>, void * </code><var>params</var><code>)</code><dd>this function should return the value of
<!-- {$f(x,\hbox{\it params})$} -->
f(x,params) for argument <var>x</var> and parameters <var>params</var>
<br><dt><code>double (* df) (double </code><var>x</var><code>, void * </code><var>params</var><code>)</code><dd>this function should return the value of the derivative of <var>f</var> with
respect to <var>x</var>,
<!-- {$f'(x,\hbox{\it params})$} -->
f'(x,params), for argument <var>x</var> and parameters <var>params</var>
<br><dt><code>void (* fdf) (double </code><var>x</var><code>, void * </code><var>params</var><code>, double * </code><var>f</var><code>, double * </code><var>df</var><code>)</code><dd>this function should set the values of the function <var>f</var> to
<!-- {$f(x,\hbox{\it params})$} -->
f(x,params)
and its derivative <var>df</var> to
<!-- {$f'(x,\hbox{\it params})$} -->
f'(x,params)
for argument <var>x</var> and parameters <var>params</var>. This function
provides an optimization of the separate functions for f(x) and
f'(x)—it is always faster to compute the function and its
derivative at the same time.
<br><dt><code>void * params</code><dd>a pointer to the parameters of the function
</dl>
</p></blockquote></div>
<p>Here is an example where
<!-- {$f(x) = \exp(2x)$} -->
f(x) = 2\exp(2x):
<pre class="example"> double
my_f (double x, void * params)
{
return exp (2 * x);
}
double
my_df (double x, void * params)
{
return 2 * exp (2 * x);
}
void
my_fdf (double x, void * params,
double * f, double * df)
{
double t = exp (2 * x);
*f = t;
*df = 2 * t; /* uses existing value */
}
gsl_function_fdf FDF;
FDF.f = &my_f;
FDF.df = &my_df;
FDF.fdf = &my_fdf;
FDF.params = 0;
</pre>
<p class="noindent">The function f(x) can be evaluated using the following macro,
<pre class="example"> #define GSL_FN_FDF_EVAL_F(FDF,x)
(*((FDF)->f))(x,(FDF)->params)
</pre>
<p class="noindent">The derivative f'(x) can be evaluated using the following macro,
<pre class="example"> #define GSL_FN_FDF_EVAL_DF(FDF,x)
(*((FDF)->df))(x,(FDF)->params)
</pre>
<p class="noindent">and both the function y = f(x) and its derivative dy = f'(x)
can be evaluated at the same time using the following macro,
<pre class="example"> #define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy)
(*((FDF)->fdf))(x,(FDF)->params,(y),(dy))
</pre>
<p class="noindent">The macro stores f(x) in its <var>y</var> argument and f'(x) in
its <var>dy</var> argument—both of these should be pointers to
<code>double</code>.
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