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<title>GNU Scientific Library &ndash; Reference Manual: Providing a function to minimize</title>

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<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Providing-a-function-to-minimize"></a>
<div class="header">
<p>
Next: <a href="Multimin-Iteration.html#Multimin-Iteration" accesskey="n" rel="next">Multimin Iteration</a>, Previous: <a href="Initializing-the-Multidimensional-Minimizer.html#Initializing-the-Multidimensional-Minimizer" accesskey="p" rel="previous">Initializing the Multidimensional Minimizer</a>, Up: <a href="Multidimensional-Minimization.html#Multidimensional-Minimization" accesskey="u" rel="up">Multidimensional Minimization</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Providing-a-function-to-minimize-1"></a>
<h3 class="section">37.4 Providing a function to minimize</h3>

<p>You must provide a parametric function of <em>n</em> variables for the
minimizers to operate on.  You may also need to provide a routine which
calculates the gradient of the function and a third routine which
calculates both the function value and the gradient together.  In order
to allow for general parameters the functions are defined by the
following data types:
</p>
<dl>
<dt><a name="index-gsl_005fmultimin_005ffunction_005ffdf"></a>Data Type: <strong>gsl_multimin_function_fdf</strong></dt>
<dd><p>This data type defines a general function of <em>n</em> variables with
parameters and the corresponding gradient vector of derivatives,
</p>
<dl compact="compact">
<dt><code>double (* f) (const gsl_vector * <var>x</var>, void * <var>params</var>)</code></dt>
<dd><p>this function should return the result
<em>f(x,params)</em> for argument <var>x</var> and parameters <var>params</var>.
If the function cannot be computed, an error value of <code>GSL_NAN</code>
should be returned.
</p>
</dd>
<dt><code>void (* df) (const gsl_vector * <var>x</var>, void * <var>params</var>, gsl_vector * <var>g</var>)</code></dt>
<dd><p>this function should store the <var>n</var>-dimensional gradient
<em>g_i = d f(x,params) / d x_i</em> in the vector <var>g</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>void (* fdf) (const gsl_vector * <var>x</var>, void * <var>params</var>, double * f, gsl_vector * <var>g</var>)</code></dt>
<dd><p>This function should set the values of the <var>f</var> and <var>g</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>g(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>.
</p>
</dd>
<dt><code>void * params</code></dt>
<dd><p>a pointer to the parameters of the function.
</p></dd>
</dl>
</dd></dl>
<dl>
<dt><a name="index-gsl_005fmultimin_005ffunction"></a>Data Type: <strong>gsl_multimin_function</strong></dt>
<dd><p>This data type defines a general function of <em>n</em> variables with
parameters,
</p>
<dl compact="compact">
<dt><code>double (* f) (const gsl_vector * <var>x</var>, void * <var>params</var>)</code></dt>
<dd><p>this function should return the result
<em>f(x,params)</em> for argument <var>x</var> and parameters <var>params</var>.
If the function cannot be computed, an error value of <code>GSL_NAN</code>
should be returned.
</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>.
</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 following example function defines a simple two-dimensional
paraboloid with five parameters,
</p>
<div class="example">
<pre class="verbatim">/* Paraboloid centered on (p[0],p[1]), with  
   scale factors (p[2],p[3]) and minimum p[4] */

double
my_f (const gsl_vector *v, void *params)
{
  double x, y;
  double *p = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  return p[2] * (x - p[0]) * (x - p[0]) +
           p[3] * (y - p[1]) * (y - p[1]) + p[4]; 
}

/* The gradient of f, df = (df/dx, df/dy). */
void 
my_df (const gsl_vector *v, void *params, 
       gsl_vector *df)
{
  double x, y;
  double *p = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
  gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));
}

/* Compute both f and df together. */
void 
my_fdf (const gsl_vector *x, void *params, 
        double *f, gsl_vector *df) 
{
  *f = my_f(x, params); 
  my_df(x, params, df);
}
</pre></div>

<p>The function can be initialized using the following code,
</p>
<div class="example">
<pre class="example">gsl_multimin_function_fdf my_func;

/* Paraboloid center at (1,2), scale factors (10, 20), 
   minimum value 30 */
double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 }; 

my_func.n = 2;  /* number of function components */
my_func.f = &amp;my_f;
my_func.df = &amp;my_df;
my_func.fdf = &amp;my_fdf;
my_func.params = (void *)p;
</pre></div>

<hr>
<div class="header">
<p>
Next: <a href="Multimin-Iteration.html#Multimin-Iteration" accesskey="n" rel="next">Multimin Iteration</a>, Previous: <a href="Initializing-the-Multidimensional-Minimizer.html#Initializing-the-Multidimensional-Minimizer" accesskey="p" rel="previous">Initializing the Multidimensional Minimizer</a>, Up: <a href="Multidimensional-Minimization.html#Multidimensional-Minimization" accesskey="u" rel="up">Multidimensional Minimization</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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