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<a name="Defining-the-ODE-System"></a>
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<h3 class="section">25.1 Defining the ODE System</h3>
<p>The routines solve the general n-dimensional first-order system,
<pre class="example"> dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
</pre>
<p class="noindent">for i = 1, \dots, n. The stepping functions rely on the vector
of derivatives f_i and the Jacobian matrix,
<!-- {$J_{ij} = \partial f_i(t, y(t)) / \partial y_j$} -->
J_{ij} = df_i(t,y(t)) / dy_j.
A system of equations is defined using the <code>gsl_odeiv_system</code>
datatype.
<div class="defun">
— Data Type: <b>gsl_odeiv_system</b><var><a name="index-gsl_005fodeiv_005fsystem-2032"></a></var><br>
<blockquote><p>This data type defines a general ODE system with arbitrary parameters.
<dl>
<dt><code>int (* function) (double t, const double y[], double dydt[], void * params)</code><dd>This function should store the vector elements
<!-- {$f_i(t,y,\hbox{\it params})$} -->
f_i(t,y,params) in the array <var>dydt</var>,
for arguments (<var>t</var>,<var>y</var>) and parameters <var>params</var>.
The function should return <code>GSL_SUCCESS</code> if the calculation
was completed successfully. Any other return value indicates
an error.
<br><dt><code>int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);</code><dd>This function should store the vector of derivative elements
<!-- {$\partial f_i(t,y,params) / \partial t$} -->
df_i(t,y,params)/dt in the array <var>dfdt</var> and the
Jacobian matrix <!-- {$J_{ij}$} -->
J_{ij} in the array
<var>dfdy</var>, regarded as a row-ordered matrix <code>J(i,j) = dfdy[i * dimension + j]</code>
where <code>dimension</code> is the dimension of the system.
The function should return <code>GSL_SUCCESS</code> if the calculation
was completed successfully. Any other return value indicates
an error.
<p>Some of the simpler solver algorithms do not make use of the Jacobian
matrix, so it is not always strictly necessary to provide it (the
<code>jacobian</code> element of the struct can be replaced by a null pointer
for those algorithms). However, it is useful to provide the Jacobian to allow
the solver algorithms to be interchanged—the best algorithms make use
of the Jacobian.
<br><dt><code>size_t dimension;</code><dd>This is the dimension of the system of equations.
<br><dt><code>void * params</code><dd>This is a pointer to the arbitrary parameters of the system.
</dl>
</p></blockquote></div>
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