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<a name="Defining-the-ODE-System"></a>
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<p>
Next: <a href="Stepping-Functions.html#Stepping-Functions" accesskey="n" rel="next">Stepping Functions</a>, Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Defining-the-ODE-System-1"></a>
<h3 class="section">27.1 Defining the ODE System</h3>

<p>The routines solve the general <em>n</em>-dimensional first-order system,
</p>
<div class="example">
<pre class="example">dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
</pre></div>

<p>for <em>i = 1, \dots, n</em>.  The stepping functions rely on the vector
of derivatives <em>f_i</em> and the Jacobian matrix, 
<em>J_{ij} = df_i(t,y(t)) / dy_j</em>. 
A system of equations is defined using the <code>gsl_odeiv2_system</code>
datatype.
</p>
<dl>
<dt><a name="index-gsl_005fodeiv2_005fsystem"></a>Data Type: <strong>gsl_odeiv2_system</strong></dt>
<dd><p>This data type defines a general ODE system with arbitrary parameters.
</p>
<dl compact="compact">
<dt><code>int (* function) (double t, const double y[], double dydt[], void * params)</code></dt>
<dd><p>This function should store the vector elements
<em>f_i(t,y,params)</em> in the array <var>dydt</var>,
for arguments (<var>t</var>,<var>y</var>) and parameters <var>params</var>.
</p>
<p>The function should return <code>GSL_SUCCESS</code> if the calculation was
completed successfully. Any other return value indicates an error. A
special return value <code>GSL_EBADFUNC</code> causes <code>gsl_odeiv2</code>
routines to immediately stop and return. If <code>function</code> 
is modified (for example contents of <var>params</var>), the user must call an
appropriate reset function (<code>gsl_odeiv2_driver_reset</code>, 
<code>gsl_odeiv2_evolve_reset</code> or <code>gsl_odeiv2_step_reset</code>) 
before continuing. Use return values
distinct from standard GSL error codes to distinguish your function as
the source of the error.
</p>
</dd>
<dt><code>int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);</code></dt>
<dd><a name="index-Jacobian-matrix_002c-ODEs"></a>
<p>This function should store the vector of derivative elements
in the array <var>dfdt</var> and the Jacobian matrix <em>J_{ij}</em> 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. 
</p>
<p>Not all of the stepper algorithms of <code>gsl_odeiv2</code> make use of the
Jacobian matrix, so it may not be necessary to provide this function
(the <code>jacobian</code> element of the struct can be replaced by a null
pointer for those algorithms).
</p>
<p>The function should return <code>GSL_SUCCESS</code> if the calculation was
completed successfully. Any other return value indicates an error. A
special return value <code>GSL_EBADFUNC</code> causes <code>gsl_odeiv2</code>
routines to immediately stop and return. If <code>jacobian</code> 
is modified (for example contents of <var>params</var>), the user must call an
appropriate reset function (<code>gsl_odeiv2_driver_reset</code>, 
<code>gsl_odeiv2_evolve_reset</code> or <code>gsl_odeiv2_step_reset</code>) 
before continuing. Use return values
distinct from standard GSL error codes to distinguish your function as
the source of the error.
</p>
</dd>
<dt><code>size_t dimension;</code></dt>
<dd><p>This is the dimension of the system of equations.
</p>
</dd>
<dt><code>void * params</code></dt>
<dd><p>This is a pointer to the arbitrary parameters of the system.
</p></dd>
</dl>
</dd></dl>

<hr>
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<p>
Next: <a href="Stepping-Functions.html#Stepping-Functions" accesskey="n" rel="next">Stepping Functions</a>, Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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