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<h3 class="section">25.2 Stepping Functions</h3>

<p>The lowest level components are the <dfn>stepping functions</dfn> which
advance a solution from time t to t+h for a fixed
step-size h and estimate the resulting local error.

<div class="defun">
&mdash; Function: gsl_odeiv_step * <b>gsl_odeiv_step_alloc</b> (<var>const gsl_odeiv_step_type * T, size_t dim</var>)<var><a name="index-gsl_005fodeiv_005fstep_005falloc-2033"></a></var><br>
<blockquote><p>This function returns a pointer to a newly allocated instance of a
stepping function of type <var>T</var> for a system of <var>dim</var> dimensions. 
</p></blockquote></div>

<div class="defun">
&mdash; Function: int <b>gsl_odeiv_step_reset</b> (<var>gsl_odeiv_step * s</var>)<var><a name="index-gsl_005fodeiv_005fstep_005freset-2034"></a></var><br>
<blockquote><p>This function resets the stepping function <var>s</var>.  It should be used
whenever the next use of <var>s</var> will not be a continuation of a
previous step. 
</p></blockquote></div>

<div class="defun">
&mdash; Function: void <b>gsl_odeiv_step_free</b> (<var>gsl_odeiv_step * s</var>)<var><a name="index-gsl_005fodeiv_005fstep_005ffree-2035"></a></var><br>
<blockquote><p>This function frees all the memory associated with the stepping function
<var>s</var>. 
</p></blockquote></div>

<div class="defun">
&mdash; Function: const char * <b>gsl_odeiv_step_name</b> (<var>const gsl_odeiv_step * s</var>)<var><a name="index-gsl_005fodeiv_005fstep_005fname-2036"></a></var><br>
<blockquote><p>This function returns a pointer to the name of the stepping function. 
For example,

     <pre class="example">          printf ("step method is '%s'\n",
                   gsl_odeiv_step_name (s));
</pre>
        <p class="noindent">would print something like <code>step method is 'rk4'</code>. 
</p></blockquote></div>

<div class="defun">
&mdash; Function: unsigned int <b>gsl_odeiv_step_order</b> (<var>const gsl_odeiv_step * s</var>)<var><a name="index-gsl_005fodeiv_005fstep_005forder-2037"></a></var><br>
<blockquote><p>This function returns the order of the stepping function on the previous
step.  This order can vary if the stepping function itself is adaptive. 
</p></blockquote></div>

<div class="defun">
&mdash; Function: int <b>gsl_odeiv_step_apply</b> (<var>gsl_odeiv_step * s, double t, double h, double y</var>[]<var>, double yerr</var>[]<var>, const double dydt_in</var>[]<var>, double dydt_out</var>[]<var>, const gsl_odeiv_system * dydt</var>)<var><a name="index-gsl_005fodeiv_005fstep_005fapply-2038"></a></var><br>
<blockquote><p>This function applies the stepping function <var>s</var> to the system of
equations defined by <var>dydt</var>, using the step size <var>h</var> to advance
the system from time <var>t</var> and state <var>y</var> to time <var>t</var>+<var>h</var>. 
The new state of the system is stored in <var>y</var> on output, with an
estimate of the absolute error in each component stored in <var>yerr</var>. 
If the argument <var>dydt_in</var> is not null it should point an array
containing the derivatives for the system at time <var>t</var> on input. This
is optional as the derivatives will be computed internally if they are
not provided, but allows the reuse of existing derivative information. 
On output the new derivatives of the system at time <var>t</var>+<var>h</var> will
be stored in <var>dydt_out</var> if it is not null.

        <p>If the user-supplied functions defined in the system <var>dydt</var> return a
status other than <code>GSL_SUCCESS</code> the step will be aborted.  In this
case, the elements of <var>y</var> will be restored to their pre-step values
and the error code from the user-supplied function will be returned. 
The step-size <var>h</var> will be set to the step-size which caused the error. 
If the function is called again with a smaller step-size, e.g. <var>h</var>/10,
it should be possible to get closer to any singularity. 
To distinguish between error codes from the user-supplied functions and
those from <code>gsl_odeiv_step_apply</code> itself, any user-defined return
values should be distinct from the standard GSL error codes. 
</p></blockquote></div>

   <p>The following algorithms are available,

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rk2</b><var><a name="index-gsl_005fodeiv_005fstep_005frk2-2039"></a></var><br>
<blockquote><p><a name="index-RK2_002c-Runge_002dKutta-method-2040"></a><a name="index-Runge_002dKutta-methods_002c-ordinary-differential-equations-2041"></a>Embedded Runge-Kutta (2, 3) method. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rk4</b><var><a name="index-gsl_005fodeiv_005fstep_005frk4-2042"></a></var><br>
<blockquote><p><a name="index-RK4_002c-Runge_002dKutta-method-2043"></a>4th order (classical) Runge-Kutta. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rkf45</b><var><a name="index-gsl_005fodeiv_005fstep_005frkf45-2044"></a></var><br>
<blockquote><p><a name="index-Fehlberg-method_002c-differential-equations-2045"></a><a name="index-RKF45_002c-Runge_002dKutta_002dFehlberg-method-2046"></a>Embedded Runge-Kutta-Fehlberg (4, 5) method.  This method is a good
general-purpose integrator. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rkck</b><var><a name="index-gsl_005fodeiv_005fstep_005frkck-2047"></a></var><br>
<blockquote><p><a name="index-Runge_002dKutta-Cash_002dKarp-method-2048"></a><a name="index-Cash_002dKarp_002c-Runge_002dKutta-method-2049"></a>Embedded Runge-Kutta Cash-Karp (4, 5) method. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rk8pd</b><var><a name="index-gsl_005fodeiv_005fstep_005frk8pd-2050"></a></var><br>
<blockquote><p><a name="index-Runge_002dKutta-Prince_002dDormand-method-2051"></a><a name="index-Prince_002dDormand_002c-Runge_002dKutta-method-2052"></a>Embedded Runge-Kutta Prince-Dormand (8,9) method. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rk2imp</b><var><a name="index-gsl_005fodeiv_005fstep_005frk2imp-2053"></a></var><br>
<blockquote><p>Implicit 2nd order Runge-Kutta at Gaussian points. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_rk4imp</b><var><a name="index-gsl_005fodeiv_005fstep_005frk4imp-2054"></a></var><br>
<blockquote><p>Implicit 4th order Runge-Kutta at Gaussian points. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_bsimp</b><var><a name="index-gsl_005fodeiv_005fstep_005fbsimp-2055"></a></var><br>
<blockquote><p><a name="index-Bulirsch_002dStoer-method-2056"></a><a name="index-Bader-and-Deuflhard_002c-Bulirsch_002dStoer-method_002e-2057"></a><a name="index-Deuflhard-and-Bader_002c-Bulirsch_002dStoer-method_002e-2058"></a>Implicit Bulirsch-Stoer method of Bader and Deuflhard.  This algorithm
requires the Jacobian. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_gear1</b><var><a name="index-gsl_005fodeiv_005fstep_005fgear1-2059"></a></var><br>
<blockquote><p><a name="index-Gear-method_002c-differential-equations-2060"></a>M=1 implicit Gear method. 
</p></blockquote></div>

<div class="defun">
&mdash; Step Type: <b>gsl_odeiv_step_gear2</b><var><a name="index-gsl_005fodeiv_005fstep_005fgear2-2061"></a></var><br>
<blockquote><p>M=2 implicit Gear method. 
</p></blockquote></div>

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