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<title>Adaptive Step-size Control - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">25.3 Adaptive Step-size Control</h3>
<p><a name="index-Adaptive-step_002dsize-control_002c-differential-equations-2062"></a>
The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.
<div class="defun">
— Function: gsl_odeiv_control * <b>gsl_odeiv_control_standard_new</b> (<var>double eps_abs, double eps_rel, double a_y, double a_dydt</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fstandard_005fnew-2063"></a></var><br>
<blockquote><p>The standard control object is a four parameter heuristic based on
absolute and relative errors <var>eps_abs</var> and <var>eps_rel</var>, and
scaling factors <var>a_y</var> and <var>a_dydt</var> for the system state
y(t) and derivatives y'(t) respectively.
<p>The step-size adjustment procedure for this method begins by computing
the desired error level D_i for each component,
<pre class="example"> D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
</pre>
<p class="noindent">and comparing it with the observed error E_i = |yerr_i|. If the
observed error <var>E</var> exceeds the desired error level <var>D</var> by more
than 10% for any component then the method reduces the step-size by an
appropriate factor,
<pre class="example"> h_new = h_old * S * (E/D)^(-1/q)
</pre>
<p class="noindent">where q is the consistency order of the method (e.g. q=4 for
4(5) embedded RK), and S is a safety factor of 0.9. The ratio
E/D is taken to be the maximum of the ratios
E_i/D_i.
<p>If the observed error E is less than 50% of the desired error
level <var>D</var> for the maximum ratio E_i/D_i then the algorithm
takes the opportunity to increase the step-size to bring the error in
line with the desired level,
<pre class="example"> h_new = h_old * S * (E/D)^(-1/(q+1))
</pre>
<p class="noindent">This encompasses all the standard error scaling methods. To avoid
uncontrolled changes in the stepsize, the overall scaling factor is
limited to the range 1/5 to 5.
</p></blockquote></div>
<div class="defun">
— Function: gsl_odeiv_control * <b>gsl_odeiv_control_y_new</b> (<var>double eps_abs, double eps_rel</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fy_005fnew-2064"></a></var><br>
<blockquote><p>This function creates a new control object which will keep the local
error on each step within an absolute error of <var>eps_abs</var> and
relative error of <var>eps_rel</var> with respect to the solution y_i(t).
This is equivalent to the standard control object with <var>a_y</var>=1 and
<var>a_dydt</var>=0.
</p></blockquote></div>
<div class="defun">
— Function: gsl_odeiv_control * <b>gsl_odeiv_control_yp_new</b> (<var>double eps_abs, double eps_rel</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fyp_005fnew-2065"></a></var><br>
<blockquote><p>This function creates a new control object which will keep the local
error on each step within an absolute error of <var>eps_abs</var> and
relative error of <var>eps_rel</var> with respect to the derivatives of the
solution y'_i(t). This is equivalent to the standard control
object with <var>a_y</var>=0 and <var>a_dydt</var>=1.
</p></blockquote></div>
<div class="defun">
— Function: gsl_odeiv_control * <b>gsl_odeiv_control_scaled_new</b> (<var>double eps_abs, double eps_rel, double a_y, double a_dydt, const double scale_abs</var>[]<var>, size_t dim</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fscaled_005fnew-2066"></a></var><br>
<blockquote><p>This function creates a new control object which uses the same algorithm
as <code>gsl_odeiv_control_standard_new</code> but with an absolute error
which is scaled for each component by the array <var>scale_abs</var>.
The formula for D_i for this control object is,
<pre class="example"> D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
</pre>
<p class="noindent">where s_i is the i-th component of the array <var>scale_abs</var>.
The same error control heuristic is used by the Matlab <span class="sc">ode</span> suite.
</p></blockquote></div>
<div class="defun">
— Function: gsl_odeiv_control * <b>gsl_odeiv_control_alloc</b> (<var>const gsl_odeiv_control_type * T</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005falloc-2067"></a></var><br>
<blockquote><p>This function returns a pointer to a newly allocated instance of a
control function of type <var>T</var>. This function is only needed for
defining new types of control functions. For most purposes the standard
control functions described above should be sufficient.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_odeiv_control_init</b> (<var>gsl_odeiv_control * c, double eps_abs, double eps_rel, double a_y, double a_dydt</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005finit-2068"></a></var><br>
<blockquote><p>This function initializes the control function <var>c</var> with the
parameters <var>eps_abs</var> (absolute error), <var>eps_rel</var> (relative
error), <var>a_y</var> (scaling factor for y) and <var>a_dydt</var> (scaling
factor for derivatives).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_odeiv_control_free</b> (<var>gsl_odeiv_control * c</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005ffree-2069"></a></var><br>
<blockquote><p>This function frees all the memory associated with the control function
<var>c</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_odeiv_control_hadjust</b> (<var>gsl_odeiv_control * c, gsl_odeiv_step * s, const double y</var>[]<var>, const double yerr</var>[]<var>, const double dydt</var>[]<var>, double * h</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fhadjust-2070"></a></var><br>
<blockquote><p>This function adjusts the step-size <var>h</var> using the control function
<var>c</var>, and the current values of <var>y</var>, <var>yerr</var> and <var>dydt</var>.
The stepping function <var>step</var> is also needed to determine the order
of the method. If the error in the y-values <var>yerr</var> is found to be
too large then the step-size <var>h</var> is reduced and the function returns
<code>GSL_ODEIV_HADJ_DEC</code>. If the error is sufficiently small then
<var>h</var> may be increased and <code>GSL_ODEIV_HADJ_INC</code> is returned. The
function returns <code>GSL_ODEIV_HADJ_NIL</code> if the step-size is
unchanged. The goal of the function is to estimate the largest
step-size which satisfies the user-specified accuracy requirements for
the current point.
</p></blockquote></div>
<div class="defun">
— Function: const char * <b>gsl_odeiv_control_name</b> (<var>const gsl_odeiv_control * c</var>)<var><a name="index-gsl_005fodeiv_005fcontrol_005fname-2071"></a></var><br>
<blockquote><p>This function returns a pointer to the name of the control function.
For example,
<pre class="example"> printf ("control method is '%s'\n",
gsl_odeiv_control_name (c));
</pre>
<p class="noindent">would print something like <code>control method is 'standard'</code>
</p></blockquote></div>
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