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<title>GNU Scientific Library &ndash; Reference Manual: Minimization Stopping Parameters</title>

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<a name="Minimization-Stopping-Parameters"></a>
<div class="header">
<p>
Next: <a href="Minimization-Algorithms.html#Minimization-Algorithms" accesskey="n" rel="next">Minimization Algorithms</a>, Previous: <a href="Minimization-Iteration.html#Minimization-Iteration" accesskey="p" rel="previous">Minimization Iteration</a>, Up: <a href="One-dimensional-Minimization.html#One-dimensional-Minimization" accesskey="u" rel="up">One dimensional Minimization</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Stopping-Parameters"></a>
<h3 class="section">35.6 Stopping Parameters</h3>
<a name="index-minimization_002c-stopping-parameters"></a>

<p>A minimization procedure should stop when one of the following
conditions is true:
</p>
<ul>
<li> A minimum has been found to within the user-specified precision.

</li><li> A user-specified maximum number of iterations has been reached.

</li><li> An error has occurred.
</li></ul>

<p>The handling of these conditions is under user control.  The function
below allows the user to test the precision of the current result.
</p>
<dl>
<dt><a name="index-gsl_005fmin_005ftest_005finterval"></a>Function: <em>int</em> <strong>gsl_min_test_interval</strong> <em>(double <var>x_lower</var>, double <var>x_upper</var>,  double <var>epsabs</var>, double <var>epsrel</var>)</em></dt>
<dd><p>This function tests for the convergence of the interval [<var>x_lower</var>,
<var>x_upper</var>] with absolute error <var>epsabs</var> and relative error
<var>epsrel</var>.  The test returns <code>GSL_SUCCESS</code> if the following
condition is achieved,
</p>
<div class="example">
<pre class="example">|a - b| &lt; epsabs + epsrel min(|a|,|b|) 
</pre></div>

<p>when the interval <em>x = [a,b]</em> does not include the origin.  If the
interval includes the origin then <em>\min(|a|,|b|)</em> is replaced by
zero (which is the minimum value of <em>|x|</em> over the interval).  This
ensures that the relative error is accurately estimated for minima close
to the origin.
</p>
<p>This condition on the interval also implies that any estimate of the
minimum <em>x_m</em> in the interval satisfies the same condition with respect
to the true minimum <em>x_m^*</em>,
</p>
<div class="example">
<pre class="example">|x_m - x_m^*| &lt; epsabs + epsrel x_m^*
</pre></div>

<p>assuming that the true minimum <em>x_m^*</em> is contained within the interval.
</p></dd></dl>





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