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<title>GNU Scientific Library – Reference Manual: Search Stopping Parameters</title>
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<a name="Search-Stopping-Parameters"></a>
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<p>
Next: <a href="Root-Bracketing-Algorithms.html#Root-Bracketing-Algorithms" accesskey="n" rel="next">Root Bracketing Algorithms</a>, Previous: <a href="Root-Finding-Iteration.html#Root-Finding-Iteration" accesskey="p" rel="previous">Root Finding Iteration</a>, Up: <a href="One-dimensional-Root_002dFinding.html#One-dimensional-Root_002dFinding" accesskey="u" rel="up">One dimensional Root-Finding</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Search-Stopping-Parameters-1"></a>
<h3 class="section">34.7 Search Stopping Parameters</h3>
<a name="index-root-finding_002c-stopping-parameters"></a>
<p>A root finding procedure should stop when one of the following conditions is
true:
</p>
<ul>
<li> A root 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 functions
below allow the user to test the precision of the current result in
several standard ways.
</p>
<dl>
<dt><a name="index-gsl_005froot_005ftest_005finterval"></a>Function: <em>int</em> <strong>gsl_root_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| < 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 roots close
to the origin.
</p>
<p>This condition on the interval also implies that any estimate of the
root <em>r</em> in the interval satisfies the same condition with respect
to the true root <em>r^*</em>,
</p>
<div class="example">
<pre class="example">|r - r^*| < epsabs + epsrel r^*
</pre></div>
<p>assuming that the true root <em>r^*</em> is contained within the interval.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005froot_005ftest_005fdelta"></a>Function: <em>int</em> <strong>gsl_root_test_delta</strong> <em>(double <var>x1</var>, double <var>x0</var>, double <var>epsabs</var>, double <var>epsrel</var>)</em></dt>
<dd>
<p>This function tests for the convergence of the sequence …, <var>x0</var>,
<var>x1</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">|x_1 - x_0| < epsabs + epsrel |x_1|
</pre></div>
<p>and returns <code>GSL_CONTINUE</code> otherwise.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005froot_005ftest_005fresidual"></a>Function: <em>int</em> <strong>gsl_root_test_residual</strong> <em>(double <var>f</var>, double <var>epsabs</var>)</em></dt>
<dd><p>This function tests the residual value <var>f</var> against the absolute
error bound <var>epsabs</var>. The test returns <code>GSL_SUCCESS</code> if the
following condition is achieved,
</p>
<div class="example">
<pre class="example">|f| < epsabs
</pre></div>
<p>and returns <code>GSL_CONTINUE</code> otherwise. This criterion is suitable
for situations where the precise location of the root, <em>x</em>, is
unimportant provided a value can be found where the residual,
<em>|f(x)|</em>, is small enough.
</p></dd></dl>
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Next: <a href="Root-Bracketing-Algorithms.html#Root-Bracketing-Algorithms" accesskey="n" rel="next">Root Bracketing Algorithms</a>, Previous: <a href="Root-Finding-Iteration.html#Root-Finding-Iteration" accesskey="p" rel="previous">Root Finding Iteration</a>, Up: <a href="One-dimensional-Root_002dFinding.html#One-dimensional-Root_002dFinding" accesskey="u" rel="up">One dimensional Root-Finding</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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