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<a name="Nonlinear-Least_002dSquares-Testing-for-Convergence"></a>
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<p>
Next: <a href="Nonlinear-Least_002dSquares-High-Level-Driver.html#Nonlinear-Least_002dSquares-High-Level-Driver" accesskey="n" rel="next">Nonlinear Least-Squares High Level Driver</a>, Previous: <a href="Nonlinear-Least_002dSquares-Iteration.html#Nonlinear-Least_002dSquares-Iteration" accesskey="p" rel="previous">Nonlinear Least-Squares Iteration</a>, Up: <a href="Nonlinear-Least_002dSquares-Fitting.html#Nonlinear-Least_002dSquares-Fitting" accesskey="u" rel="up">Nonlinear Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Testing-for-Convergence"></a>
<h3 class="section">39.8 Testing for Convergence</h3>
<a name="index-nonlinear-fitting_002c-stopping-parameters_002c-convergence"></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 functions
below allow the user to test the current estimate of the best-fit
parameters in several standard ways.
</p>
<dl>
<dt><a name="index-gsl_005fmultifit_005fnlinear_005ftest"></a>Function: <em>int</em> <strong>gsl_multifit_nlinear_test</strong> <em>(const double <var>xtol</var>, const double <var>gtol</var>, const double <var>ftol</var>, int * <var>info</var>, const gsl_multifit_nlinear_workspace * <var>w</var>)</em></dt>
<dt><a name="index-gsl_005fmultilarge_005fnlinear_005ftest"></a>Function: <em>int</em> <strong>gsl_multilarge_nlinear_test</strong> <em>(const double <var>xtol</var>, const double <var>gtol</var>, const double <var>ftol</var>, int * <var>info</var>, const gsl_multilarge_nlinear_workspace * <var>w</var>)</em></dt>
<dd><p>These functions test for convergence of the minimization method
using the following criteria:
</p>
<ul>
<li> Testing for a small step size relative to the current parameter vector
<div class="example">
<pre class="example">|\delta_i| <= xtol (|x_i| + xtol)
</pre></div>
<p>for each <em>0 <= i < p</em>. Each element of the step vector <em>\delta</em>
is tested individually in case the different parameters have widely
different scales. Adding <var>xtol</var> to <em>|x_i|</em> helps the test avoid
breaking down in situations where the true solution value <em>x_i = 0</em>.
If this test succeeds, <var>info</var> is set to 1 and the function
returns <code>GSL_SUCCESS</code>.
</p>
<p>A general guideline for selecting the step tolerance is to choose
<em>xtol = 10^{-d}</em> where <em>d</em> is the number of accurate
decimal digits desired in the solution <em>x</em>. See Dennis and
Schnabel for more information.
</p>
</li><li> Testing for a small gradient (<em>g = \nabla \Phi(x) = J^T f</em>)
indicating a local function minimum:
<div class="example">
<pre class="example">||g||_inf <= gtol
</pre></div>
<p>This expression tests whether the ratio
<em>(\nabla \Phi)_i x_i / \Phi</em> is small. Testing this scaled gradient
is a better than <em>\nabla \Phi</em> alone since it is a dimensionless
quantity and so independent of the scale of the problem. The
<code>max</code> arguments help ensure the test doesn’t break down in
regions where <em>x_i</em> or <em>\Phi(x)</em> are close to 0.
If this test succeeds, <var>info</var> is set to 2 and the function
returns <code>GSL_SUCCESS</code>.
</p>
<p>A general guideline for choosing the gradient tolerance is to set
<code>gtol = GSL_DBL_EPSILON^(1/3)</code>. See Dennis and Schnabel for
more information.
</p>
</li></ul>
<p>If none of the tests succeed, <var>info</var> is set to 0 and the
function returns <code>GSL_CONTINUE</code>, indicating further iterations
are required.
</p>
</dd></dl>
<hr>
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Next: <a href="Nonlinear-Least_002dSquares-High-Level-Driver.html#Nonlinear-Least_002dSquares-High-Level-Driver" accesskey="n" rel="next">Nonlinear Least-Squares High Level Driver</a>, Previous: <a href="Nonlinear-Least_002dSquares-Iteration.html#Nonlinear-Least_002dSquares-Iteration" accesskey="p" rel="previous">Nonlinear Least-Squares Iteration</a>, Up: <a href="Nonlinear-Least_002dSquares-Fitting.html#Nonlinear-Least_002dSquares-Fitting" accesskey="u" rel="up">Nonlinear Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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