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

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<a name="Minimization-Caveats"></a>
<div class="header">
<p>
Next: <a href="Initializing-the-Minimizer.html#Initializing-the-Minimizer" accesskey="n" rel="next">Initializing the Minimizer</a>, Previous: <a href="Minimization-Overview.html#Minimization-Overview" accesskey="p" rel="previous">Minimization Overview</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="Caveats-1"></a>
<h3 class="section">35.2 Caveats</h3>
<a name="index-minimization_002c-caveats"></a>

<p>Note that minimization functions can only search for one minimum at a
time.  When there are several minima in the search area, the first
minimum to be found will be returned; however it is difficult to predict
which of the minima this will be. <em>In most cases, no error will be
reported if you try to find a minimum in an area where there is more
than one.</em>
</p>
<p>With all minimization algorithms it can be difficult to determine the
location of the minimum to full numerical precision.  The behavior of the
function in the region of the minimum <em>x^*</em> can be approximated by
a Taylor expansion,
</p>
<div class="example">
<pre class="example">y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2
</pre></div>

<p>and the second term of this expansion can be lost when added to the
first term at finite precision.  This magnifies the error in locating
<em>x^*</em>, making it proportional to <em>\sqrt \epsilon</em> (where
<em>\epsilon</em> is the relative accuracy of the floating point numbers).
For functions with higher order minima, such as <em>x^4</em>, the
magnification of the error is correspondingly worse.  The best that can
be achieved is to converge to the limit of numerical accuracy in the
function values, rather than the location of the minimum itself.
</p>



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