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<title>Termination Condition Functors</title>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.roots_noderiv.root_termination"></a><a class="link" href="root_termination.html" title="Termination Condition Functors">Termination
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Condition Functors</a>
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</h3></div></div></div>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="keyword">struct</span> <span class="identifier">eps_tolerance</span>
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<span class="special">{</span>
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<span class="identifier">eps_tolerance</span><span class="special">();</span>
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<span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
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<span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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<code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
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termination condition used with these root finding functions. Its <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>
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will return true when the relative distance between <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span> is less than four times the machine epsilon for
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T, or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span>
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to the number of bits of precision you want in the result. The minimal tolerance
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of <span class="emphasis"><em>four times the machine epsilon of type T</em></span> is required
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to ensure that we get back a bracketing interval, since this must clearly
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be at greater than one epsilon in size. While in theory a maximum distance
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of twice machine epsilon is possible to achieve, in practice this results
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in a great deal of "thrashing" given that the function whose root
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is being found can only ever be accurate to 1 epsilon at best.
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</p>
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<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
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<span class="special">{</span>
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<span class="identifier">equal_floor</span><span class="special">();</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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This termination condition is used when you want to find an integer result
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that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
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as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
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</p>
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<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
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<span class="special">{</span>
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<span class="identifier">equal_ceil</span><span class="special">();</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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This termination condition is used when you want to find an integer result
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that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate
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as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
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</p>
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<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
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<span class="special">{</span>
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<span class="identifier">equal_nearest_integer</span><span class="special">();</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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This termination condition is used when you want to find an integer result
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that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
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as soon as both ends of the interval round to the same nearest integer.
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</p>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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