ceph/src/boost/libs/math/doc/html/math_toolkit/roots/roots_noderiv/root_termination.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Termination Condition Functors</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../roots_noderiv.html" title="Root Finding Without Derivatives">
   9 <link rel="prev" href="brent.html" title="Brent-Decker Algorithm">
  10 <link rel="next" href="implementation.html" title="Implementation">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="brent.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../roots_noderiv.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="implementation.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.roots.roots_noderiv.root_termination"></a><a class="link" href="root_termination.html" title="Termination Condition Functors">Termination
  28         Condition Functors</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  31 <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span>
  32 <span class="special">{</span>
  33    <span class="identifier">eps_tolerance</span><span class="special">();</span>
  34    <span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
  35    <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">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  36 <span class="special">};</span>
  37 </pre>
  38 <p>
  39           <code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
  40           termination condition used with these root finding functions. Its <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>
  41           will return true when the relative distance between <span class="emphasis"><em>a</em></span>
  42           and <span class="emphasis"><em>b</em></span> is less than four times the machine epsilon
  43           for T, or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span>
  44           to the number of bits of precision you want in the result. The minimal
  45           tolerance of <span class="emphasis"><em>four times the machine epsilon of type T</em></span>
  46           is required to ensure that we get back a bracketing interval, since this
  47           must clearly be at greater than one epsilon in size. While in theory a
  48           maximum distance of twice machine epsilon is possible to achieve, in practice
  49           this results in a great deal of "thrashing" given that the function
  50           whose root is being found can only ever be accurate to 1 epsilon at best.
  51         </p>
  52 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
  53 <span class="special">{</span>
  54    <span class="identifier">equal_floor</span><span class="special">();</span>
  55    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  56 <span class="special">};</span>
  57 </pre>
  58 <p>
  59           This termination condition is used when you want to find an integer result
  60           that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
  61           as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
  62         </p>
  63 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
  64 <span class="special">{</span>
  65    <span class="identifier">equal_ceil</span><span class="special">();</span>
  66    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  67 <span class="special">};</span>
  68 </pre>
  69 <p>
  70           This termination condition is used when you want to find an integer result
  71           that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate
  72           as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
  73         </p>
  74 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
  75 <span class="special">{</span>
  76    <span class="identifier">equal_nearest_integer</span><span class="special">();</span>
  77    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  78 <span class="special">};</span>
  79 </pre>
  80 <p>
  81           This termination condition is used when you want to find an integer result
  82           that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
  83           as soon as both ends of the interval round to the same nearest integer.
  84         </p>
  85 </div>
  86 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
  87 <td align="left"></td>
  88 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
  89       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
  90       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
  91       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
  92         Distributed under the Boost Software License, Version 1.0. (See accompanying
  93         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>)
  94       </p>
  95 </div></td>
  96 </tr></table>
  97 <hr>
  98 <div class="spirit-nav">
  99 <a accesskey="p" href="brent.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../roots_noderiv.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="implementation.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
 100 </div>
 101 </body>
 102 </html>