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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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">&</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> | |
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"><</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> | |
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"><</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> | |
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"><</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> | |
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> | |
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88 | <td align="right"><div class="copyright-footer">Copyright © 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 Råde, Gautam Sewani, | |
91 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
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