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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h4 class="title"> | |
27 | <a name="math_toolkit.roots.roots_noderiv.bisect"></a><a class="link" href="bisect.html" title="Bisection">Bisection</a> | |
28 | </h4></div></div></div> | |
29 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
30 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
31 | <span class="identifier">bisect</span><span class="special">(</span> <span class="comment">// Unlimited iterations.</span> | |
32 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
33 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
34 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
35 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span> | |
36 | ||
37 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
38 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
39 | <span class="identifier">bisect</span><span class="special">(</span> <span class="comment">// Limited iterations.</span> | |
40 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
41 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
42 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
43 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
44 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> | |
45 | ||
46 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
47 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
48 | <span class="identifier">bisect</span><span class="special">(</span> <span class="comment">// Specified policy.</span> | |
49 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
50 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
51 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
52 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
53 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span> | |
54 | <span class="keyword">const</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
55 | </pre> | |
56 | <p> | |
57 | These functions locate the root using <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>. | |
58 | </p> | |
59 | <p> | |
60 | <code class="computeroutput"><span class="identifier">bisect</span></code> function arguments | |
61 | are: | |
62 | </p> | |
63 | <div class="variablelist"> | |
64 | <p class="title"><b></b></p> | |
65 | <dl class="variablelist"> | |
66 | <dt><span class="term">f</span></dt> | |
67 | <dd><p> | |
68 | A unary functor which is the function <span class="emphasis"><em>f(x)</em></span> whose | |
69 | root is to be found. | |
70 | </p></dd> | |
71 | <dt><span class="term">min</span></dt> | |
72 | <dd><p> | |
73 | The left bracket of the interval known to contain the root. | |
74 | </p></dd> | |
75 | <dt><span class="term">max</span></dt> | |
76 | <dd><p> | |
77 | The right bracket of the interval known to contain the root.<br> | |
78 | It is a precondition that <span class="emphasis"><em>min < max</em></span> and | |
79 | <span class="emphasis"><em>f(min)*f(max) <= 0</em></span>, the function raises an | |
80 | <a class="link" href="../../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a> | |
81 | if these preconditions are violated. The action taken on error is | |
82 | controlled by the <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> template argument: | |
83 | the default behavior is to throw a <span class="emphasis"><em>boost::math::evaluation_error</em></span>. | |
84 | If the <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> is changed to not throw | |
85 | then it returns <span class="emphasis"><em>std::pair<T>(min, min)</em></span>. | |
86 | </p></dd> | |
87 | <dt><span class="term">tol</span></dt> | |
88 | <dd><p> | |
89 | A binary functor that specifies the termination condition: the function | |
90 | will return the current brackets enclosing the root when <span class="emphasis"><em>tol(min, | |
91 | max)</em></span> becomes true. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined | |
92 | termination functors</a>. | |
93 | </p></dd> | |
94 | <dt><span class="term">max_iter</span></dt> | |
95 | <dd><p> | |
96 | The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to | |
97 | make while searching for the root. On exit, this is updated to the | |
98 | actual number of invocations performed. | |
99 | </p></dd> | |
100 | </dl> | |
101 | </div> | |
102 | <p> | |
103 | The final <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and | |
104 | can be used to control the behaviour of the function: how it handles errors, | |
105 | what level of precision to use etc. Refer to the <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
106 | documentation for more details</a>. | |
107 | </p> | |
108 | <p> | |
109 | <span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span> | |
110 | that bracket the root so that: | |
111 | </p> | |
112 | <pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span> | |
113 | </pre> | |
114 | <p> | |
115 | and either | |
116 | </p> | |
117 | <pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span> | |
118 | </pre> | |
119 | <p> | |
120 | or | |
121 | </p> | |
122 | <pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span> | |
123 | </pre> | |
124 | <p> | |
125 | where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span> | |
126 | passed to the function. | |
127 | </p> | |
128 | <p> | |
129 | In other words, it's up to the caller to verify whether termination occurred | |
130 | as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations | |
131 | (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span> | |
132 | when the function returns), rather than because the termination condition | |
133 | <span class="emphasis"><em>tol</em></span> was satisfied. | |
134 | </p> | |
135 | </div> | |
136 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
137 | <td align="left"></td> | |
138 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
139 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
140 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
141 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
142 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
143 | 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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