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

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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">&lt;</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">&gt;</span>
  30 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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">&gt;</span>
  38 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</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">&amp;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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 &lt; max</em></span> and
  79                 <span class="emphasis"><em>f(min)*f(max) &lt;= 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&#160;15.&#160;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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> is changed to not throw
  85                 then it returns <span class="emphasis"><em>std::pair&lt;T&gt;(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&#160;15.&#160;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&#160;15.&#160;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">&lt;=</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">&gt;=</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 &#169; 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 R&#229;de, 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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