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  27 <a name="math_toolkit.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without Derivatives</a>
  28 </h2></div></div></div>
  29 <div class="toc"><dl class="toc">
  30 <dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt>
  31 <dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket and
  32       Solve Root</a></span></dt>
  33 <dt><span class="section"><a href="roots_noderiv/TOMS748.html">Algorithm TOMS 748:
  34       Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
  35 <dt><span class="section"><a href="roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
  36 <dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination
  37       Condition Functors</a></span></dt>
  38 <dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt>
  39 </dl></div>
  40 <h5>
  41 <a name="math_toolkit.roots_noderiv.h0"></a>
  42       <span class="phrase"><a name="math_toolkit.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.synopsis">Synopsis</a>
  43     </h5>
  44 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  45 </pre>
  46 <pre class="programlisting"> <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
  47  <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span>
  48  <span class="comment">// Bisection</span>
  49  <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>
  50  <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>
  51     <span class="identifier">bisect</span><span class="special">(</span>
  52        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  53        <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  54        <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  55        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  56        <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>
  57
  58  <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>
  59  <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>
  60     <span class="identifier">bisect</span><span class="special">(</span>
  61        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  62        <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  63        <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  64        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
  65
  66  <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  67  <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>
  68     <span class="identifier">bisect</span><span class="special">(</span>
  69        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  70        <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  71        <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  72        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  73        <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>
  74        <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  75
  76  <span class="comment">// Bracket and Solve Root</span>
  77  <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>
  78  <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>
  79     <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
  80        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  81        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
  82        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
  83        <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
  84        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  85        <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>
  86
  87  <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  88  <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>
  89     <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
  90        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  91        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
  92        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
  93        <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
  94        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  95        <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>
  96        <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  97
  98 <span class="comment">// TOMS 748 algorithm</span>
  99  <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>
 100  <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>
 101     <span class="identifier">toms748_solve</span><span class="special">(</span>
 102        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
 103        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 104        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
 105        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
 106        <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>
 107
 108  <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 109  <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>
 110     <span class="identifier">toms748_solve</span><span class="special">(</span>
 111        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
 112        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 113        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
 114        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
 115        <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>
 116        <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 117
 118  <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>
 119  <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>
 120     <span class="identifier">toms748_solve</span><span class="special">(</span>
 121        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
 122        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 123        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
 124        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
 125        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
 126        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
 127        <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>
 128
 129  <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 130  <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>
 131     <span class="identifier">toms748_solve</span><span class="special">(</span>
 132        <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
 133        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 134        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
 135        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
 136        <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
 137        <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
 138        <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>
 139        <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 140
 141  <span class="comment">// Termination conditions:</span>
 142  <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>
 143  <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>
 144
 145  <span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
 146  <span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
 147  <span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>
 148
 149  <span class="special">}}}</span> <span class="comment">// boost::math::tools namespaces</span>
 150 </pre>
 151 <h5>
 152 <a name="math_toolkit.roots_noderiv.h1"></a>
 153       <span class="phrase"><a name="math_toolkit.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.description">Description</a>
 154     </h5>
 155 <p>
 156       These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span> -
 157       <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
 158     </p>
 159 <p>
 160       The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
 161       functions use <a class="link" href="roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a>
 162       by Alefeld, Potra and Shi that is asymptotically the most efficient known,
 163       and has been shown to be optimal for a certain classes of smooth functions.
 164       Variants with and without <a class="link" href="../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a> are provided.
 165     </p>
 166 <p>
 167       Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
 168       is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
 169       routine which can be useful in its own right in some situations, or alternatively
 170       for narrowing down the range containing the root, prior to calling a more advanced
 171       algorithm.
 172     </p>
 173 <p>
 174       All the algorithms in this section reduce the diameter of the enclosing interval
 175       with the same asymptotic efficiency with which they locate the root. This is
 176       in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
 177       significantly reduce the enclosing interval, even though they rapidly approach
 178       the root. This is also in contrast to some other derivative-free methods (for
 179       example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a>
 180       which only reduces the enclosing interval on the final step. Therefore these
 181       methods return a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> containing the enclosing interval found,
 182       and accept a function object specifying the termination condition.
 183     </p>
 184 <p>
 185       Three function objects are provided for ready-made termination conditions:
 186     </p>
 187 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 188 <li class="listitem">
 189           <span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative
 190           error in the enclosing interval is below a certain threshold.
 191         </li>
 192 <li class="listitem">
 193           <span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span> are
 194           useful for certain statistical applications where the result is known to
 195           be an integer.
 196         </li>
 197 <li class="listitem">
 198           Other user-defined termination conditions are likely to be used only rarely,
 199           but may be useful in some specific circumstances.
 200         </li>
 201 </ul></div>
 202 </div>
 203 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 204 <td align="left"></td>
 205 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 206       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 207       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 208       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 209       Daryle Walker and Xiaogang Zhang<p>
 210         Distributed under the Boost Software License, Version 1.0. (See accompanying
 211         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>)
 212       </p>
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