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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Root Finding Without Derivatives</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.roots.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without
Derivatives</a>
</h3></div></div></div>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket
and Solve Root</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/TOMS748.html">Algorithm
TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/brent.html">Brent-Decker
Algorithm</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination
Condition Functors</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt>
</dl></div>
<h5>
<a name="math_toolkit.roots.roots_noderiv.h0"></a>
<span class="phrase"><a name="math_toolkit.roots.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots.roots_noderiv.synopsis">Synopsis</a>
</h5>
<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>
</pre>
<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>
<span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span>
<span class="comment">// Bisection</span>
<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>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
<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>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<span class="comment">// Bracket and Solve Root</span>
<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>
<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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<span class="comment">// TOMS 748 algorithm</span>
<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>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</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="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</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="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<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>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</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="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</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="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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>
<span class="comment">// Termination conditions:</span>
<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">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>
<span class="special">}}}</span> <span class="comment">// boost::math::tools namespaces</span>
</pre>
<h5>
<a name="math_toolkit.roots.roots_noderiv.h1"></a>
<span class="phrase"><a name="math_toolkit.roots.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots.roots_noderiv.description">Description</a>
</h5>
<p>
These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span>
- <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
</p>
<p>
The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
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> by Alefeld, Potra and Shi that is asymptotically the
most efficient known, and has been shown to be optimal for a certain classes
of smooth functions. Variants with and without <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
are provided.
</p>
<p>
Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
routine which can be useful in its own right in some situations, or alternatively
for narrowing down the range containing the root, prior to calling a more
advanced algorithm.
</p>
<p>
All the algorithms in this section reduce the diameter of the enclosing interval
with the same asymptotic efficiency with which they locate the root. This
is in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
significantly reduce the enclosing interval, even though they rapidly approach
the root. This is also in contrast to some other derivative-free methods
(for example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a>
which only reduces the enclosing interval on the final step. Therefore these
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, and accept a function object specifying the termination condition.
</p>
<p>
Three function objects are provided for ready-made termination conditions:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative
error in the enclosing interval is below a certain threshold.
</li>
<li class="listitem">
<span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span>
are useful for certain statistical applications where the result is known
to be an integer.
</li>
<li class="listitem">
Other user-defined termination conditions are likely to be used only
rarely, but may be useful in some specific circumstances.
</li>
</ul></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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>)
</p>
</div></td>
</tr></table>
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