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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.roots_noderiv.bisect"></a><a class="link" href="bisect.html" title="Bisection">Bisection</a>
</h3></div></div></div>
<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>
<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="comment">// Unlimited iterations.</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">&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="comment">// Limited iterations.</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">,</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>
<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="comment">// Specified policy.</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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
These functions locate the root using <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>.
</p>
<p>
<code class="computeroutput"><span class="identifier">bisect</span></code> function arguments
are:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
A unary functor (or C++ lambda) which is the function <span class="emphasis"><em>f(x)</em></span>
whose root is to be found.
</p></dd>
<dt><span class="term">min</span></dt>
<dd><p>
The left bracket of the interval known to contain the root.
</p></dd>
<dt><span class="term">max</span></dt>
<dd><p>
The right bracket of the interval known to contain the root.<br>
It is a precondition that <span class="emphasis"><em>min &lt; max</em></span> and <span class="emphasis"><em>f(min)*f(max)
&lt;= 0</em></span>, the function raises an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
if these preconditions are violated. The action taken on error is controlled
by the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> template argument: the
default behavior is to throw a <span class="emphasis"><em>boost::math::evaluation_error</em></span>.
If the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> is changed to not throw
then it returns <span class="emphasis"><em>std::pair&lt;T&gt;(min, min)</em></span>.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor (or C++ lambda) that specifies the termination condition:
the function will return the current brackets enclosing the root when
<span class="emphasis"><em>tol(min, max)</em></span> becomes true. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
termination functors</a>.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to make
while searching for the root. On exit, this is updated to the actual
number of invocations performed.
</p></dd>
</dl>
</div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
that bracket the root so that:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
f(r.first) * f(r.second) &lt;= 0
</p></blockquote></div>
<p>
and either
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
tol(r.first, r.second) == true
</p></blockquote></div>
<p>
or
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
max_iter &gt;= m
</p></blockquote></div>
<p>
where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
passed to the function.
</p>
<p>
In other words, it's up to the caller to verify whether termination occurred
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
when the function returns), rather than because the termination condition
<span class="emphasis"><em>tol</em></span> was satisfied.
</p>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
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