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<title>Root Finding Without Derivatives: Bisection, Bracket and TOMS748</title>
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.roots2"></a><a class="link" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">Root Finding Without Derivatives:
Bisection, Bracket and TOMS748</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.roots2.h0"></a>
<span class="phrase"><a name="math_toolkit.roots2.synopsis"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.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">// 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">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.roots2.h1"></a>
<span class="phrase"><a name="math_toolkit.roots2.description"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.description">Description</a>
</h5>
<p>
These functions <code class="computeroutput"><span class="identifier">bisect</span><span class="special">,</span>
<span class="identifier">bracket_and_solve_root</span></code>, and <code class="computeroutput"><span class="identifier">toms748_solve</span></code> solve the root of some function
<span class="emphasis"><em>f(x)</em></span> <span class="bold"><strong>without the need for any derivatives
of <span class="emphasis"><em>f(x)</em></span></strong></span>.
</p>
<p>
(Algorithms that can use the derivative of <span class="emphasis"><em>f(x)</em></span> are nearly
always faster and <a class="link" href="roots.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schroeder">root-finding with derivatives</a>
should be used whenever possible).
</p>
<p>
Variants of these functions allow control via a tolerance for stopping iteration,
setting a maximum number of iterations, or provision of a guess.
</p>
<p>
Variants with and without <a class="link" href="../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> are provided
to allow:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Control of the precision used via <a class="link" href="pol_ref/precision_pol.html" title="Precision Policies">precision
policy</a>.
</li>
<li class="listitem">
Handling of errors via <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error
handling policies</a>.
</li>
</ul></div>
<p>
The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
functions use <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
Algorithm 748: enclosing zeros of continuous functions</a> that is asymptotically
the most efficient known, and have been shown to be optimal for a certain classes
of smooth functions.
</p>
<p>
Alternatively, there is a simple <a href="http://en.wikipedia.org/wiki/Bisection_method" target="_top">bisection</a>
routine <code class="computeroutput"><span class="identifier">bisect</span></code> which can be
useful in its own right in some situations, or alternatively for narrowing
down the range containing the root, to get a good guess 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 the methods of <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent
or Dekker)</a> which only reduce the enclosing interval on the final step.
</p>
<p>
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. 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>, <span class="emphasis"><em>equal_ceil</em></span> and
<span class="emphasis"><em>equal_nearest_integer</em></span> are useful for certain statistical
applications where the result is known to be an integer.
</li>
</ul></div>
<p>
Other user-defined termination conditions are likely to be used only rarely,
but might be useful in some specific circumstances.
</p>
<h3>
<a name="math_toolkit.roots2.h2"></a>
<span class="phrase"><a name="math_toolkit.roots2.bisection"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.bisection">Bisection</a>
</h3>
<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="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>
</pre>
<p>
These functions locate the root using <a href="http://en.wikipedia.org/wiki/Bisection_method" target="_top">bisection</a>:
their 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 which is the function whose root is to be found.
</p></dd>
<dt><span class="term">min</span></dt>
<dd><p>
The left bracket of an interval known to contain the root.
</p></dd>
<dt><span class="term">max</span></dt>
<dd><p>
The right bracket of an interval known to contain the root. 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 signals <a class="link" href="error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
if these preconditions are violated. The action taken is controlled by
the <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error
handling policies</a>. A best guess may be returned, perhaps significantly
wrong.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor 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 <code class="computeroutput"><span class="keyword">true</span></code>.
</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. <br> On exit this is set to actual number
of invocations performed.
</p></dd>
</dl>
</div>
<p>
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 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;15.&#160;Policies: Controlling Precision, Error Handling etc">policy documentation
for more details</a>.
</p>
<p>
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so that:
</p>
<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>
</pre>
<p>
and either
</p>
<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>
</pre>
<p>
or
</p>
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">&gt;=</span> <span class="identifier">m</span>
</pre>
<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>
<h3>
<a name="math_toolkit.roots2.h3"></a>
<span class="phrase"><a name="math_toolkit.roots2.bracket_and_solve"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.bracket_and_solve">Bracket
and solve</a>
</h3>
<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">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>
</pre>
<p>
<code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> is a
convenience function that calls <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
Algorithm 748: enclosing zeros of continuous functions</a> internally to
find the root of <span class="emphasis"><em>f(x)</em></span>. It's usable only when:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Derivative of <span class="emphasis"><em>f(x)</em></span> is a monotonic function.
</li>
<li class="listitem">
The location of the root is known approximately (a guess).
</li>
<li class="listitem">
It is known whether the root occurs for positive or negative <span class="emphasis"><em>x</em></span>.
</li>
</ul></div>
<p>
The parameters 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 that is the function whose root is to be solved. <span class="emphasis"><em>f(x)</em></span>
must be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>.
</p></dd>
<dt><span class="term">guess</span></dt>
<dd><p>
An initial approximation to the root.
</p></dd>
<dt><span class="term">factor</span></dt>
<dd><p>
A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
until two values are found that bracket the root. A value such as 2 is
a typical choice for <span class="emphasis"><em>factor</em></span>.
</p></dd>
<dt><span class="term">rising</span></dt>
<dd><p>
Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if <span class="emphasis"><em>f(x)</em></span>
is falling on <span class="emphasis"><em>x</em></span>. This value is used along with the
result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
is above or below the root.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor that determines the termination condition for the search
for the root. <span class="emphasis"><em>tol</em></span> is passed the current brackets
at each step, when it returns <code class="computeroutput"><span class="keyword">true</span></code>
then the current brackets are returned as the result.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of function invocations to perform in the search for
the root. <br> On exit <span class="emphasis"><em>max_iter</em></span> is set to the
actual number of invocations performed.
</p></dd>
</dl>
</div>
<p>
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 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;15.&#160;Policies: Controlling Precision, Error Handling etc">policy documentation
for more details</a>.
</p>
<p>
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so that:
</p>
<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>
</pre>
<p>
and either
</p>
<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>
</pre>
<p>
or
</p>
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">&gt;=</span> <span class="identifier">m</span>
</pre>
<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 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>
<h3>
<a name="math_toolkit.roots2.h4"></a>
<span class="phrase"><a name="math_toolkit.roots2.algorithm_toms_748_alefeld_potra"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.algorithm_toms_748_alefeld_potra">Algorithm
TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a>
</h3>
<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">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>
</pre>
<p>
The functions <code class="computeroutput"><span class="identifier">toms748_solve</span></code>
implement <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
748: enclosing zeros of continuous functions</a> : using a mixture of cubic,
quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>.
Function variants differ only by whether values for <span class="emphasis"><em>f(a)</em></span>
and <span class="emphasis"><em>f(b)</em></span> are already available, and if special policies
apply.
</p>
<p>
The <code class="computeroutput"><span class="identifier">toms748_solve</span></code> parameters
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 that is the function whose root is to be solved. f(x)
need not be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>
and may have multiple roots.
</p></dd>
<dt><span class="term">a</span></dt>
<dd><p>
The lower bound for the initial bracket of the root.
</p></dd>
<dt><span class="term">b</span></dt>
<dd><p>
The upper bound for the initial bracket of the root. It is a precondition
that <span class="emphasis"><em>a &lt; b</em></span> and that <span class="emphasis"><em>a</em></span> and
<span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a)
* f(b) &lt; 0</em></span>.
</p></dd>
<dt><span class="term">fa</span></dt>
<dd><p>
Optional: the value of <span class="emphasis"><em>f(a)</em></span>.
</p></dd>
<dt><span class="term">fb</span></dt>
<dd><p>
Optional: the value of <span class="emphasis"><em>f(b)</em></span>.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor that determines the termination condition for the search
for the root. <span class="emphasis"><em>tol</em></span> is passed the current brackets
at each step, when it returns <code class="computeroutput"><span class="keyword">true</span></code>,
then the current brackets are returned as the result.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of function invocations to perform in the search for
the root. <br> On exit <span class="emphasis"><em>max_iter</em></span> is set to actual
number of function invocations used.
</p></dd>
</dl>
</div>
<p>
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 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;15.&#160;Policies: Controlling Precision, Error Handling etc">policy documentation
for more details</a>.
</p>
<p>
<code class="computeroutput"><span class="identifier">toms748_solve</span></code> returns a pair
of values <span class="emphasis"><em>r</em></span> that bracket the root so that:
</p>
<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>
</pre>
<p>
and either
</p>
<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>
</pre>
<p>
or
</p>
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">&gt;=</span> <span class="identifier">m</span>
</pre>
<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>
against its previous value passed as parameter), rather than because the termination
condition <span class="emphasis"><em>tol</em></span> was satisfied.
</p>
<pre class="programlisting"><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="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
<span class="keyword">bool</span> <span class="keyword">operator</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="special">;</span>
<span class="special">};</span>
</pre>
<p>
<code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
termination condition used with these root-finding functions. Its <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code> will
return true when the relative distance between <span class="emphasis"><em>a</em></span> and
<span class="emphasis"><em>b</em></span> is less than twice the <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
epsilon</a> for T, or 2<sup>1-bits</sup>, whichever is the larger. In other words,
you set <span class="emphasis"><em>bits</em></span> to the number of bits of precision you want
in the result. The minimal tolerance of twice the <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
epsilon</a> of <code class="computeroutput"><span class="identifier">T</span></code>, typically
<code class="computeroutput"><span class="number">2</span> <span class="special">*</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span></code>, is required to ensure that we get
back a bracketing interval &gt; zero, since this must clearly be at least 1
<a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>
in size.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
<span class="special">{</span>
<span class="identifier">equal_floor</span><span class="special">();</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">bool</span> <span class="keyword">operator</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="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
<span class="special">{</span>
<span class="identifier">equal_ceil</span><span class="special">();</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">bool</span> <span class="keyword">operator</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="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate as
soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
<span class="special">{</span>
<span class="identifier">equal_nearest_integer</span><span class="special">();</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">bool</span> <span class="keyword">operator</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="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
as soon as both ends of the interval round to the same nearest integer.
</p>
<h5>
<a name="math_toolkit.roots2.h5"></a>
<span class="phrase"><a name="math_toolkit.roots2.implementation"></a></span><a class="link" href="roots2.html#math_toolkit.roots2.implementation">Implementation</a>
</h5>
<p>
The implementation of the bisection algorithm is extremely straightforward
and not detailed here. <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
Algorithm 748: enclosing zeros of continuous functions</a> is described
in detail in:
</p>
<p>
<span class="emphasis"><em>Algorithm 748: Enclosing Zeros of Continuous Functions, G. E. Alefeld,
F. A. Potra and Yixun Shi, ACM Transactions on Mathematica1 Software, Vol.
21. No. 3. September 1995. Pages 327-344.</em></span>
</p>
<p>
The implementation here is a faithful translation of this paper into C++.
</p>
</div>
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<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, 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>
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