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<title>Elliptic Integrals of the Second Kind - Legendre Form</title>
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<div class="section math_toolkit_special_ellint_ellint_2">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.ellint.ellint_2"></a><a class="link" href="ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">Elliptic Integrals
of the Second Kind - Legendre Form</a>
</h4></div></div></div>
<h6>
<a name="math_toolkit.special.ellint.ellint_2.h0"></a>
<span><a name="math_toolkit.special.ellint.ellint_2.synopsis"></a></span><a class="link" href="ellint_2.html#math_toolkit.special.ellint.ellint_2.synopsis">Synopsis</a>
</h6>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ellint_2</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<p>
</p>
<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">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h6>
<a name="math_toolkit.special.ellint.ellint_2.h1"></a>
<span><a name="math_toolkit.special.ellint.ellint_2.description"></a></span><a class="link" href="ellint_2.html#math_toolkit.special.ellint.ellint_2.description">Description</a>
</h6>
<p>
These two functions evaluate the incomplete elliptic integral of the second
kind <span class="emphasis"><em>E(φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>E(k)
= E(π/2, k)</em></span>.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/ellint_2.png" align="middle"></span>
</p>
<p>
The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a> when T1 and T2 are different types:
when they are the same type then the result is the same type as the arguments.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the incomplete elliptic integral of the second kind <span class="emphasis"><em>E(φ,
k)</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/ellint3.png"></span>
</p>
<p>
Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the complete elliptic integral of the second kind <span class="emphasis"><em>E(k)</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/ellint7.png"></span>
</p>
<p>
Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>.
</p>
<h6>
<a name="math_toolkit.special.ellint.ellint_2.h2"></a>
<span><a name="math_toolkit.special.ellint.ellint_2.accuracy"></a></span><a class="link" href="ellint_2.html#math_toolkit.special.ellint.ellint_2.accuracy">Accuracy</a>
</h6>
<p>
These functions are computed using only basic arithmetic operations, so
there isn't much variation in accuracy over differing platforms. Note that
only results for the widest floating point type on the system are given
as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
All values are relative errors in units of epsilon.
</p>
<div class="table">
<a name="math_toolkit.special.ellint.ellint_2.errors_rates_in_the_elliptic_integrals_of_the_second_kind"></a><p class="title"><b>Table 45. Errors Rates in the Elliptic Integrals of the Second Kind</b></p>
<div class="table-contents"><table class="table" summary="Errors Rates in the Elliptic Integrals of the Second Kind">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
F(φ, k)
</p>
</th>
<th>
<p>
K(k)
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32 / Visual C++ 8.0
</p>
</td>
<td>
<p>
Peak=4.6 Mean=1.2
</p>
</td>
<td>
<p>
Peak=3.5 Mean=1.0
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Red Hat Linux / G++ 3.4
</p>
</td>
<td>
<p>
Peak=4.3 Mean=1.1
</p>
</td>
<td>
<p>
Peak=4.6 Mean=1.2
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HP-UX / HP aCC 6
</p>
</td>
<td>
<p>
Peak=5.8 Mean=2.2
</p>
</td>
<td>
<p>
Peak=10.8 Mean=2.3
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h6>
<a name="math_toolkit.special.ellint.ellint_2.h3"></a>
<span><a name="math_toolkit.special.ellint.ellint_2.testing"></a></span><a class="link" href="ellint_2.html#math_toolkit.special.ellint.ellint_2.testing">Testing</a>
</h6>
<p>
The tests use a mixture of spot test values calculated using the online
calculator at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
and random test data generated using NTL::RR at 1000-bit precision and
this implementation.
</p>
<h6>
<a name="math_toolkit.special.ellint.ellint_2.h4"></a>
<span><a name="math_toolkit.special.ellint.ellint_2.implementation"></a></span><a class="link" href="ellint_2.html#math_toolkit.special.ellint.ellint_2.implementation">Implementation</a>
</h6>
<p>
These functions are implemented in terms of Carlson's integrals using the
relations:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/ellint21.png"></span>
</p>
<p>
and
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/ellint22.png"></span>
</p>
</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 © 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<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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