<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Elliptic Integrals of the Third Kind - Legendre Form</title> <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../index.html" title="Math Toolkit"> <link rel="up" href="../ellint.html" title="Elliptic Integrals"> <link rel="prev" href="ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form"> <link rel="next" href="../zetas.html" title="Zeta Functions"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../index.html">Home</a></td> <td align="center"><a 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title="Elliptic Integrals of the Third Kind - Legendre Form"> Elliptic Integrals of the Third Kind - Legendre Form</a> </h4></div></div></div> <a name="math_toolkit.special.ellint.ellint_3.synopsis"></a><h6> <a name="id1143416"></a> <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.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_3</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> <span class="keyword">class</span> <span class="identifier">T3</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_3</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">n</span><span class="special">,</span> <span class="identifier">T3</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> <span class="identifier">T3</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_3</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">n</span><span class="special">,</span> <span class="identifier">T3</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">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_3</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">n</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_3</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">n</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> <a name="math_toolkit.special.ellint.ellint_3.description"></a><h6> <a name="id1143926"></a> <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.description">Description</a> </h6> <p> These two functions evaluate the incomplete elliptic integral of the third kind <span class="emphasis"><em>Π(n, φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>Π(n, k) = E(n, π/2, k)</em></span>. </p> <p> <span class="inlinemediaobject"><img src="../../../../graphs/ellint_3.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 the arguments are of 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> <span class="keyword">class</span> <span class="identifier">T3</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_3</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">n</span><span class="special">,</span> <span class="identifier">T3</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> <span class="identifier">T3</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_3</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">n</span><span class="special">,</span> <span class="identifier">T3</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 third kind <span class="emphasis"><em>Π(n, φ, k)</em></span>: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint4.png"></span> </p> <p> Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> and <span class="emphasis"><em>n < 1/sin<sup>2</sup>(φ)</em></span>, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> (outside this range the result would be complex). </p> <p> </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> <p> </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_3</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">n</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_3</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">n</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 first kind <span class="emphasis"><em>Π(n, k)</em></span>: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint8.png"></span> </p> <p> Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> and <span class="emphasis"><em>n < 1</em></span>, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> (outside this range the result would be complex). </p> <p> </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> <p> </p> <a name="math_toolkit.special.ellint.ellint_3.accuracy"></a><h6> <a name="id1144528"></a> <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.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="id1144546"></a><p class="title"><b>Table 43. Errors Rates in the Elliptic Integrals of the Third Kind</b></p> <div class="table-contents"><table class="table" summary="Errors Rates in the Elliptic Integrals of the Third Kind"> <colgroup> <col> <col> <col> <col> </colgroup> <thead><tr> <th> <p> Significand Size </p> </th> <th> <p> Platform and Compiler </p> </th> <th> <p> Π(n, φ, k) </p> </th> <th> <p> Π(n, k) </p> </th> </tr></thead> <tbody> <tr> <td> <p> 53 </p> </td> <td> <p> Win32 / Visual C++ 8.0 </p> </td> <td> <p> Peak=29 Mean=2.2 </p> </td> <td> <p> Peak=3 Mean=0.8 </p> </td> </tr> <tr> <td> <p> 64 </p> </td> <td> <p> Red Hat Linux / G++ 3.4 </p> </td> <td> <p> Peak=14 Mean=1.3 </p> </td> <td> <p> Peak=2.3 Mean=0.8 </p> </td> </tr> <tr> <td> <p> 113 </p> </td> <td> <p> HP-UX / HP aCC 6 </p> </td> <td> <p> Peak=10 Mean=1.4 </p> </td> <td> <p> Peak=4.2 Mean=1.1 </p> </td> </tr> </tbody> </table></div> </div> <br class="table-break"><a name="math_toolkit.special.ellint.ellint_3.testing"></a><h6> <a name="id1144713"></a> <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.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> <a name="math_toolkit.special.ellint.ellint_3.implementation"></a><h6> <a name="id1144734"></a> <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.implementation">Implementation</a> </h6> <p> The implementation for Π(n, φ, k) first siphons off the special cases: </p> <p> <span class="emphasis"><em>Π(0, φ, k) = F(φ, k)</em></span> </p> <p> <span class="emphasis"><em>Π(n, π/2, k) = Π(n, k)</em></span> </p> <p> and </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint23.png"></span> </p> <p> Then if n < 0 the relations (A&S 17.7.15/16): </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint24.png"></span> </p> <p> are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1]. </p> <p> Then the relations: </p> <p> <span class="emphasis"><em>Π(n, -φ, k) = -Π(n, φ, k)</em></span> </p> <p> <span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k) ; n <= 1</em></span> </p> <p> <span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1</em></span> <sup>[<a name="id1144848" href="#ftn.id1144848" class="footnote">1</a>]</sup> </p> <p> are used to move φ to the range [0, π/2]. </p> <p> The functions are then implemented in terms of Carlson's integrals using the relations: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint25.png"></span> </p> <p> and </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ellint26.png"></span> </p> <div class="footnotes"> <br><hr width="100" align="left"> <div class="footnote"><p><sup>[<a name="ftn.id1144848" href="#id1144848" class="para">1</a>] </sup> I haven't been able to find a literature reference for this relation, but it appears to be the convention used by Mathematica. Intuitively the first <span class="emphasis"><em>2 * m * Π(n, k)</em></span> terms cancel out as the derivative alternates between +∞ and -∞. </p></div> </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 © 2006 , 2007, 2008, 2009 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> <hr> <div class="spirit-nav"> <a accesskey="p" href="ellint_2.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../zetas.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>