devn00b
/
EQ2EMu


			
				
					
						
						
							123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382
							<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Elliptic Integral D - Legendre Form</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
<link rel="up" href="../ellint.html" title="Elliptic Integrals">
<link rel="prev" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">
<link rel="next" href="jacobi_zeta.html" title="Jacobi Zeta Function">
</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 href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="ellint_3.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="jacobi_zeta.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.ellint.ellint_d"></a><a class="link" href="ellint_d.html" title="Elliptic Integral D - Legendre Form">Elliptic Integral D - Legendre
      Form</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.ellint.ellint_d.h0"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.synopsis"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.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">special_functions</span><span class="special">/</span><span class="identifier">ellint_d</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">template</span> <span class="special">&lt;</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">&gt;</span>
<a class="link" href="../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_d</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">&lt;</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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../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_d</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="Chapter&#160;20.&#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">T1</span><span class="special">&gt;</span>
<a class="link" href="../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_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</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>
<a class="link" href="../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_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.ellint.ellint_d.h1"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.description"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.description">Description</a>
      </h5>
<p>
        These two functions evaluate the incomplete elliptic integral <span class="emphasis"><em>D(&#966;,
        k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(&#960;/2, k)</em></span>.
      </p>
<p>
        The return type of these functions is computed using the <a class="link" href="../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">&lt;</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">&gt;</span>
<a class="link" href="../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_d</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">&lt;</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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../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">phi</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>
        Returns the incomplete elliptic integral:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>

        </p></blockquote></div>
<p>
        Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) &lt; 1</em></span>, otherwise returns the result
        of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<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>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</span>
<a class="link" href="../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_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</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>
<a class="link" href="../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_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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>
        Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(&#960;/2, k)</em></span>
      </p>
<p>
        Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span> otherwise returns the result
        of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<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>
<h5>
<a name="math_toolkit.ellint.ellint_d.h2"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.accuracy"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.accuracy">Accuracy</a>
      </h5>
<p>
        These functions are trivially computed in terms of other elliptic integrals
        and generally have very low error rates (a few epsilon) unless parameter
        &#966;
is very large, in which case the usual trigonometric function argument-reduction
        issues apply.
      </p>
<div class="table">
<a name="math_toolkit.ellint.ellint_d.table_ellint_d_complete_"></a><p class="title"><b>Table&#160;8.66.&#160;Error rates for ellint_d (complete)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_d (complete)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Elliptic Integral E: Mathworld Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral D: Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27&#949; (Mean = 0.355&#949;)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_d.table_ellint_d"></a><p class="title"><b>Table&#160;8.67.&#160;Error rates for ellint_d</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_d">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Elliptic Integral E: Mathworld Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 0.862&#949; (Mean = 0.568&#949;))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.862&#949; (Mean = 0.457&#949;)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral D: Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 3.01&#949; (Mean = 0.928&#949;))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.87&#949; (Mean = 0.805&#949;)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
        The following error plot are based on an exhaustive search of the functions
        domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
        precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
        <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__80_bit_long_double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d____float128.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_d.h3"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.testing"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.testing">Testing</a>
      </h5>
<p>
        The tests use a mixture of spot test values calculated using values calculated
        at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random
        test data generated using MPFR at 1000-bit precision and a deliberately naive
        implementation in terms of the Legendre integrals.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_d.h4"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.implementation"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.implementation">Implementation</a>
      </h5>
<p>
        The implementation for D(&#966;, k) first performs argument reduction using the
        relations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>D(-&#966;, k) = -D(&#966;, k)</em></span></span>
        </p></blockquote></div>
<p>
        and
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>D(n&#960;+&#966;, k) = 2nD(k) + D(&#966;, k)</em></span></span>
        </p></blockquote></div>
<p>
        to move &#966; to the range [0, &#960;/2].
      </p>
<p>
        The functions are then implemented in terms of Carlson's integral R<sub>D</sub>
using
        the relation:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>

        </p></blockquote></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 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>)
      </p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="ellint_3.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="jacobi_zeta.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>