1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Heuman Lambda Function</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../ellint.html" title="Elliptic Integrals"> 9<link rel="prev" href="jacobi_zeta.html" title="Jacobi Zeta Function"> 10<link rel="next" href="../jacobi.html" title="Jacobi Elliptic Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="jacobi_zeta.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.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.ellint.heuman_lambda"></a><a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">Heuman Lambda Function</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.ellint.heuman_lambda.h0"></a> 31 <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.synopsis"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.synopsis">Synopsis</a> 32 </h5> 33<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">heuman_lambda</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<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> 36 37<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> 38<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">heuman_lambda</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> 39 40<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<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">heuman_lambda</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 42 43<span class="special">}}</span> <span class="comment">// namespaces</span> 44</pre> 45<h5> 46<a name="math_toolkit.ellint.heuman_lambda.h1"></a> 47 <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.description"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.description">Description</a> 48 </h5> 49<p> 50 This function evaluates the Heuman Lambda Function <span class="emphasis"><em>Λ<sub>0</sub>(φ, k)</em></span> 51 </p> 52<div class="blockquote"><blockquote class="blockquote"><p> 53 <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span> 54 55 </p></blockquote></div> 56<p> 57 The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 58 type calculation rules</em></span></a> when the arguments are of different 59 types: when they are the same type then the result is the same type as the 60 arguments. 61 </p> 62<p> 63 Requires <span class="emphasis"><em>-1 <= k <= 1</em></span>, otherwise returns the result 64 of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 65 (outside this range the result would be complex). 66 </p> 67<p> 68 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 69 be used to control the behaviour of the function: how it handles errors, 70 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 71 documentation for more details</a>. 72 </p> 73<p> 74 Note that there is no complete analogue of this function (where φ = π / 2) as 75 this takes the value 1 for all <span class="emphasis"><em>k</em></span>. 76 </p> 77<h5> 78<a name="math_toolkit.ellint.heuman_lambda.h2"></a> 79 <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.accuracy"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.accuracy">Accuracy</a> 80 </h5> 81<p> 82 These functions are trivially computed in terms of other elliptic integrals 83 and generally have very low error rates (a few epsilon) unless parameter 84 φ 85is very large, in which case the usual trigonometric function argument-reduction 86 issues apply. 87 </p> 88<div class="table"> 89<a name="math_toolkit.ellint.heuman_lambda.table_heuman_lambda"></a><p class="title"><b>Table 8.69. Error rates for heuman_lambda</b></p> 90<div class="table-contents"><table class="table" summary="Error rates for heuman_lambda"> 91<colgroup> 92<col> 93<col> 94<col> 95<col> 96<col> 97</colgroup> 98<thead><tr> 99<th> 100 </th> 101<th> 102 <p> 103 GNU C++ version 7.1.0<br> linux<br> double 104 </p> 105 </th> 106<th> 107 <p> 108 GNU C++ version 7.1.0<br> linux<br> long double 109 </p> 110 </th> 111<th> 112 <p> 113 Sun compiler version 0x5150<br> Sun Solaris<br> long double 114 </p> 115 </th> 116<th> 117 <p> 118 Microsoft Visual C++ version 14.1<br> Win32<br> double 119 </p> 120 </th> 121</tr></thead> 122<tbody> 123<tr> 124<td> 125 <p> 126 Elliptic Integral Jacobi Zeta: Mathworld Data 127 </p> 128 </td> 129<td> 130 <p> 131 <span class="blue">Max = 0ε (Mean = 0ε)</span> 132 </p> 133 </td> 134<td> 135 <p> 136 <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span> 137 </p> 138 </td> 139<td> 140 <p> 141 <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span> 142 </p> 143 </td> 144<td> 145 <p> 146 <span class="blue">Max = 1.08ε (Mean = 0.734ε)</span> 147 </p> 148 </td> 149</tr> 150<tr> 151<td> 152 <p> 153 Elliptic Integral Heuman Lambda: Random Data 154 </p> 155 </td> 156<td> 157 <p> 158 <span class="blue">Max = 0ε (Mean = 0ε)</span> 159 </p> 160 </td> 161<td> 162 <p> 163 <span class="blue">Max = 3.82ε (Mean = 0.609ε)</span> 164 </p> 165 </td> 166<td> 167 <p> 168 <span class="blue">Max = 3.82ε (Mean = 0.608ε)</span> 169 </p> 170 </td> 171<td> 172 <p> 173 <span class="blue">Max = 2.12ε (Mean = 0.588ε)</span> 174 </p> 175 </td> 176</tr> 177</tbody> 178</table></div> 179</div> 180<br class="table-break"><h5> 181<a name="math_toolkit.ellint.heuman_lambda.h3"></a> 182 <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.testing"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.testing">Testing</a> 183 </h5> 184<p> 185 The tests use a mixture of spot test values calculated using values calculated 186 at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random 187 test data generated using MPFR at 1000-bit precision and a deliberately naive 188 implementation in terms of the Legendre integrals. 189 </p> 190<h5> 191<a name="math_toolkit.ellint.heuman_lambda.h4"></a> 192 <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.implementation"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.implementation">Implementation</a> 193 </h5> 194<p> 195 The function is then implemented in terms of Carlson's integrals R<sub>J</sub> and R<sub>F</sub> 196using 197 the relation: 198 </p> 199<div class="blockquote"><blockquote class="blockquote"><p> 200 <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span> 201 202 </p></blockquote></div> 203<p> 204 This relation fails for <span class="emphasis"><em>|φ| >= π/2</em></span> in which case the 205 definition in terms of the Jacobi Zeta is used. 206 </p> 207</div> 208<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 209<td align="left"></td> 210<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 211 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 212 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 213 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 214 Daryle Walker and Xiaogang Zhang<p> 215 Distributed under the Boost Software License, Version 1.0. (See accompanying 216 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>) 217 </p> 218</div></td> 219</tr></table> 220<hr> 221<div class="spirit-nav"> 222<a accesskey="p" href="jacobi_zeta.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.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 223</div> 224</body> 225</html> 226
