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   4 <title>Heuman Lambda Function</title>
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  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">&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">heuman_lambda</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&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>
  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">&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>
  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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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>&#923;<sub>0</sub>(&#966;, 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 &lt;= k &lt;= 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&#160;20.&#160;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&#160;20.&#160;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 &#966; = &#960; / 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         &#966;
  85 is 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&#160;8.69.&#160;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 = 0&#949; (Mean = 0&#949;)</span>
 132                 </p>
 133               </td>
 134 <td>
 135                 <p>
 136                   <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
 137                 </p>
 138               </td>
 139 <td>
 140                 <p>
 141                   <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
 142                 </p>
 143               </td>
 144 <td>
 145                 <p>
 146                   <span class="blue">Max = 1.08&#949; (Mean = 0.734&#949;)</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 = 0&#949; (Mean = 0&#949;)</span>
 159                 </p>
 160               </td>
 161 <td>
 162                 <p>
 163                   <span class="blue">Max = 3.82&#949; (Mean = 0.609&#949;)</span>
 164                 </p>
 165               </td>
 166 <td>
 167                 <p>
 168                   <span class="blue">Max = 3.82&#949; (Mean = 0.608&#949;)</span>
 169                 </p>
 170               </td>
 171 <td>
 172                 <p>
 173                   <span class="blue">Max = 2.12&#949; (Mean = 0.588&#949;)</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>
 196 using
 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>|&#966;| &gt;= &#960;/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 &#169; 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       R&#229;de, 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>
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