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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;15.&#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;15.&#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 <p>
  53         <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
  54       </p>
  55 <p>
  56         The return type of this function is computed using the __arg_pomotion_rules
  57         when the arguments are of different types: when they are the same type then
  58         the result is the same type as the arguments.
  59       </p>
  60 <p>
  61         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span>, otherwise returns the result
  62         of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  63         (outside this range the result would be complex).
  64       </p>
  65 <p>
  66         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  67         be used to control the behaviour of the function: how it handles errors,
  68         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  69         documentation for more details</a>.
  70       </p>
  71 <p>
  72         Note that there is no complete analogue of this function (where &#966; = &#960; / 2) as
  73         this takes the value 1 for all <span class="emphasis"><em>k</em></span>.
  74       </p>
  75 <h5>
  76 <a name="math_toolkit.ellint.heuman_lambda.h2"></a>
  77         <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>
  78       </h5>
  79 <p>
  80         These functions are trivially computed in terms of other elliptic integrals
  81         and generally have very low error rates (a few epsilon) unless parameter
  82         &#966;
  83 is very large, in which case the usual trigonometric function argument-reduction
  84         issues apply.
  85       </p>
  86 <div class="table">
  87 <a name="math_toolkit.ellint.heuman_lambda.table_heuman_lambda"></a><p class="title"><b>Table&#160;6.69.&#160;Error rates for heuman_lambda</b></p>
  88 <div class="table-contents"><table class="table" summary="Error rates for heuman_lambda">
  89 <colgroup>
  90 <col>
  91 <col>
  92 <col>
  93 <col>
  94 <col>
  95 </colgroup>
  96 <thead><tr>
  97 <th>
  98               </th>
  99 <th>
 100                 <p>
 101                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 102                 </p>
 103               </th>
 104 <th>
 105                 <p>
 106                   GNU C++ version 5.1.0<br> linux<br> double
 107                 </p>
 108               </th>
 109 <th>
 110                 <p>
 111                   GNU C++ version 5.1.0<br> linux<br> long double
 112                 </p>
 113               </th>
 114 <th>
 115                 <p>
 116                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 117                 </p>
 118               </th>
 119 </tr></thead>
 120 <tbody>
 121 <tr>
 122 <td>
 123                 <p>
 124                   Elliptic Integral Jacobi Zeta: Mathworld Data
 125                 </p>
 126               </td>
 127 <td>
 128                 <p>
 129                   <span class="blue">Max = 1.08&#949; (Mean = 0.734&#949;)</span>
 130                 </p>
 131               </td>
 132 <td>
 133                 <p>
 134                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 135                 </p>
 136               </td>
 137 <td>
 138                 <p>
 139                   <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
 140                 </p>
 141               </td>
 142 <td>
 143                 <p>
 144                   <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
 145                 </p>
 146               </td>
 147 </tr>
 148 <tr>
 149 <td>
 150                 <p>
 151                   Elliptic Integral Heuman Lambda: Random Data
 152                 </p>
 153               </td>
 154 <td>
 155                 <p>
 156                   <span class="blue">Max = 2.12&#949; (Mean = 0.59&#949;)</span>
 157                 </p>
 158               </td>
 159 <td>
 160                 <p>
 161                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 162                 </p>
 163               </td>
 164 <td>
 165                 <p>
 166                   <span class="blue">Max = 3.82&#949; (Mean = 0.609&#949;)</span>
 167                 </p>
 168               </td>
 169 <td>
 170                 <p>
 171                   <span class="blue">Max = 3.82&#949; (Mean = 0.608&#949;)</span>
 172                 </p>
 173               </td>
 174 </tr>
 175 </tbody>
 176 </table></div>
 177 </div>
 178 <br class="table-break"><h5>
 179 <a name="math_toolkit.ellint.heuman_lambda.h3"></a>
 180         <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>
 181       </h5>
 182 <p>
 183         The tests use a mixture of spot test values calculated using values calculated
 184         at wolframalpha.com, and random test data generated using MPFR at 1000-bit
 185         precision and a deliberately naive implementation in terms of the Legendre
 186         integrals.
 187       </p>
 188 <h5>
 189 <a name="math_toolkit.ellint.heuman_lambda.h4"></a>
 190         <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>
 191       </h5>
 192 <p>
 193         The function is then implemented in terms of Carlson's integrals R<sub>J</sub> and R<sub>F</sub>
 194 using
 195         the relation:
 196       </p>
 197 <p>
 198         <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
 199       </p>
 200 <p>
 201         This relation fails for <span class="emphasis"><em>|&#966;| &gt;= &#960;/2</em></span> in which case the
 202         definition in terms of the Jacobi Zeta is used.
 203       </p>
 204 </div>
 205 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 206 <td align="left"></td>
 207 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 208       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 209       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 210       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 211         Distributed under the Boost Software License, Version 1.0. (See accompanying
 212         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>)
 213       </p>
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