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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;
83is 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>
194using
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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