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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 15. 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 15. 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 | <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 <= k <= 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 15. 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 15. 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 φ = π / 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 | φ | |
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 6.69. 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 = 1.08ε (Mean = 0.734ε)</span> | |
130 | </p> | |
131 | </td> | |
132 | <td> | |
133 | <p> | |
134 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
135 | </p> | |
136 | </td> | |
137 | <td> | |
138 | <p> | |
139 | <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span> | |
140 | </p> | |
141 | </td> | |
142 | <td> | |
143 | <p> | |
144 | <span class="blue">Max = 1.89ε (Mean = 0.887ε)</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 = 2.12ε (Mean = 0.59ε)</span> | |
157 | </p> | |
158 | </td> | |
159 | <td> | |
160 | <p> | |
161 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
162 | </p> | |
163 | </td> | |
164 | <td> | |
165 | <p> | |
166 | <span class="blue">Max = 3.82ε (Mean = 0.609ε)</span> | |
167 | </p> | |
168 | </td> | |
169 | <td> | |
170 | <p> | |
171 | <span class="blue">Max = 3.82ε (Mean = 0.608ε)</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>|φ| >= π/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 © 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 Råde, 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> | |
214 | </div></td> | |
215 | </tr></table> | |
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