]>
Commit | Line | Data |
---|---|---|
7c673cae FG |
1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
4 | <title>Elliptic Integrals of the Second Kind - Legendre Form</title> | |
5 | <link rel="stylesheet" href="../../math.css" type="text/css"> | |
6 | <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> | |
7 | <link rel="home" href="../../index.html" title="Math Toolkit 2.5.1"> | |
8 | <link rel="up" href="../ellint.html" title="Elliptic Integrals"> | |
9 | <link rel="prev" href="ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form"> | |
10 | <link rel="next" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form"> | |
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="ellint_1.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="ellint_3.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.ellint_2"></a><a class="link" href="ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">Elliptic Integrals of the | |
28 | Second Kind - Legendre Form</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.ellint.ellint_2.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.ellint.ellint_2.synopsis"></a></span><a class="link" href="ellint_2.html#math_toolkit.ellint.ellint_2.synopsis">Synopsis</a> | |
33 | </h5> | |
34 | <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">ellint_2</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
35 | </pre> | |
36 | <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> | |
37 | ||
38 | <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> | |
39 | <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">ellint_2</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> | |
40 | ||
41 | <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> | |
42 | <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">ellint_2</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> | |
43 | ||
44 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
45 | <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">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span> | |
46 | ||
47 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</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> | |
48 | <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">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</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> | |
49 | ||
50 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
51 | </pre> | |
52 | <h5> | |
53 | <a name="math_toolkit.ellint.ellint_2.h1"></a> | |
54 | <span class="phrase"><a name="math_toolkit.ellint.ellint_2.description"></a></span><a class="link" href="ellint_2.html#math_toolkit.ellint.ellint_2.description">Description</a> | |
55 | </h5> | |
56 | <p> | |
57 | These two functions evaluate the incomplete elliptic integral of the second | |
58 | kind <span class="emphasis"><em>E(φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>E(k) | |
59 | = E(π/2, k)</em></span>. | |
60 | </p> | |
61 | <p> | |
62 | <span class="inlinemediaobject"><img src="../../../graphs/ellint_2.svg" align="middle"></span> | |
63 | </p> | |
64 | <p> | |
65 | The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result | |
66 | type calculation rules</em></span></a> when T1 and T2 are different types: | |
67 | when they are the same type then the result is the same type as the arguments. | |
68 | </p> | |
69 | <pre class="programlisting"><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> | |
70 | <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">ellint_2</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> | |
71 | ||
72 | <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> | |
73 | <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">ellint_2</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> | |
74 | </pre> | |
75 | <p> | |
76 | Returns the incomplete elliptic integral of the second kind <span class="emphasis"><em>E(φ, | |
77 | k)</em></span>: | |
78 | </p> | |
79 | <p> | |
80 | <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span> | |
81 | </p> | |
82 | <p> | |
83 | Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
84 | </p> | |
85 | <p> | |
86 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
87 | be used to control the behaviour of the function: how it handles errors, | |
88 | 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 | |
89 | documentation for more details</a>. | |
90 | </p> | |
91 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
92 | <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">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span> | |
93 | ||
94 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
95 | <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">ellint_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</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> | |
96 | </pre> | |
97 | <p> | |
98 | Returns the complete elliptic integral of the second kind <span class="emphasis"><em>E(k)</em></span>: | |
99 | </p> | |
100 | <p> | |
101 | <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span> | |
102 | </p> | |
103 | <p> | |
104 | Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
105 | </p> | |
106 | <p> | |
107 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
108 | be used to control the behaviour of the function: how it handles errors, | |
109 | 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 | |
110 | documentation for more details</a>. | |
111 | </p> | |
112 | <h5> | |
113 | <a name="math_toolkit.ellint.ellint_2.h2"></a> | |
114 | <span class="phrase"><a name="math_toolkit.ellint.ellint_2.accuracy"></a></span><a class="link" href="ellint_2.html#math_toolkit.ellint.ellint_2.accuracy">Accuracy</a> | |
115 | </h5> | |
116 | <p> | |
117 | These functions are computed using only basic arithmetic operations, so there | |
118 | isn't much variation in accuracy over differing platforms. Note that only | |
119 | results for the widest floating point type on the system are given as narrower | |
120 | types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively | |
121 | zero error</a>. All values are relative errors in units of epsilon. | |
122 | </p> | |
123 | <div class="table"> | |
124 | <a name="math_toolkit.ellint.ellint_2.table_ellint_2"></a><p class="title"><b>Table 6.64. Error rates for ellint_2</b></p> | |
125 | <div class="table-contents"><table class="table" summary="Error rates for ellint_2"> | |
126 | <colgroup> | |
127 | <col> | |
128 | <col> | |
129 | <col> | |
130 | <col> | |
131 | <col> | |
132 | </colgroup> | |
133 | <thead><tr> | |
134 | <th> | |
135 | </th> | |
136 | <th> | |
137 | <p> | |
138 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
139 | </p> | |
140 | </th> | |
141 | <th> | |
142 | <p> | |
143 | GNU C++ version 5.1.0<br> linux<br> long double | |
144 | </p> | |
145 | </th> | |
146 | <th> | |
147 | <p> | |
148 | GNU C++ version 5.1.0<br> linux<br> double | |
149 | </p> | |
150 | </th> | |
151 | <th> | |
152 | <p> | |
153 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
154 | </p> | |
155 | </th> | |
156 | </tr></thead> | |
157 | <tbody> | |
158 | <tr> | |
159 | <td> | |
160 | <p> | |
161 | Elliptic Integral E: Mathworld Data | |
162 | </p> | |
163 | </td> | |
164 | <td> | |
165 | <p> | |
166 | <span class="blue">Max = 1.31ε (Mean = 0.727ε)</span> | |
167 | </p> | |
168 | </td> | |
169 | <td> | |
170 | <p> | |
171 | <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span><br> <br> | |
172 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
173 | = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_ellint_2__tr1_cmath__Elliptic_Integral_E_Mathworld_Data">And | |
174 | other failures.</a>)</span> | |
175 | </p> | |
176 | </td> | |
177 | <td> | |
178 | <p> | |
179 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
180 | 1.16:</em></span> Max = 0.63ε (Mean = 0.325ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
181 | <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ellint_2_Cephes_Elliptic_Integral_E_Mathworld_Data">And | |
182 | other failures.</a>)</span> | |
183 | </p> | |
184 | </td> | |
185 | <td> | |
186 | <p> | |
187 | <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span> | |
188 | </p> | |
189 | </td> | |
190 | </tr> | |
191 | <tr> | |
192 | <td> | |
193 | <p> | |
194 | Elliptic Integral E: Random Data | |
195 | </p> | |
196 | </td> | |
197 | <td> | |
198 | <p> | |
199 | <span class="blue">Max = 2.23ε (Mean = 0.639ε)</span> | |
200 | </p> | |
201 | </td> | |
202 | <td> | |
203 | <p> | |
204 | <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span><br> <br> | |
205 | (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 3.08e+04ε (Mean = | |
206 | 3.84e+03ε)) | |
207 | </p> | |
208 | </td> | |
209 | <td> | |
210 | <p> | |
211 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
212 | 1.16:</em></span> Max = 4.4ε (Mean = 1.16ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
213 | Max = 8.92ε (Mean = 1.48ε)) | |
214 | </p> | |
215 | </td> | |
216 | <td> | |
217 | <p> | |
218 | <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span> | |
219 | </p> | |
220 | </td> | |
221 | </tr> | |
222 | </tbody> | |
223 | </table></div> | |
224 | </div> | |
225 | <br class="table-break"><h5> | |
226 | <a name="math_toolkit.ellint.ellint_2.h3"></a> | |
227 | <span class="phrase"><a name="math_toolkit.ellint.ellint_2.testing"></a></span><a class="link" href="ellint_2.html#math_toolkit.ellint.ellint_2.testing">Testing</a> | |
228 | </h5> | |
229 | <p> | |
230 | The tests use a mixture of spot test values calculated using the online calculator | |
231 | at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, | |
232 | and random test data generated using NTL::RR at 1000-bit precision and this | |
233 | implementation. | |
234 | </p> | |
235 | <h5> | |
236 | <a name="math_toolkit.ellint.ellint_2.h4"></a> | |
237 | <span class="phrase"><a name="math_toolkit.ellint.ellint_2.implementation"></a></span><a class="link" href="ellint_2.html#math_toolkit.ellint.ellint_2.implementation">Implementation</a> | |
238 | </h5> | |
239 | <p> | |
240 | These functions are implemented in terms of Carlson's integrals using the | |
241 | relations: | |
242 | </p> | |
243 | <p> | |
244 | <span class="inlinemediaobject"><img src="../../../equations/ellint21.svg"></span> | |
245 | </p> | |
246 | <p> | |
247 | and | |
248 | </p> | |
249 | <p> | |
250 | <span class="inlinemediaobject"><img src="../../../equations/ellint22.svg"></span> | |
251 | </p> | |
252 | </div> | |
253 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
254 | <td align="left"></td> | |
255 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
256 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
257 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
258 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
259 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
260 | 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>) | |
261 | </p> | |
262 | </div></td> | |
263 | </tr></table> | |
264 | <hr> | |
265 | <div class="spirit-nav"> | |
266 | <a accesskey="p" href="ellint_1.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="ellint_3.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
267 | </div> | |
268 | </body> | |
269 | </html> |