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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.ellint_d"></a><a class="link" href="ellint_d.html" title="Elliptic Integral D - Legendre Form">Elliptic Integral D - Legendre | |
28 | Form</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.ellint.ellint_d.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.ellint.ellint_d.synopsis"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.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_d</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_d</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_d</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">T1</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_d</span><span class="special">(</span><span class="identifier">T1</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">T1</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_d</span><span class="special">(</span><span class="identifier">T1</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_d.h1"></a> | |
54 | <span class="phrase"><a name="math_toolkit.ellint.ellint_d.description"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.description">Description</a> | |
55 | </h5> | |
56 | <p> | |
57 | These two functions evaluate the incomplete elliptic integral <span class="emphasis"><em>D(φ, | |
58 | k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(π/2, k)</em></span>. | |
59 | </p> | |
60 | <p> | |
61 | The return type of these functions is computed using the __arg_pomotion_rules | |
62 | when the arguments are of different types: when they are the same type then | |
63 | the result is the same type as the arguments. | |
64 | </p> | |
65 | <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> | |
66 | <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_d</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> | |
67 | ||
68 | <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> | |
69 | <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_3</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> | |
70 | </pre> | |
71 | <p> | |
72 | Returns the incomplete elliptic integral: | |
73 | </p> | |
74 | <p> | |
75 | <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> | |
76 | </p> | |
77 | <p> | |
78 | Requires <span class="emphasis"><em>-1 <= k <= 1</em></span>, otherwise returns the result | |
79 | of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
80 | (outside this range the result would be complex). | |
81 | </p> | |
82 | <p> | |
83 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
84 | be used to control the behaviour of the function: how it handles errors, | |
85 | 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 | |
86 | documentation for more details</a>. | |
87 | </p> | |
88 | <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> | |
89 | <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_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">);</span> | |
90 | ||
91 | <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> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><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_d</span><span class="special">(</span><span class="identifier">T1</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> | |
93 | </pre> | |
94 | <p> | |
95 | Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(π/2, k)</em></span> | |
96 | </p> | |
97 | <p> | |
98 | Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> otherwise returns the result | |
99 | of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
100 | (outside this range the result would be complex). | |
101 | </p> | |
102 | <p> | |
103 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
104 | be used to control the behaviour of the function: how it handles errors, | |
105 | 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 | |
106 | documentation for more details</a>. | |
107 | </p> | |
108 | <h5> | |
109 | <a name="math_toolkit.ellint.ellint_d.h2"></a> | |
110 | <span class="phrase"><a name="math_toolkit.ellint.ellint_d.accuracy"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.accuracy">Accuracy</a> | |
111 | </h5> | |
112 | <p> | |
113 | These functions are trivially computed in terms of other elliptic integrals | |
114 | and generally have very low error rates (a few epsilon) unless parameter | |
115 | φ | |
116 | is very large, in which case the usual trigonometric function argument-reduction | |
117 | issues apply. | |
118 | </p> | |
119 | <div class="table"> | |
120 | <a name="math_toolkit.ellint.ellint_d.table_ellint_d_complete_"></a><p class="title"><b>Table 6.66. Error rates for ellint_d (complete)</b></p> | |
121 | <div class="table-contents"><table class="table" summary="Error rates for ellint_d (complete)"> | |
122 | <colgroup> | |
123 | <col> | |
124 | <col> | |
125 | <col> | |
126 | <col> | |
127 | <col> | |
128 | </colgroup> | |
129 | <thead><tr> | |
130 | <th> | |
131 | </th> | |
132 | <th> | |
133 | <p> | |
134 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
135 | </p> | |
136 | </th> | |
137 | <th> | |
138 | <p> | |
139 | GNU C++ version 5.1.0<br> linux<br> double | |
140 | </p> | |
141 | </th> | |
142 | <th> | |
143 | <p> | |
144 | GNU C++ version 5.1.0<br> linux<br> long double | |
145 | </p> | |
146 | </th> | |
147 | <th> | |
148 | <p> | |
149 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
150 | </p> | |
151 | </th> | |
152 | </tr></thead> | |
153 | <tbody> | |
154 | <tr> | |
155 | <td> | |
156 | <p> | |
157 | Elliptic Integral E: Mathworld Data | |
158 | </p> | |
159 | </td> | |
160 | <td> | |
161 | <p> | |
162 | <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span> | |
163 | </p> | |
164 | </td> | |
165 | <td> | |
166 | <p> | |
167 | <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span> | |
168 | </p> | |
169 | </td> | |
170 | <td> | |
171 | <p> | |
172 | <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span> | |
173 | </p> | |
174 | </td> | |
175 | <td> | |
176 | <p> | |
177 | <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span> | |
178 | </p> | |
179 | </td> | |
180 | </tr> | |
181 | <tr> | |
182 | <td> | |
183 | <p> | |
184 | Elliptic Integral D: Random Data | |
185 | </p> | |
186 | </td> | |
187 | <td> | |
188 | <p> | |
189 | <span class="blue">Max = 1.27ε (Mean = 0.355ε)</span> | |
190 | </p> | |
191 | </td> | |
192 | <td> | |
193 | <p> | |
194 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
195 | </p> | |
196 | </td> | |
197 | <td> | |
198 | <p> | |
199 | <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span> | |
200 | </p> | |
201 | </td> | |
202 | <td> | |
203 | <p> | |
204 | <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span> | |
205 | </p> | |
206 | </td> | |
207 | </tr> | |
208 | </tbody> | |
209 | </table></div> | |
210 | </div> | |
211 | <br class="table-break"><div class="table"> | |
212 | <a name="math_toolkit.ellint.ellint_d.table_ellint_d"></a><p class="title"><b>Table 6.67. Error rates for ellint_d</b></p> | |
213 | <div class="table-contents"><table class="table" summary="Error rates for ellint_d"> | |
214 | <colgroup> | |
215 | <col> | |
216 | <col> | |
217 | <col> | |
218 | <col> | |
219 | <col> | |
220 | </colgroup> | |
221 | <thead><tr> | |
222 | <th> | |
223 | </th> | |
224 | <th> | |
225 | <p> | |
226 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
227 | </p> | |
228 | </th> | |
229 | <th> | |
230 | <p> | |
231 | GNU C++ version 5.1.0<br> linux<br> double | |
232 | </p> | |
233 | </th> | |
234 | <th> | |
235 | <p> | |
236 | GNU C++ version 5.1.0<br> linux<br> long double | |
237 | </p> | |
238 | </th> | |
239 | <th> | |
240 | <p> | |
241 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
242 | </p> | |
243 | </th> | |
244 | </tr></thead> | |
245 | <tbody> | |
246 | <tr> | |
247 | <td> | |
248 | <p> | |
249 | Elliptic Integral E: Mathworld Data | |
250 | </p> | |
251 | </td> | |
252 | <td> | |
253 | <p> | |
254 | <span class="blue">Max = 0.862ε (Mean = 0.457ε)</span> | |
255 | </p> | |
256 | </td> | |
257 | <td> | |
258 | <p> | |
259 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
260 | 1.16:</em></span> Max = 0.862ε (Mean = 0.568ε)) | |
261 | </p> | |
262 | </td> | |
263 | <td> | |
264 | <p> | |
265 | <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span> | |
266 | </p> | |
267 | </td> | |
268 | <td> | |
269 | <p> | |
270 | <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span> | |
271 | </p> | |
272 | </td> | |
273 | </tr> | |
274 | <tr> | |
275 | <td> | |
276 | <p> | |
277 | Elliptic Integral D: Random Data | |
278 | </p> | |
279 | </td> | |
280 | <td> | |
281 | <p> | |
282 | <span class="blue">Max = 2.87ε (Mean = 0.805ε)</span> | |
283 | </p> | |
284 | </td> | |
285 | <td> | |
286 | <p> | |
287 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
288 | 1.16:</em></span> Max = 3.01ε (Mean = 0.928ε)) | |
289 | </p> | |
290 | </td> | |
291 | <td> | |
292 | <p> | |
293 | <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span> | |
294 | </p> | |
295 | </td> | |
296 | <td> | |
297 | <p> | |
298 | <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span> | |
299 | </p> | |
300 | </td> | |
301 | </tr> | |
302 | </tbody> | |
303 | </table></div> | |
304 | </div> | |
305 | <br class="table-break"><h5> | |
306 | <a name="math_toolkit.ellint.ellint_d.h3"></a> | |
307 | <span class="phrase"><a name="math_toolkit.ellint.ellint_d.testing"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.testing">Testing</a> | |
308 | </h5> | |
309 | <p> | |
310 | The tests use a mixture of spot test values calculated using values calculated | |
311 | at wolframalpha.com, and random test data generated using MPFR at 1000-bit | |
312 | precision and a deliberately naive implementation in terms of the Legendre | |
313 | integrals. | |
314 | </p> | |
315 | <h5> | |
316 | <a name="math_toolkit.ellint.ellint_d.h4"></a> | |
317 | <span class="phrase"><a name="math_toolkit.ellint.ellint_d.implementation"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.implementation">Implementation</a> | |
318 | </h5> | |
319 | <p> | |
320 | The implementation for D(φ, k) first performs argument reduction using the | |
321 | relations: | |
322 | </p> | |
323 | <p> | |
324 | <span class="emphasis"><em>D(-φ, k) = -D(φ, k)</em></span> | |
325 | </p> | |
326 | <p> | |
327 | and | |
328 | </p> | |
329 | <p> | |
330 | <span class="emphasis"><em>D(nπ+φ, k) = 2nD(k) + D(φ, k)</em></span> | |
331 | </p> | |
332 | <p> | |
333 | to move φ   to the range [0, π/2]. | |
334 | </p> | |
335 | <p> | |
336 | The functions are then implemented in terms of Carlson's integral R<sub>D</sub> | |
337 | using | |
338 | the relation: | |
339 | </p> | |
340 | <p> | |
341 | <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> | |
342 | </p> | |
343 | </div> | |
344 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
345 | <td align="left"></td> | |
346 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
347 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
348 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
349 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
350 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
351 | 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>) | |
352 | </p> | |
353 | </div></td> | |
354 | </tr></table> | |
355 | <hr> | |
356 | <div class="spirit-nav"> | |
357 | <a accesskey="p" href="ellint_3.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="jacobi_zeta.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
358 | </div> | |
359 | </body> | |
360 | </html> |