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4<title>Elliptic Integral D - Legendre Form</title>
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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">&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">ellint_d</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&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>
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">&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>
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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
43
44<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</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">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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(&#966;,
58 k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(&#960;/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">&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>
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">&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>
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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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 &lt;= k &lt;= 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&#160;15.&#160;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&#160;15.&#160;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">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</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">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
93</pre>
94<p>
95 Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(&#960;/2, k)</em></span>
96 </p>
97<p>
98 Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 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&#160;15.&#160;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&#160;15.&#160;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 &#966;
116is 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&#160;6.66.&#160;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 = 0.637&#949; (Mean = 0.368&#949;)</span>
163 </p>
164 </td>
165<td>
166 <p>
167 <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</span>
168 </p>
169 </td>
170<td>
171 <p>
172 <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
173 </p>
174 </td>
175<td>
176 <p>
177 <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</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 = 1.27&#949; (Mean = 0.355&#949;)</span>
190 </p>
191 </td>
192<td>
193 <p>
194 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
195 </p>
196 </td>
197<td>
198 <p>
199 <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
200 </p>
201 </td>
202<td>
203 <p>
204 <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</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&#160;6.67.&#160;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 = 0.862&#949; (Mean = 0.457&#949;)</span>
255 </p>
256 </td>
257<td>
258 <p>
259 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
260 1.16:</em></span> Max = 0.862&#949; (Mean = 0.568&#949;))
261 </p>
262 </td>
263<td>
264 <p>
265 <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
266 </p>
267 </td>
268<td>
269 <p>
270 <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</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 = 2.87&#949; (Mean = 0.805&#949;)</span>
283 </p>
284 </td>
285<td>
286 <p>
287 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
288 1.16:</em></span> Max = 3.01&#949; (Mean = 0.928&#949;))
289 </p>
290 </td>
291<td>
292 <p>
293 <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
294 </p>
295 </td>
296<td>
297 <p>
298 <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</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(&#966;, k) first performs argument reduction using the
321 relations:
322 </p>
323<p>
324 <span class="emphasis"><em>D(-&#966;, k) = -D(&#966;, k)</em></span>
325 </p>
326<p>
327 and
328 </p>
329<p>
330 <span class="emphasis"><em>D(n&#960;+&#966;, k) = 2nD(k) + D(&#966;, k)</em></span>
331 </p>
332<p>
333 to move &#966; &#160; to the range [0, &#960;/2].
334 </p>
335<p>
336 The functions are then implemented in terms of Carlson's integral R<sub>D</sub>
337using
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 &#169; 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 R&#229;de, 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>
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