1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Elliptic Integral D - Legendre Form</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../ellint.html" title="Elliptic Integrals"> 9<link rel="prev" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form"> 10<link rel="next" href="jacobi_zeta.html" title="Jacobi Zeta Function"> 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_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> 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 21. 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 21. 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 21. 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 21. 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 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 62 type calculation rules</em></span></a> when the arguments are of different 63 types: when they are the same type then the result is the same type as the 64 arguments. 65 </p> 66<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> 67<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> 68 69<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><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_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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 71</pre> 72<p> 73 Returns the incomplete elliptic integral: 74 </p> 75<div class="blockquote"><blockquote class="blockquote"><p> 76 <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> 77 78 </p></blockquote></div> 79<p> 80 Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) < 1</em></span>, otherwise returns the result 81 of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 82 (outside this range the result would be complex). 83 </p> 84<p> 85 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 86 be used to control the behaviour of the function: how it handles errors, 87 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 88 documentation for more details</a>. 89 </p> 90<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> 91<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> 92 93<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 94<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 95</pre> 96<p> 97 Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(π/2, k)</em></span> 98 </p> 99<p> 100 Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> otherwise returns the result 101 of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 102 (outside this range the result would be complex). 103 </p> 104<p> 105 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 106 be used to control the behaviour of the function: how it handles errors, 107 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 108 documentation for more details</a>. 109 </p> 110<h5> 111<a name="math_toolkit.ellint.ellint_d.h2"></a> 112 <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> 113 </h5> 114<p> 115 These functions are trivially computed in terms of other elliptic integrals 116 and generally have very low error rates (a few epsilon) unless parameter 117 φ 118is very large, in which case the usual trigonometric function argument-reduction 119 issues apply. 120 </p> 121<div class="table"> 122<a name="math_toolkit.ellint.ellint_d.table_ellint_d_complete_"></a><p class="title"><b>Table 8.66. Error rates for ellint_d (complete)</b></p> 123<div class="table-contents"><table class="table" summary="Error rates for ellint_d (complete)"> 124<colgroup> 125<col> 126<col> 127<col> 128<col> 129<col> 130</colgroup> 131<thead><tr> 132<th> 133 </th> 134<th> 135 <p> 136 GNU C++ version 7.1.0<br> linux<br> double 137 </p> 138 </th> 139<th> 140 <p> 141 GNU C++ version 7.1.0<br> linux<br> long double 142 </p> 143 </th> 144<th> 145 <p> 146 Sun compiler version 0x5150<br> Sun Solaris<br> long double 147 </p> 148 </th> 149<th> 150 <p> 151 Microsoft Visual C++ version 14.1<br> Win32<br> double 152 </p> 153 </th> 154</tr></thead> 155<tbody> 156<tr> 157<td> 158 <p> 159 Elliptic Integral E: Mathworld Data 160 </p> 161 </td> 162<td> 163 <p> 164 <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span> 165 </p> 166 </td> 167<td> 168 <p> 169 <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span> 170 </p> 171 </td> 172<td> 173 <p> 174 <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span> 175 </p> 176 </td> 177<td> 178 <p> 179 <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span> 180 </p> 181 </td> 182</tr> 183<tr> 184<td> 185 <p> 186 Elliptic Integral D: Random Data 187 </p> 188 </td> 189<td> 190 <p> 191 <span class="blue">Max = 0ε (Mean = 0ε)</span> 192 </p> 193 </td> 194<td> 195 <p> 196 <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span> 197 </p> 198 </td> 199<td> 200 <p> 201 <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span> 202 </p> 203 </td> 204<td> 205 <p> 206 <span class="blue">Max = 1.27ε (Mean = 0.355ε)</span> 207 </p> 208 </td> 209</tr> 210</tbody> 211</table></div> 212</div> 213<br class="table-break"><div class="table"> 214<a name="math_toolkit.ellint.ellint_d.table_ellint_d"></a><p class="title"><b>Table 8.67. Error rates for ellint_d</b></p> 215<div class="table-contents"><table class="table" summary="Error rates for ellint_d"> 216<colgroup> 217<col> 218<col> 219<col> 220<col> 221<col> 222</colgroup> 223<thead><tr> 224<th> 225 </th> 226<th> 227 <p> 228 GNU C++ version 7.1.0<br> linux<br> double 229 </p> 230 </th> 231<th> 232 <p> 233 GNU C++ version 7.1.0<br> linux<br> long double 234 </p> 235 </th> 236<th> 237 <p> 238 Sun compiler version 0x5150<br> Sun Solaris<br> long double 239 </p> 240 </th> 241<th> 242 <p> 243 Microsoft Visual C++ version 14.1<br> Win32<br> double 244 </p> 245 </th> 246</tr></thead> 247<tbody> 248<tr> 249<td> 250 <p> 251 Elliptic Integral E: Mathworld Data 252 </p> 253 </td> 254<td> 255 <p> 256 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 257 2.1:</em></span> Max = 0.862ε (Mean = 0.568ε)) 258 </p> 259 </td> 260<td> 261 <p> 262 <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span> 263 </p> 264 </td> 265<td> 266 <p> 267 <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span> 268 </p> 269 </td> 270<td> 271 <p> 272 <span class="blue">Max = 0.862ε (Mean = 0.457ε)</span> 273 </p> 274 </td> 275</tr> 276<tr> 277<td> 278 <p> 279 Elliptic Integral D: Random Data 280 </p> 281 </td> 282<td> 283 <p> 284 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 285 2.1:</em></span> Max = 3.01ε (Mean = 0.928ε)) 286 </p> 287 </td> 288<td> 289 <p> 290 <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span> 291 </p> 292 </td> 293<td> 294 <p> 295 <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span> 296 </p> 297 </td> 298<td> 299 <p> 300 <span class="blue">Max = 2.87ε (Mean = 0.805ε)</span> 301 </p> 302 </td> 303</tr> 304</tbody> 305</table></div> 306</div> 307<br class="table-break"><p> 308 The following error plot are based on an exhaustive search of the functions 309 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code> 310 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span> 311 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>. 312 </p> 313<div class="blockquote"><blockquote class="blockquote"><p> 314 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__double.svg" align="middle"></span> 315 316 </p></blockquote></div> 317<div class="blockquote"><blockquote class="blockquote"><p> 318 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__80_bit_long_double.svg" align="middle"></span> 319 320 </p></blockquote></div> 321<div class="blockquote"><blockquote class="blockquote"><p> 322 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d____float128.svg" align="middle"></span> 323 324 </p></blockquote></div> 325<h5> 326<a name="math_toolkit.ellint.ellint_d.h3"></a> 327 <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> 328 </h5> 329<p> 330 The tests use a mixture of spot test values calculated using values calculated 331 at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random 332 test data generated using MPFR at 1000-bit precision and a deliberately naive 333 implementation in terms of the Legendre integrals. 334 </p> 335<h5> 336<a name="math_toolkit.ellint.ellint_d.h4"></a> 337 <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> 338 </h5> 339<p> 340 The implementation for D(φ, k) first performs argument reduction using the 341 relations: 342 </p> 343<div class="blockquote"><blockquote class="blockquote"><p> 344 <span class="serif_italic"><span class="emphasis"><em>D(-φ, k) = -D(φ, k)</em></span></span> 345 </p></blockquote></div> 346<p> 347 and 348 </p> 349<div class="blockquote"><blockquote class="blockquote"><p> 350 <span class="serif_italic"><span class="emphasis"><em>D(nπ+φ, k) = 2nD(k) + D(φ, k)</em></span></span> 351 </p></blockquote></div> 352<p> 353 to move φ to the range [0, π/2]. 354 </p> 355<p> 356 The functions are then implemented in terms of Carlson's integral R<sub>D</sub> 357using 358 the relation: 359 </p> 360<div class="blockquote"><blockquote class="blockquote"><p> 361 <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> 362 363 </p></blockquote></div> 364</div> 365<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 366<td align="left"></td> 367<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 368 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 369 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 370 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 371 Daryle Walker and Xiaogang Zhang<p> 372 Distributed under the Boost Software License, Version 1.0. (See accompanying 373 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>) 374 </p> 375</div></td> 376</tr></table> 377<hr> 378<div class="spirit-nav"> 379<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> 380</div> 381</body> 382</html> 383
