1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Elliptic Integrals of the Third Kind - 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_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form"> 10<link rel="next" href="ellint_d.html" title="Elliptic Integral D - 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_2.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_d.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_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals of the 28 Third Kind - Legendre Form</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.ellint.ellint_3.h0"></a> 32 <span class="phrase"><a name="math_toolkit.ellint.ellint_3.synopsis"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.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_3</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> <span class="keyword">class</span> <span class="identifier">T3</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_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">n</span><span class="special">,</span> <span class="identifier">T3</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> <span class="identifier">T3</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_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">n</span><span class="special">,</span> <span class="identifier">T3</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> <span class="keyword">class</span> <span class="identifier">T2</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_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">n</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> <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> 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_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">n</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_3.h1"></a> 54 <span class="phrase"><a name="math_toolkit.ellint.ellint_3.description"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.description">Description</a> 55 </h5> 56<p> 57 These two functions evaluate the incomplete elliptic integral of the third 58 kind <span class="emphasis"><em>Π(n, φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>Π(n, 59 k) = E(n, π/2, k)</em></span>. 60 </p> 61<div class="blockquote"><blockquote class="blockquote"><p> 62 <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.svg" align="middle"></span> 63 64 </p></blockquote></div> 65<p> 66 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 67 type calculation rules</em></span></a> when the arguments are of different 68 types: when they are the same type then the result is the same type as the 69 arguments. 70 </p> 71<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> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> 72<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">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span> 73 74<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> <span class="identifier">T3</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> 75<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">n</span><span class="special">,</span> <span class="identifier">T3</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> 76</pre> 77<p> 78 Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>Π(n, 79 φ, k)</em></span>: 80 </p> 81<div class="blockquote"><blockquote class="blockquote"><p> 82 <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span> 83 84 </p></blockquote></div> 85<p> 86 Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) < 1</em></span> and <span class="emphasis"><em>n < 1/sin<sup>2</sup>(φ)</em></span>, 87 otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 88 (outside this range the result would be complex). 89 </p> 90<p> 91 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 92 be used to control the behaviour of the function: how it handles errors, 93 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 94 documentation for more details</a>. 95 </p> 96<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> 97<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">n</span><span class="special">);</span> 98 99<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> 100<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">n</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> 101</pre> 102<p> 103 Returns the complete elliptic integral of the first kind <span class="emphasis"><em>Π(n, k)</em></span>: 104 </p> 105<div class="blockquote"><blockquote class="blockquote"><p> 106 <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span> 107 108 </p></blockquote></div> 109<p> 110 Requires <span class="emphasis"><em>|k| < 1</em></span> and <span class="emphasis"><em>n < 1</em></span>, 111 otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 112 (outside this range the result would be complex). 113 </p> 114<p> 115 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 116 be used to control the behaviour of the function: how it handles errors, 117 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 118 documentation for more details</a>. 119 </p> 120<h5> 121<a name="math_toolkit.ellint.ellint_3.h2"></a> 122 <span class="phrase"><a name="math_toolkit.ellint.ellint_3.accuracy"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.accuracy">Accuracy</a> 123 </h5> 124<p> 125 These functions are computed using only basic arithmetic operations, so there 126 isn't much variation in accuracy over differing platforms. Note that only 127 results for the widest floating point type on the system are given as narrower 128 types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 129 zero error</a>. All values are relative errors in units of epsilon. 130 </p> 131<div class="table"> 132<a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table 8.65. Error rates for ellint_3</b></p> 133<div class="table-contents"><table class="table" summary="Error rates for ellint_3"> 134<colgroup> 135<col> 136<col> 137<col> 138<col> 139<col> 140</colgroup> 141<thead><tr> 142<th> 143 </th> 144<th> 145 <p> 146 GNU C++ version 7.1.0<br> linux<br> long double 147 </p> 148 </th> 149<th> 150 <p> 151 GNU C++ version 7.1.0<br> linux<br> double 152 </p> 153 </th> 154<th> 155 <p> 156 Sun compiler version 0x5150<br> Sun Solaris<br> long double 157 </p> 158 </th> 159<th> 160 <p> 161 Microsoft Visual C++ version 14.1<br> Win32<br> double 162 </p> 163 </th> 164</tr></thead> 165<tbody> 166<tr> 167<td> 168 <p> 169 Elliptic Integral PI: Mathworld Data 170 </p> 171 </td> 172<td> 173 <p> 174 <span class="blue">Max = 475ε (Mean = 86.3ε)</span><br> <br> 175 (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = +INFε (Mean 176 = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Mathworld_Data">And 177 other failures.</a>)</span> 178 </p> 179 </td> 180<td> 181 <p> 182 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 183 2.1:</em></span> Max = 1.48e+05ε (Mean = 2.54e+04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_3_GSL_2_1_Elliptic_Integral_PI_Mathworld_Data">And 184 other failures.</a>) 185 </p> 186 </td> 187<td> 188 <p> 189 <span class="blue">Max = 475ε (Mean = 86.3ε)</span> 190 </p> 191 </td> 192<td> 193 <p> 194 <span class="blue">Max = 565ε (Mean = 102ε)</span> 195 </p> 196 </td> 197</tr> 198<tr> 199<td> 200 <p> 201 Elliptic Integral PI: Random Data 202 </p> 203 </td> 204<td> 205 <p> 206 <span class="blue">Max = 4.54ε (Mean = 0.895ε)</span><br> <br> 207 (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = 3.37e+20ε (Mean 208 = 3.47e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Random_Data">And 209 other failures.</a>)</span> 210 </p> 211 </td> 212<td> 213 <p> 214 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 215 2.1:</em></span> Max = 633ε (Mean = 50.1ε)) 216 </p> 217 </td> 218<td> 219 <p> 220 <span class="blue">Max = 4.49ε (Mean = 0.885ε)</span> 221 </p> 222 </td> 223<td> 224 <p> 225 <span class="blue">Max = 8.33ε (Mean = 0.971ε)</span> 226 </p> 227 </td> 228</tr> 229<tr> 230<td> 231 <p> 232 Elliptic Integral PI: Large Random Data 233 </p> 234 </td> 235<td> 236 <p> 237 <span class="blue">Max = 3.7ε (Mean = 0.893ε)</span><br> <br> 238 (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = 2.52e+18ε (Mean 239 = 4.83e+17ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Large_Random_Data">And 240 other failures.</a>)</span> 241 </p> 242 </td> 243<td> 244 <p> 245 <span class="blue">Max = 0.557ε (Mean = 0.0389ε)</span><br> 246 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 40.1ε (Mean = 7.77ε)) 247 </p> 248 </td> 249<td> 250 <p> 251 <span class="blue">Max = 3.7ε (Mean = 0.892ε)</span> 252 </p> 253 </td> 254<td> 255 <p> 256 <span class="blue">Max = 2.86ε (Mean = 0.944ε)</span> 257 </p> 258 </td> 259</tr> 260</tbody> 261</table></div> 262</div> 263<br class="table-break"><h5> 264<a name="math_toolkit.ellint.ellint_3.h3"></a> 265 <span class="phrase"><a name="math_toolkit.ellint.ellint_3.testing"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.testing">Testing</a> 266 </h5> 267<p> 268 The tests use a mixture of spot test values calculated using the online calculator 269 at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, 270 and random test data generated using NTL::RR at 1000-bit precision and this 271 implementation. 272 </p> 273<h5> 274<a name="math_toolkit.ellint.ellint_3.h4"></a> 275 <span class="phrase"><a name="math_toolkit.ellint.ellint_3.implementation"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.implementation">Implementation</a> 276 </h5> 277<p> 278 The implementation for Π(n, φ, k) first siphons off the special cases: 279 </p> 280<div class="blockquote"><blockquote class="blockquote"><p> 281 <span class="serif_italic"><span class="emphasis"><em>Π(0, φ, k) = F(φ, k)</em></span></span> 282 </p></blockquote></div> 283<div class="blockquote"><blockquote class="blockquote"><p> 284 <span class="serif_italic"><span class="emphasis"><em>Π(n, π/2, k) = Π(n, k)</em></span></span> 285 </p></blockquote></div> 286<p> 287 and 288 </p> 289<div class="blockquote"><blockquote class="blockquote"><p> 290 <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span> 291 292 </p></blockquote></div> 293<p> 294 Then if n < 0 the relations (A&S 17.7.15/16): 295 </p> 296<div class="blockquote"><blockquote class="blockquote"><p> 297 <span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span> 298 299 </p></blockquote></div> 300<p> 301 are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1]. 302 </p> 303<p> 304 Then the relations: 305 </p> 306<div class="blockquote"><blockquote class="blockquote"><p> 307 <span class="serif_italic"><span class="emphasis"><em>Π(n, -φ, k) = -Π(n, φ, k)</em></span></span> 308 </p></blockquote></div> 309<div class="blockquote"><blockquote class="blockquote"><p> 310 <span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k) 311 ; n <= 1</em></span></span> 312 </p></blockquote></div> 313<div class="blockquote"><blockquote class="blockquote"><p> 314 <span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1</em></span> 315 316<a href="#ftn.math_toolkit.ellint.ellint_3.f0" class="footnote" name="math_toolkit.ellint.ellint_3.f0"><sup class="footnote">[1]</sup></a></span> 317 </p></blockquote></div> 318<p> 319 are used to move φ to the range [0, π/2]. 320 </p> 321<p> 322 The functions are then implemented in terms of Carlson's integrals using 323 the relations: 324 </p> 325<div class="blockquote"><blockquote class="blockquote"><p> 326 <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span> 327 328 </p></blockquote></div> 329<p> 330 and 331 </p> 332<div class="blockquote"><blockquote class="blockquote"><p> 333 <span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span> 334 335 </p></blockquote></div> 336<div class="footnotes"> 337<br><hr style="width:100; text-align:left;margin-left: 0"> 338<div id="ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><p><a href="#math_toolkit.ellint.ellint_3.f0" class="para"><sup class="para">[1] </sup></a> 339 I haven't been able to find a literature reference for this relation, 340 but it appears to be the convention used by Mathematica. Intuitively 341 the first <span class="emphasis"><em>2 * m * Π(n, k)</em></span> terms cancel out as the 342 derivative alternates between +∞ and -∞. 343 </p></div> 344</div> 345</div> 346<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 347<td align="left"></td> 348<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 349 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 350 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 351 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 352 Daryle Walker and Xiaogang Zhang<p> 353 Distributed under the Boost Software License, Version 1.0. (See accompanying 354 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>) 355 </p> 356</div></td> 357</tr></table> 358<hr> 359<div class="spirit-nav"> 360<a accesskey="p" href="ellint_2.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_d.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 361</div> 362</body> 363</html> 364
