1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 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 V1.79.1"> 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_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 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_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 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">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 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_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 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_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<div class="blockquote"><blockquote class="blockquote"><p> 62 <span class="inlinemediaobject"><img src="../../../graphs/ellint_2.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 T1 and T2 are different types: 68 when they are the same type then the result is the same type as the arguments. 69 </p> 70<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> 71<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> 72 73<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> 74<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 75</pre> 76<p> 77 Returns the incomplete elliptic integral of the second kind <span class="emphasis"><em>E(φ, 78 k)</em></span>: 79 </p> 80<div class="blockquote"><blockquote class="blockquote"><p> 81 <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span> 82 83 </p></blockquote></div> 84<p> 85 Requires k<sup>2</sup>sin<sup>2</sup>(phi) < 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 86 </p> 87<p> 88 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 89 be used to control the behaviour of the function: how it handles errors, 90 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 91 documentation for more details</a>. 92 </p> 93<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> 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_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span> 95 96<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</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_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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 98</pre> 99<p> 100 Returns the complete elliptic integral of the second kind <span class="emphasis"><em>E(k)</em></span>: 101 </p> 102<div class="blockquote"><blockquote class="blockquote"><p> 103 <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span> 104 105 </p></blockquote></div> 106<p> 107 Requires |k| < 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 108 </p> 109<p> 110 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 111 be used to control the behaviour of the function: how it handles errors, 112 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 113 documentation for more details</a>. 114 </p> 115<h5> 116<a name="math_toolkit.ellint.ellint_2.h2"></a> 117 <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> 118 </h5> 119<p> 120 These functions are computed using only basic arithmetic operations, so there 121 isn't much variation in accuracy over differing platforms. Note that only 122 results for the widest floating point type on the system are given as narrower 123 types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 124 zero error</a>. All values are relative errors in units of epsilon. 125 </p> 126<div class="table"> 127<a name="math_toolkit.ellint.ellint_2.table_ellint_2"></a><p class="title"><b>Table 8.64. Error rates for ellint_2</b></p> 128<div class="table-contents"><table class="table" summary="Error rates for ellint_2"> 129<colgroup> 130<col> 131<col> 132<col> 133<col> 134<col> 135</colgroup> 136<thead><tr> 137<th> 138 </th> 139<th> 140 <p> 141 GNU C++ version 7.1.0<br> linux<br> double 142 </p> 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 Sun compiler version 0x5150<br> Sun Solaris<br> long double 152 </p> 153 </th> 154<th> 155 <p> 156 Microsoft Visual C++ version 14.1<br> Win32<br> double 157 </p> 158 </th> 159</tr></thead> 160<tbody> 161<tr> 162<td> 163 <p> 164 Elliptic Integral E: Mathworld Data 165 </p> 166 </td> 167<td> 168 <p> 169 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 170 2.1:</em></span> Max = 0.63ε (Mean = 0.325ε)) 171 </p> 172 </td> 173<td> 174 <p> 175 <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span><br> <br> 176 (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = +INFε (Mean 177 = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_2__cmath__Elliptic_Integral_E_Mathworld_Data">And 178 other failures.</a>)</span> 179 </p> 180 </td> 181<td> 182 <p> 183 <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span> 184 </p> 185 </td> 186<td> 187 <p> 188 <span class="blue">Max = 1.31ε (Mean = 0.727ε)</span> 189 </p> 190 </td> 191</tr> 192<tr> 193<td> 194 <p> 195 Elliptic Integral E: Random Data 196 </p> 197 </td> 198<td> 199 <p> 200 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 201 2.1:</em></span> Max = 4.4ε (Mean = 1.16ε)) 202 </p> 203 </td> 204<td> 205 <p> 206 <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span><br> <br> 207 (<span class="emphasis"><em><cmath>:</em></span> Max = 3.08e+04ε (Mean = 3.84e+03ε)) 208 </p> 209 </td> 210<td> 211 <p> 212 <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span> 213 </p> 214 </td> 215<td> 216 <p> 217 <span class="blue">Max = 2.23ε (Mean = 0.639ε)</span> 218 </p> 219 </td> 220</tr> 221<tr> 222<td> 223 <p> 224 Elliptic Integral E: Small Angles 225 </p> 226 </td> 227<td> 228 <p> 229 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 230 2.1:</em></span> Max = 0.5ε (Mean = 0.118ε)) 231 </p> 232 </td> 233<td> 234 <p> 235 <span class="blue">Max = 1ε (Mean = 0.283ε)</span><br> <br> 236 (<span class="emphasis"><em><cmath>:</em></span> Max = 2ε (Mean = 0.333ε)) 237 </p> 238 </td> 239<td> 240 <p> 241 <span class="blue">Max = 1ε (Mean = 0.283ε)</span> 242 </p> 243 </td> 244<td> 245 <p> 246 <span class="blue">Max = 1ε (Mean = 0.421ε)</span> 247 </p> 248 </td> 249</tr> 250</tbody> 251</table></div> 252</div> 253<br class="table-break"><p> 254 The following error plot are based on an exhaustive search of the functions 255 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code> 256 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span> 257 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>. 258 </p> 259<div class="blockquote"><blockquote class="blockquote"><p> 260 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_e__double.svg" align="middle"></span> 261 262 </p></blockquote></div> 263<div class="blockquote"><blockquote class="blockquote"><p> 264 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_e__80_bit_long_double.svg" align="middle"></span> 265 266 </p></blockquote></div> 267<div class="blockquote"><blockquote class="blockquote"><p> 268 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_e____float128.svg" align="middle"></span> 269 270 </p></blockquote></div> 271<h5> 272<a name="math_toolkit.ellint.ellint_2.h3"></a> 273 <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> 274 </h5> 275<p> 276 The tests use a mixture of spot test values calculated using the online calculator 277 at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, 278 and random test data generated using NTL::RR at 1000-bit precision and this 279 implementation. 280 </p> 281<h5> 282<a name="math_toolkit.ellint.ellint_2.h4"></a> 283 <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> 284 </h5> 285<p> 286 These functions are implemented in terms of Carlson's integrals using the 287 relations: 288 </p> 289<div class="blockquote"><blockquote class="blockquote"><p> 290 <span class="inlinemediaobject"><img src="../../../equations/ellint21.svg"></span> 291 292 </p></blockquote></div> 293<p> 294 and 295 </p> 296<div class="blockquote"><blockquote class="blockquote"><p> 297 <span class="inlinemediaobject"><img src="../../../equations/ellint22.svg"></span> 298 299 </p></blockquote></div> 300</div> 301<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 302<td align="left"></td> 303<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 304 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 305 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 306 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 307 Daryle Walker and Xiaogang Zhang<p> 308 Distributed under the Boost Software License, Version 1.0. (See accompanying 309 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>) 310 </p> 311</div></td> 312</tr></table> 313<hr> 314<div class="spirit-nav"> 315<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> 316</div> 317</body> 318</html> 319
