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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Elliptic Integrals of the Second Kind - Legendre Form</title> | |
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| 13 | <table cellpadding="2" width="100%"><tr> | |
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| 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 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_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 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">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 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_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 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_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 | <p> | |
| 62 | <span class="inlinemediaobject"><img src="../../../graphs/ellint_2.svg" align="middle"></span> | |
| 63 | </p> | |
| 64 | <p> | |
| 65 | 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 | |
| 66 | type calculation rules</em></span></a> when T1 and T2 are different types: | |
| 67 | when they are the same type then the result is the same type as the arguments. | |
| 68 | </p> | |
| 69 | <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> | |
| 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_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> | |
| 71 | ||
| 72 | <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> | |
| 73 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 74 | </pre> | |
| 75 | <p> | |
| 76 | Returns the incomplete elliptic integral of the second kind <span class="emphasis"><em>E(φ, | |
| 77 | k)</em></span>: | |
| 78 | </p> | |
| 79 | <p> | |
| 80 | <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span> | |
| 81 | </p> | |
| 82 | <p> | |
| 83 | Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
| 84 | </p> | |
| 85 | <p> | |
| 86 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 87 | be used to control the behaviour of the function: how it handles errors, | |
| 88 | 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 | |
| 89 | documentation for more details</a>. | |
| 90 | </p> | |
| 91 | <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> | |
| 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_2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span> | |
| 93 | ||
| 94 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
| 95 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 96 | </pre> | |
| 97 | <p> | |
| 98 | Returns the complete elliptic integral of the second kind <span class="emphasis"><em>E(k)</em></span>: | |
| 99 | </p> | |
| 100 | <p> | |
| 101 | <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span> | |
| 102 | </p> | |
| 103 | <p> | |
| 104 | Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
| 105 | </p> | |
| 106 | <p> | |
| 107 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 108 | be used to control the behaviour of the function: how it handles errors, | |
| 109 | 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 | |
| 110 | documentation for more details</a>. | |
| 111 | </p> | |
| 112 | <h5> | |
| 113 | <a name="math_toolkit.ellint.ellint_2.h2"></a> | |
| 114 | <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> | |
| 115 | </h5> | |
| 116 | <p> | |
| 117 | These functions are computed using only basic arithmetic operations, so there | |
| 118 | isn't much variation in accuracy over differing platforms. Note that only | |
| 119 | results for the widest floating point type on the system are given as narrower | |
| 120 | types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively | |
| 121 | zero error</a>. All values are relative errors in units of epsilon. | |
| 122 | </p> | |
| 123 | <div class="table"> | |
| 124 | <a name="math_toolkit.ellint.ellint_2.table_ellint_2"></a><p class="title"><b>Table 6.64. Error rates for ellint_2</b></p> | |
| 125 | <div class="table-contents"><table class="table" summary="Error rates for ellint_2"> | |
| 126 | <colgroup> | |
| 127 | <col> | |
| 128 | <col> | |
| 129 | <col> | |
| 130 | <col> | |
| 131 | <col> | |
| 132 | </colgroup> | |
| 133 | <thead><tr> | |
| 134 | <th> | |
| 135 | </th> | |
| 136 | <th> | |
| 137 | <p> | |
| 138 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 139 | </p> | |
| 140 | </th> | |
| 141 | <th> | |
| 142 | <p> | |
| 143 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 144 | </p> | |
| 145 | </th> | |
| 146 | <th> | |
| 147 | <p> | |
| 148 | GNU C++ version 5.1.0<br> linux<br> double | |
| 149 | </p> | |
| 150 | </th> | |
| 151 | <th> | |
| 152 | <p> | |
| 153 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 154 | </p> | |
| 155 | </th> | |
| 156 | </tr></thead> | |
| 157 | <tbody> | |
| 158 | <tr> | |
| 159 | <td> | |
| 160 | <p> | |
| 161 | Elliptic Integral E: Mathworld Data | |
| 162 | </p> | |
| 163 | </td> | |
| 164 | <td> | |
| 165 | <p> | |
| 166 | <span class="blue">Max = 1.31ε (Mean = 0.727ε)</span> | |
| 167 | </p> | |
| 168 | </td> | |
| 169 | <td> | |
| 170 | <p> | |
| 171 | <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span><br> <br> | |
| 172 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
| 173 | = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_ellint_2__tr1_cmath__Elliptic_Integral_E_Mathworld_Data">And | |
| 174 | other failures.</a>)</span> | |
| 175 | </p> | |
| 176 | </td> | |
| 177 | <td> | |
| 178 | <p> | |
| 179 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
| 180 | 1.16:</em></span> Max = 0.63ε (Mean = 0.325ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
| 181 | <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ellint_2_Cephes_Elliptic_Integral_E_Mathworld_Data">And | |
| 182 | other failures.</a>)</span> | |
| 183 | </p> | |
| 184 | </td> | |
| 185 | <td> | |
| 186 | <p> | |
| 187 | <span class="blue">Max = 0.656ε (Mean = 0.317ε)</span> | |
| 188 | </p> | |
| 189 | </td> | |
| 190 | </tr> | |
| 191 | <tr> | |
| 192 | <td> | |
| 193 | <p> | |
| 194 | Elliptic Integral E: Random Data | |
| 195 | </p> | |
| 196 | </td> | |
| 197 | <td> | |
| 198 | <p> | |
| 199 | <span class="blue">Max = 2.23ε (Mean = 0.639ε)</span> | |
| 200 | </p> | |
| 201 | </td> | |
| 202 | <td> | |
| 203 | <p> | |
| 204 | <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span><br> <br> | |
| 205 | (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 3.08e+04ε (Mean = | |
| 206 | 3.84e+03ε)) | |
| 207 | </p> | |
| 208 | </td> | |
| 209 | <td> | |
| 210 | <p> | |
| 211 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
| 212 | 1.16:</em></span> Max = 4.4ε (Mean = 1.16ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
| 213 | Max = 8.92ε (Mean = 1.48ε)) | |
| 214 | </p> | |
| 215 | </td> | |
| 216 | <td> | |
| 217 | <p> | |
| 218 | <span class="blue">Max = 2.05ε (Mean = 0.632ε)</span> | |
| 219 | </p> | |
| 220 | </td> | |
| 221 | </tr> | |
| 222 | </tbody> | |
| 223 | </table></div> | |
| 224 | </div> | |
| 225 | <br class="table-break"><h5> | |
| 226 | <a name="math_toolkit.ellint.ellint_2.h3"></a> | |
| 227 | <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> | |
| 228 | </h5> | |
| 229 | <p> | |
| 230 | The tests use a mixture of spot test values calculated using the online calculator | |
| 231 | at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, | |
| 232 | and random test data generated using NTL::RR at 1000-bit precision and this | |
| 233 | implementation. | |
| 234 | </p> | |
| 235 | <h5> | |
| 236 | <a name="math_toolkit.ellint.ellint_2.h4"></a> | |
| 237 | <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> | |
| 238 | </h5> | |
| 239 | <p> | |
| 240 | These functions are implemented in terms of Carlson's integrals using the | |
| 241 | relations: | |
| 242 | </p> | |
| 243 | <p> | |
| 244 | <span class="inlinemediaobject"><img src="../../../equations/ellint21.svg"></span> | |
| 245 | </p> | |
| 246 | <p> | |
| 247 | and | |
| 248 | </p> | |
| 249 | <p> | |
| 250 | <span class="inlinemediaobject"><img src="../../../equations/ellint22.svg"></span> | |
| 251 | </p> | |
| 252 | </div> | |
| 253 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 254 | <td align="left"></td> | |
| 255 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 256 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 257 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 258 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 259 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 260 | 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>) | |
| 261 | </p> | |
| 262 | </div></td> | |
| 263 | </tr></table> | |
| 264 | <hr> | |
| 265 | <div class="spirit-nav"> | |
| 266 | <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> | |
| 267 | </div> | |
| 268 | </body> | |
| 269 | </html> |