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  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">&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_2</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_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">&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;20.&#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_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&#160;20.&#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">T</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_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">&lt;</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&#160;20.&#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_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&#160;20.&#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_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(&#966;, k)</em></span> and its complete counterpart <span class="emphasis"><em>E(k)
  59         = E(&#960;/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">&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>
  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">&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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  75 </pre>
  76 <p>
  77         Returns the incomplete elliptic integral of the second kind <span class="emphasis"><em>E(&#966;,
  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) &lt; 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&#160;20.&#160;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&#160;20.&#160;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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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| &lt; 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&#160;20.&#160;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&#160;20.&#160;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&#160;8.64.&#160;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 = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 170                   2.1:</em></span> Max = 0.63&#949; (Mean = 0.325&#949;))
 171                 </p>
 172               </td>
 173 <td>
 174                 <p>
 175                   <span class="blue">Max = 0.656&#949; (Mean = 0.317&#949;)</span><br> <br>
 176                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = +INF&#949; (Mean
 177                   = +INF&#949;) <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 = 0.656&#949; (Mean = 0.317&#949;)</span>
 184                 </p>
 185               </td>
 186 <td>
 187                 <p>
 188                   <span class="blue">Max = 1.31&#949; (Mean = 0.727&#949;)</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 = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 201                   2.1:</em></span> Max = 4.4&#949; (Mean = 1.16&#949;))
 202                 </p>
 203               </td>
 204 <td>
 205                 <p>
 206                   <span class="blue">Max = 2.05&#949; (Mean = 0.632&#949;)</span><br> <br>
 207                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 3.08e+04&#949; (Mean = 3.84e+03&#949;))
 208                 </p>
 209               </td>
 210 <td>
 211                 <p>
 212                   <span class="blue">Max = 2.05&#949; (Mean = 0.632&#949;)</span>
 213                 </p>
 214               </td>
 215 <td>
 216                 <p>
 217                   <span class="blue">Max = 2.23&#949; (Mean = 0.639&#949;)</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 = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 230                   2.1:</em></span> Max = 0.5&#949; (Mean = 0.118&#949;))
 231                 </p>
 232               </td>
 233 <td>
 234                 <p>
 235                   <span class="blue">Max = 1&#949; (Mean = 0.283&#949;)</span><br> <br>
 236                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2&#949; (Mean = 0.333&#949;))
 237                 </p>
 238               </td>
 239 <td>
 240                 <p>
 241                   <span class="blue">Max = 1&#949; (Mean = 0.283&#949;)</span>
 242                 </p>
 243               </td>
 244 <td>
 245                 <p>
 246                   <span class="blue">Max = 1&#949; (Mean = 0.421&#949;)</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 &#169; 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       R&#229;de, 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>
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