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  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">&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_d</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_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">&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_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&#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">T1</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_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">&lt;</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&#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_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&#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_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(&#966;,
  58         k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(&#960;/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">&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>
  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">&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>
  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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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) &lt; 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&#160;20.&#160;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&#160;20.&#160;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">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><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_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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  95 </pre>
  96 <p>
  97         Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(&#960;/2, k)</em></span>
  98       </p>
  99 <p>
 100         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 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&#160;20.&#160;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&#160;20.&#160;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         &#966;
 118 is 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&#160;8.66.&#160;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 = 0.637&#949; (Mean = 0.368&#949;)</span>
 165                 </p>
 166               </td>
 167 <td>
 168                 <p>
 169                   <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
 170                 </p>
 171               </td>
 172 <td>
 173                 <p>
 174                   <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
 175                 </p>
 176               </td>
 177 <td>
 178                 <p>
 179                   <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</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 = 0&#949; (Mean = 0&#949;)</span>
 192                 </p>
 193               </td>
 194 <td>
 195                 <p>
 196                   <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
 197                 </p>
 198               </td>
 199 <td>
 200                 <p>
 201                   <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
 202                 </p>
 203               </td>
 204 <td>
 205                 <p>
 206                   <span class="blue">Max = 1.27&#949; (Mean = 0.355&#949;)</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&#160;8.67.&#160;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 = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 257                   2.1:</em></span> Max = 0.862&#949; (Mean = 0.568&#949;))
 258                 </p>
 259               </td>
 260 <td>
 261                 <p>
 262                   <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
 263                 </p>
 264               </td>
 265 <td>
 266                 <p>
 267                   <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
 268                 </p>
 269               </td>
 270 <td>
 271                 <p>
 272                   <span class="blue">Max = 0.862&#949; (Mean = 0.457&#949;)</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 = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 285                   2.1:</em></span> Max = 3.01&#949; (Mean = 0.928&#949;))
 286                 </p>
 287               </td>
 288 <td>
 289                 <p>
 290                   <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
 291                 </p>
 292               </td>
 293 <td>
 294                 <p>
 295                   <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
 296                 </p>
 297               </td>
 298 <td>
 299                 <p>
 300                   <span class="blue">Max = 2.87&#949; (Mean = 0.805&#949;)</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(&#966;, 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(-&#966;, k) = -D(&#966;, 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(n&#960;+&#966;, k) = 2nD(k) + D(&#966;, k)</em></span></span>
 351         </p></blockquote></div>
 352 <p>
 353         to move &#966; to the range [0, &#960;/2].
 354       </p>
 355 <p>
 356         The functions are then implemented in terms of Carlson's integral R<sub>D</sub>
 357 using
 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 &#169; 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       R&#229;de, 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>
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