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  24 </div>
  25 <div class="section">
  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;15.&#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;15.&#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;15.&#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;15.&#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 __arg_pomotion_rules
  62         when the arguments are of different types: when they are the same type then
  63         the result is the same type as the arguments.
  64       </p>
  65 <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>
  66 <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>
  67
  68 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  69 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  70 </pre>
  71 <p>
  72         Returns the incomplete elliptic integral:
  73       </p>
  74 <p>
  75         <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
  76       </p>
  77 <p>
  78         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span>, otherwise returns the result
  79         of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  80         (outside this range the result would be complex).
  81       </p>
  82 <p>
  83         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  84         be used to control the behaviour of the function: how it handles errors,
  85         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  86         documentation for more details</a>.
  87       </p>
  88 <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>
  89 <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>
  90
  91 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  93 </pre>
  94 <p>
  95         Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(&#960;/2, k)</em></span>
  96       </p>
  97 <p>
  98         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span> otherwise returns the result
  99         of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 100         (outside this range the result would be complex).
 101       </p>
 102 <p>
 103         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 104         be used to control the behaviour of the function: how it handles errors,
 105         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 106         documentation for more details</a>.
 107       </p>
 108 <h5>
 109 <a name="math_toolkit.ellint.ellint_d.h2"></a>
 110         <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>
 111       </h5>
 112 <p>
 113         These functions are trivially computed in terms of other elliptic integrals
 114         and generally have very low error rates (a few epsilon) unless parameter
 115         &#966;
 116 is very large, in which case the usual trigonometric function argument-reduction
 117         issues apply.
 118       </p>
 119 <div class="table">
 120 <a name="math_toolkit.ellint.ellint_d.table_ellint_d_complete_"></a><p class="title"><b>Table&#160;6.66.&#160;Error rates for ellint_d (complete)</b></p>
 121 <div class="table-contents"><table class="table" summary="Error rates for ellint_d (complete)">
 122 <colgroup>
 123 <col>
 124 <col>
 125 <col>
 126 <col>
 127 <col>
 128 </colgroup>
 129 <thead><tr>
 130 <th>
 131               </th>
 132 <th>
 133                 <p>
 134                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 135                 </p>
 136               </th>
 137 <th>
 138                 <p>
 139                   GNU C++ version 5.1.0<br> linux<br> double
 140                 </p>
 141               </th>
 142 <th>
 143                 <p>
 144                   GNU C++ version 5.1.0<br> linux<br> long double
 145                 </p>
 146               </th>
 147 <th>
 148                 <p>
 149                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 150                 </p>
 151               </th>
 152 </tr></thead>
 153 <tbody>
 154 <tr>
 155 <td>
 156                 <p>
 157                   Elliptic Integral E: Mathworld Data
 158                 </p>
 159               </td>
 160 <td>
 161                 <p>
 162                   <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</span>
 163                 </p>
 164               </td>
 165 <td>
 166                 <p>
 167                   <span class="blue">Max = 0.637&#949; (Mean = 0.368&#949;)</span>
 168                 </p>
 169               </td>
 170 <td>
 171                 <p>
 172                   <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
 173                 </p>
 174               </td>
 175 <td>
 176                 <p>
 177                   <span class="blue">Max = 1.27&#949; (Mean = 0.735&#949;)</span>
 178                 </p>
 179               </td>
 180 </tr>
 181 <tr>
 182 <td>
 183                 <p>
 184                   Elliptic Integral D: Random Data
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   <span class="blue">Max = 1.27&#949; (Mean = 0.355&#949;)</span>
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 195                 </p>
 196               </td>
 197 <td>
 198                 <p>
 199                   <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
 200                 </p>
 201               </td>
 202 <td>
 203                 <p>
 204                   <span class="blue">Max = 1.27&#949; (Mean = 0.334&#949;)</span>
 205                 </p>
 206               </td>
 207 </tr>
 208 </tbody>
 209 </table></div>
 210 </div>
 211 <br class="table-break"><div class="table">
 212 <a name="math_toolkit.ellint.ellint_d.table_ellint_d"></a><p class="title"><b>Table&#160;6.67.&#160;Error rates for ellint_d</b></p>
 213 <div class="table-contents"><table class="table" summary="Error rates for ellint_d">
 214 <colgroup>
 215 <col>
 216 <col>
 217 <col>
 218 <col>
 219 <col>
 220 </colgroup>
 221 <thead><tr>
 222 <th>
 223               </th>
 224 <th>
 225                 <p>
 226                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 227                 </p>
 228               </th>
 229 <th>
 230                 <p>
 231                   GNU C++ version 5.1.0<br> linux<br> double
 232                 </p>
 233               </th>
 234 <th>
 235                 <p>
 236                   GNU C++ version 5.1.0<br> linux<br> long double
 237                 </p>
 238               </th>
 239 <th>
 240                 <p>
 241                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 242                 </p>
 243               </th>
 244 </tr></thead>
 245 <tbody>
 246 <tr>
 247 <td>
 248                 <p>
 249                   Elliptic Integral E: Mathworld Data
 250                 </p>
 251               </td>
 252 <td>
 253                 <p>
 254                   <span class="blue">Max = 0.862&#949; (Mean = 0.457&#949;)</span>
 255                 </p>
 256               </td>
 257 <td>
 258                 <p>
 259                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 260                   1.16:</em></span> Max = 0.862&#949; (Mean = 0.568&#949;))
 261                 </p>
 262               </td>
 263 <td>
 264                 <p>
 265                   <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
 266                 </p>
 267               </td>
 268 <td>
 269                 <p>
 270                   <span class="blue">Max = 1.3&#949; (Mean = 0.813&#949;)</span>
 271                 </p>
 272               </td>
 273 </tr>
 274 <tr>
 275 <td>
 276                 <p>
 277                   Elliptic Integral D: Random Data
 278                 </p>
 279               </td>
 280 <td>
 281                 <p>
 282                   <span class="blue">Max = 2.87&#949; (Mean = 0.805&#949;)</span>
 283                 </p>
 284               </td>
 285 <td>
 286                 <p>
 287                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 288                   1.16:</em></span> Max = 3.01&#949; (Mean = 0.928&#949;))
 289                 </p>
 290               </td>
 291 <td>
 292                 <p>
 293                   <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
 294                 </p>
 295               </td>
 296 <td>
 297                 <p>
 298                   <span class="blue">Max = 2.51&#949; (Mean = 0.883&#949;)</span>
 299                 </p>
 300               </td>
 301 </tr>
 302 </tbody>
 303 </table></div>
 304 </div>
 305 <br class="table-break"><h5>
 306 <a name="math_toolkit.ellint.ellint_d.h3"></a>
 307         <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>
 308       </h5>
 309 <p>
 310         The tests use a mixture of spot test values calculated using values calculated
 311         at wolframalpha.com, and random test data generated using MPFR at 1000-bit
 312         precision and a deliberately naive implementation in terms of the Legendre
 313         integrals.
 314       </p>
 315 <h5>
 316 <a name="math_toolkit.ellint.ellint_d.h4"></a>
 317         <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>
 318       </h5>
 319 <p>
 320         The implementation for D(&#966;, k) first performs argument reduction using the
 321         relations:
 322       </p>
 323 <p>
 324         <span class="emphasis"><em>D(-&#966;, k) = -D(&#966;, k)</em></span>
 325       </p>
 326 <p>
 327         and
 328       </p>
 329 <p>
 330         <span class="emphasis"><em>D(n&#960;+&#966;, k) = 2nD(k) + D(&#966;, k)</em></span>
 331       </p>
 332 <p>
 333         to move &#966; &#160; to the range [0, &#960;/2].
 334       </p>
 335 <p>
 336         The functions are then implemented in terms of Carlson's integral R<sub>D</sub>
 337 using
 338         the relation:
 339       </p>
 340 <p>
 341         <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
 342       </p>
 343 </div>
 344 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 345 <td align="left"></td>
 346 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 347       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 348       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 349       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 350         Distributed under the Boost Software License, Version 1.0. (See accompanying
 351         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>)
 352       </p>
 353 </div></td>
 354 </tr></table>
 355 <hr>
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