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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.ellint.ellint_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals of the
  28       Third Kind - Legendre Form</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.ellint.ellint_3.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.ellint.ellint_3.synopsis"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.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_3</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">,</span> <span class="keyword">class</span> <span class="identifier">T3</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_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">n</span><span class="special">,</span> <span class="identifier">T3</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> <span class="identifier">T3</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_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">n</span><span class="special">,</span> <span class="identifier">T3</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">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_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">n</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> <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>
  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_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">n</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_3.h1"></a>
  54         <span class="phrase"><a name="math_toolkit.ellint.ellint_3.description"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.description">Description</a>
  55       </h5>
  56 <p>
  57         These two functions evaluate the incomplete elliptic integral of the third
  58         kind <span class="emphasis"><em>&#928;(n, &#966;, k)</em></span> and its complete counterpart <span class="emphasis"><em>&#928;(n,
  59         k) = E(n, &#960;/2, k)</em></span>.
  60       </p>
  61 <div class="blockquote"><blockquote class="blockquote"><p>
  62           <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.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 the arguments are of different
  68         types: when they are the same type then the result is the same type as the
  69         arguments.
  70       </p>
  71 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  72 <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">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span>
  73
  74 <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> <span class="identifier">T3</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>
  75 <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">n</span><span class="special">,</span> <span class="identifier">T3</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>
  76 </pre>
  77 <p>
  78         Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>&#928;(n,
  79         &#966;, k)</em></span>:
  80       </p>
  81 <div class="blockquote"><blockquote class="blockquote"><p>
  82           <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>
  83
  84         </p></blockquote></div>
  85 <p>
  86         Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1/sin<sup>2</sup>(&#966;)</em></span>,
  87         otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  88         (outside this range the result would be complex).
  89       </p>
  90 <p>
  91         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
  92         be used to control the behaviour of the function: how it handles errors,
  93         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
  94         documentation for more details</a>.
  95       </p>
  96 <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>
  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_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">n</span><span class="special">);</span>
  98
  99 <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>
 100 <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">n</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>
 101 </pre>
 102 <p>
 103         Returns the complete elliptic integral of the first kind <span class="emphasis"><em>&#928;(n, k)</em></span>:
 104       </p>
 105 <div class="blockquote"><blockquote class="blockquote"><p>
 106           <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
 107
 108         </p></blockquote></div>
 109 <p>
 110         Requires <span class="emphasis"><em>|k| &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1</em></span>,
 111         otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 112         (outside this range the result would be complex).
 113       </p>
 114 <p>
 115         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
 116         be used to control the behaviour of the function: how it handles errors,
 117         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
 118         documentation for more details</a>.
 119       </p>
 120 <h5>
 121 <a name="math_toolkit.ellint.ellint_3.h2"></a>
 122         <span class="phrase"><a name="math_toolkit.ellint.ellint_3.accuracy"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.accuracy">Accuracy</a>
 123       </h5>
 124 <p>
 125         These functions are computed using only basic arithmetic operations, so there
 126         isn't much variation in accuracy over differing platforms. Note that only
 127         results for the widest floating point type on the system are given as narrower
 128         types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 129         zero error</a>. All values are relative errors in units of epsilon.
 130       </p>
 131 <div class="table">
 132 <a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table&#160;8.65.&#160;Error rates for ellint_3</b></p>
 133 <div class="table-contents"><table class="table" summary="Error rates for ellint_3">
 134 <colgroup>
 135 <col>
 136 <col>
 137 <col>
 138 <col>
 139 <col>
 140 </colgroup>
 141 <thead><tr>
 142 <th>
 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                   GNU C++ version 7.1.0<br> linux<br> double
 152                 </p>
 153               </th>
 154 <th>
 155                 <p>
 156                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 157                 </p>
 158               </th>
 159 <th>
 160                 <p>
 161                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 162                 </p>
 163               </th>
 164 </tr></thead>
 165 <tbody>
 166 <tr>
 167 <td>
 168                 <p>
 169                   Elliptic Integral PI: Mathworld Data
 170                 </p>
 171               </td>
 172 <td>
 173                 <p>
 174                   <span class="blue">Max = 475&#949; (Mean = 86.3&#949;)</span><br> <br>
 175                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = +INF&#949; (Mean
 176                   = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Mathworld_Data">And
 177                   other failures.</a>)</span>
 178                 </p>
 179               </td>
 180 <td>
 181                 <p>
 182                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 183                   2.1:</em></span> Max = 1.48e+05&#949; (Mean = 2.54e+04&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_3_GSL_2_1_Elliptic_Integral_PI_Mathworld_Data">And
 184                   other failures.</a>)
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   <span class="blue">Max = 475&#949; (Mean = 86.3&#949;)</span>
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 565&#949; (Mean = 102&#949;)</span>
 195                 </p>
 196               </td>
 197 </tr>
 198 <tr>
 199 <td>
 200                 <p>
 201                   Elliptic Integral PI: Random Data
 202                 </p>
 203               </td>
 204 <td>
 205                 <p>
 206                   <span class="blue">Max = 4.54&#949; (Mean = 0.895&#949;)</span><br> <br>
 207                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 3.37e+20&#949; (Mean
 208                   = 3.47e+19&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Random_Data">And
 209                   other failures.</a>)</span>
 210                 </p>
 211               </td>
 212 <td>
 213                 <p>
 214                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 215                   2.1:</em></span> Max = 633&#949; (Mean = 50.1&#949;))
 216                 </p>
 217               </td>
 218 <td>
 219                 <p>
 220                   <span class="blue">Max = 4.49&#949; (Mean = 0.885&#949;)</span>
 221                 </p>
 222               </td>
 223 <td>
 224                 <p>
 225                   <span class="blue">Max = 8.33&#949; (Mean = 0.971&#949;)</span>
 226                 </p>
 227               </td>
 228 </tr>
 229 <tr>
 230 <td>
 231                 <p>
 232                   Elliptic Integral PI: Large Random Data
 233                 </p>
 234               </td>
 235 <td>
 236                 <p>
 237                   <span class="blue">Max = 3.7&#949; (Mean = 0.893&#949;)</span><br> <br>
 238                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.52e+18&#949; (Mean
 239                   = 4.83e+17&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Large_Random_Data">And
 240                   other failures.</a>)</span>
 241                 </p>
 242               </td>
 243 <td>
 244                 <p>
 245                   <span class="blue">Max = 0.557&#949; (Mean = 0.0389&#949;)</span><br>
 246                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 40.1&#949; (Mean = 7.77&#949;))
 247                 </p>
 248               </td>
 249 <td>
 250                 <p>
 251                   <span class="blue">Max = 3.7&#949; (Mean = 0.892&#949;)</span>
 252                 </p>
 253               </td>
 254 <td>
 255                 <p>
 256                   <span class="blue">Max = 2.86&#949; (Mean = 0.944&#949;)</span>
 257                 </p>
 258               </td>
 259 </tr>
 260 </tbody>
 261 </table></div>
 262 </div>
 263 <br class="table-break"><h5>
 264 <a name="math_toolkit.ellint.ellint_3.h3"></a>
 265         <span class="phrase"><a name="math_toolkit.ellint.ellint_3.testing"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.testing">Testing</a>
 266       </h5>
 267 <p>
 268         The tests use a mixture of spot test values calculated using the online calculator
 269         at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
 270         and random test data generated using NTL::RR at 1000-bit precision and this
 271         implementation.
 272       </p>
 273 <h5>
 274 <a name="math_toolkit.ellint.ellint_3.h4"></a>
 275         <span class="phrase"><a name="math_toolkit.ellint.ellint_3.implementation"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.implementation">Implementation</a>
 276       </h5>
 277 <p>
 278         The implementation for &#928;(n, &#966;, k) first siphons off the special cases:
 279       </p>
 280 <div class="blockquote"><blockquote class="blockquote"><p>
 281           <span class="serif_italic"><span class="emphasis"><em>&#928;(0, &#966;, k) = F(&#966;, k)</em></span></span>
 282         </p></blockquote></div>
 283 <div class="blockquote"><blockquote class="blockquote"><p>
 284           <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#960;/2, k) = &#928;(n, k)</em></span></span>
 285         </p></blockquote></div>
 286 <p>
 287         and
 288       </p>
 289 <div class="blockquote"><blockquote class="blockquote"><p>
 290           <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span>
 291
 292         </p></blockquote></div>
 293 <p>
 294         Then if n &lt; 0 the relations (A&amp;S 17.7.15/16):
 295       </p>
 296 <div class="blockquote"><blockquote class="blockquote"><p>
 297           <span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span>
 298
 299         </p></blockquote></div>
 300 <p>
 301         are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1].
 302       </p>
 303 <p>
 304         Then the relations:
 305       </p>
 306 <div class="blockquote"><blockquote class="blockquote"><p>
 307           <span class="serif_italic"><span class="emphasis"><em>&#928;(n, -&#966;, k) = -&#928;(n, &#966;, k)</em></span></span>
 308         </p></blockquote></div>
 309 <div class="blockquote"><blockquote class="blockquote"><p>
 310           <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) + 2m&#928;(n, k)
 311           ; n &lt;= 1</em></span></span>
 312         </p></blockquote></div>
 313 <div class="blockquote"><blockquote class="blockquote"><p>
 314           <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) ; n &gt; 1</em></span>
 315           &#8198;&#8198;&#8198;  &#8198;&#8198;&#8198;
 316 <a href="#ftn.math_toolkit.ellint.ellint_3.f0" class="footnote" name="math_toolkit.ellint.ellint_3.f0"><sup class="footnote">[1]</sup></a></span>
 317         </p></blockquote></div>
 318 <p>
 319         are used to move &#966; to the range [0, &#960;/2].
 320       </p>
 321 <p>
 322         The functions are then implemented in terms of Carlson's integrals using
 323         the relations:
 324       </p>
 325 <div class="blockquote"><blockquote class="blockquote"><p>
 326           <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span>
 327
 328         </p></blockquote></div>
 329 <p>
 330         and
 331       </p>
 332 <div class="blockquote"><blockquote class="blockquote"><p>
 333           <span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span>
 334
 335         </p></blockquote></div>
 336 <div class="footnotes">
 337 <br><hr style="width:100; text-align:left;margin-left: 0">
 338 <div id="ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><p><a href="#math_toolkit.ellint.ellint_3.f0" class="para"><sup class="para">[1] </sup></a>
 339             I haven't been able to find a literature reference for this relation,
 340             but it appears to be the convention used by Mathematica. Intuitively
 341             the first <span class="emphasis"><em>2 * m * &#928;(n, k)</em></span> terms cancel out as the
 342             derivative alternates between +&#8734; and -&#8734;.
 343           </p></div>
 344 </div>
 345 </div>
 346 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 347 <td align="left"></td>
 348 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 349       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 350       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 351       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 352       Daryle Walker and Xiaogang Zhang<p>
 353         Distributed under the Boost Software License, Version 1.0. (See accompanying
 354         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>)
 355       </p>
 356 </div></td>
 357 </tr></table>
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