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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.jacobi.jacobi_elliptic"></a><a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">Jacobi Elliptic
  28       SN, CN and DN</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.jacobi.jacobi_elliptic.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.synopsis"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.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">jacobi_elliptic</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">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</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">jacobi_elliptic</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pcn</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pdn</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">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><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">jacobi_elliptic</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pcn</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pdn</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span>
  43
  44 <span class="special">}}</span> <span class="comment">// namespaces</span>
  45 </pre>
  46 <h5>
  47 <a name="math_toolkit.jacobi.jacobi_elliptic.h1"></a>
  48         <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.description"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.description">Description</a>
  49       </h5>
  50 <p>
  51         The function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>
  52         calculates the three copolar Jacobi elliptic functions <span class="emphasis"><em>sn(u, k)</em></span>,
  53         <span class="emphasis"><em>cn(u, k)</em></span> and <span class="emphasis"><em>dn(u, k)</em></span>. The returned
  54         value is <span class="emphasis"><em>sn(u, k)</em></span>, and if provided, <code class="computeroutput"><span class="special">*</span><span class="identifier">pcn</span></code> is set to <span class="emphasis"><em>cn(u, k)</em></span>,
  55         and <code class="computeroutput"><span class="special">*</span><span class="identifier">pdn</span></code>
  56         is set to <span class="emphasis"><em>dn(u, k)</em></span>.
  57       </p>
  58 <p>
  59         The functions are defined as follows, given:
  60       </p>
  61 <div class="blockquote"><blockquote class="blockquote"><p>
  62           <span class="inlinemediaobject"><img src="../../../equations/jacobi1.svg"></span>
  63
  64         </p></blockquote></div>
  65 <p>
  66         The the angle <span class="emphasis"><em>&#966;</em></span> is called the <span class="emphasis"><em>amplitude</em></span>
  67         and:
  68       </p>
  69 <div class="blockquote"><blockquote class="blockquote"><p>
  70           <span class="inlinemediaobject"><img src="../../../equations/jacobi2.svg"></span>
  71
  72         </p></blockquote></div>
  73 <div class="note"><table border="0" summary="Note">
  74 <tr>
  75 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
  76 <th align="left">Note</th>
  77 </tr>
  78 <tr><td align="left" valign="top"><p>
  79           <span class="emphasis"><em>&#966;</em></span> is called the amplitude. <span class="emphasis"><em>k</em></span> is
  80           called the elliptic modulus.
  81         </p></td></tr>
  82 </table></div>
  83 <div class="caution"><table border="0" summary="Caution">
  84 <tr>
  85 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
  86 <th align="left">Caution</th>
  87 </tr>
  88 <tr><td align="left" valign="top">
  89 <p>
  90           Rather like other elliptic functions, the Jacobi functions are expressed
  91           in a variety of different ways. In particular, the parameter <span class="emphasis"><em>k</em></span>
  92           (the modulus) may also be expressed using a modular angle &#945;, or a parameter
  93           <span class="emphasis"><em>m</em></span>. These are related by:
  94         </p>
  95 <div class="blockquote"><blockquote class="blockquote"><p>
  96             <span class="serif_italic">k = sin &#945;</span>
  97           </p></blockquote></div>
  98 <div class="blockquote"><blockquote class="blockquote"><p>
  99             <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>&#945;</span>
 100           </p></blockquote></div>
 101 <p>
 102           So that the function <span class="emphasis"><em>sn</em></span> (for example) may be expressed
 103           as either:
 104         </p>
 105 <div class="blockquote"><blockquote class="blockquote"><p>
 106             <span class="serif_italic">sn(u, k)</span>
 107           </p></blockquote></div>
 108 <div class="blockquote"><blockquote class="blockquote"><p>
 109             <span class="serif_italic">sn(u \ &#945;)</span>
 110           </p></blockquote></div>
 111 <div class="blockquote"><blockquote class="blockquote"><p>
 112             <span class="serif_italic">sn(u | m)</span>
 113           </p></blockquote></div>
 114 <p>
 115           To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
 116           of the parameter m</em></span>, or 1 - m, where:
 117         </p>
 118 <div class="blockquote"><blockquote class="blockquote"><p>
 119             <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>&#945;</span>
 120           </p></blockquote></div>
 121 <p>
 122           This implementation uses <span class="emphasis"><em>k</em></span> throughout, and makes this
 123           the first argument to the functions: this is for alignment with the elliptic
 124           integrals which match the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
 125           Report on C++ Library Extensions</a>. However, you should be extra
 126           careful when using these functions!
 127         </p>
 128 </td></tr>
 129 </table></div>
 130 <p>
 131         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
 132         be used to control the behaviour of the function: how it handles errors,
 133         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
 134         documentation for more details</a>.
 135       </p>
 136 <p>
 137         The following graphs illustrate how these functions change as <span class="emphasis"><em>k</em></span>
 138         changes: for small <span class="emphasis"><em>k</em></span> these are sine waves, while as
 139         <span class="emphasis"><em>k</em></span> tends to 1 they become hyperbolic functions:
 140       </p>
 141 <div class="blockquote"><blockquote class="blockquote"><p>
 142           <span class="inlinemediaobject"><img src="../../../graphs/jacobi_sn.svg" align="middle"></span>
 143
 144         </p></blockquote></div>
 145 <div class="blockquote"><blockquote class="blockquote"><p>
 146           <span class="inlinemediaobject"><img src="../../../graphs/jacobi_cn.svg" align="middle"></span>
 147
 148         </p></blockquote></div>
 149 <div class="blockquote"><blockquote class="blockquote"><p>
 150           <span class="inlinemediaobject"><img src="../../../graphs/jacobi_dn.svg" align="middle"></span>
 151
 152         </p></blockquote></div>
 153 <h5>
 154 <a name="math_toolkit.jacobi.jacobi_elliptic.h2"></a>
 155         <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.accuracy"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.accuracy">Accuracy</a>
 156       </h5>
 157 <p>
 158         These functions are computed using only basic arithmetic operations and trigomometric
 159         functions, so there isn't much variation in accuracy over differing platforms.
 160         Typically errors are trivially small for small angles, and as is typical
 161         for cyclic functions, grow as the angle increases. Note that only results
 162         for the widest floating-point type on the system are given as narrower types
 163         have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
 164         error</a>. All values are relative errors in units of epsilon.
 165       </p>
 166 <div class="table">
 167 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_cn"></a><p class="title"><b>Table&#160;8.70.&#160;Error rates for jacobi_cn</b></p>
 168 <div class="table-contents"><table class="table" summary="Error rates for jacobi_cn">
 169 <colgroup>
 170 <col>
 171 <col>
 172 <col>
 173 <col>
 174 <col>
 175 </colgroup>
 176 <thead><tr>
 177 <th>
 178               </th>
 179 <th>
 180                 <p>
 181                   GNU C++ version 7.1.0<br> linux<br> double
 182                 </p>
 183               </th>
 184 <th>
 185                 <p>
 186                   GNU C++ version 7.1.0<br> linux<br> long double
 187                 </p>
 188               </th>
 189 <th>
 190                 <p>
 191                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 192                 </p>
 193               </th>
 194 <th>
 195                 <p>
 196                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 197                 </p>
 198               </th>
 199 </tr></thead>
 200 <tbody>
 201 <tr>
 202 <td>
 203                 <p>
 204                   Jacobi Elliptic: Mathworld Data
 205                 </p>
 206               </td>
 207 <td>
 208                 <p>
 209                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 210                   2.1:</em></span> Max = 17.3&#949; (Mean = 4.29&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
 211                   other failures.</a>)
 212                 </p>
 213               </td>
 214 <td>
 215                 <p>
 216                   <span class="blue">Max = 71.6&#949; (Mean = 19.3&#949;)</span>
 217                 </p>
 218               </td>
 219 <td>
 220                 <p>
 221                   <span class="blue">Max = 71.6&#949; (Mean = 19.4&#949;)</span>
 222                 </p>
 223               </td>
 224 <td>
 225                 <p>
 226                   <span class="blue">Max = 45.8&#949; (Mean = 11.4&#949;)</span>
 227                 </p>
 228               </td>
 229 </tr>
 230 <tr>
 231 <td>
 232                 <p>
 233                   Jacobi Elliptic: Random Data
 234                 </p>
 235               </td>
 236 <td>
 237                 <p>
 238                   <span class="blue">Max = 0.816&#949; (Mean = 0.0563&#949;)</span><br>
 239                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.43&#949; (Mean = 0.803&#949;))
 240                 </p>
 241               </td>
 242 <td>
 243                 <p>
 244                   <span class="blue">Max = 1.68&#949; (Mean = 0.443&#949;)</span>
 245                 </p>
 246               </td>
 247 <td>
 248                 <p>
 249                   <span class="blue">Max = 1.68&#949; (Mean = 0.454&#949;)</span>
 250                 </p>
 251               </td>
 252 <td>
 253                 <p>
 254                   <span class="blue">Max = 1.83&#949; (Mean = 0.455&#949;)</span>
 255                 </p>
 256               </td>
 257 </tr>
 258 <tr>
 259 <td>
 260                 <p>
 261                   Jacobi Elliptic: Random Small Values
 262                 </p>
 263               </td>
 264 <td>
 265                 <p>
 266                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 267                   2.1:</em></span> Max = 55.2&#949; (Mean = 1.64&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
 268                   other failures.</a>)
 269                 </p>
 270               </td>
 271 <td>
 272                 <p>
 273                   <span class="blue">Max = 10.4&#949; (Mean = 0.594&#949;)</span>
 274                 </p>
 275               </td>
 276 <td>
 277                 <p>
 278                   <span class="blue">Max = 10.4&#949; (Mean = 0.602&#949;)</span>
 279                 </p>
 280               </td>
 281 <td>
 282                 <p>
 283                   <span class="blue">Max = 26.2&#949; (Mean = 1.17&#949;)</span>
 284                 </p>
 285               </td>
 286 </tr>
 287 <tr>
 288 <td>
 289                 <p>
 290                   Jacobi Elliptic: Modulus near 1
 291                 </p>
 292               </td>
 293 <td>
 294                 <p>
 295                   <span class="blue">Max = 0.919&#949; (Mean = 0.127&#949;)</span><br> <br>
 296                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
 297                   other failures.</a>)
 298                 </p>
 299               </td>
 300 <td>
 301                 <p>
 302                   <span class="blue">Max = 675&#949; (Mean = 87.1&#949;)</span>
 303                 </p>
 304               </td>
 305 <td>
 306                 <p>
 307                   <span class="blue">Max = 675&#949; (Mean = 86.8&#949;)</span>
 308                 </p>
 309               </td>
 310 <td>
 311                 <p>
 312                   <span class="blue">Max = 513&#949; (Mean = 126&#949;)</span>
 313                 </p>
 314               </td>
 315 </tr>
 316 <tr>
 317 <td>
 318                 <p>
 319                   Jacobi Elliptic: Large Phi
 320                 </p>
 321               </td>
 322 <td>
 323                 <p>
 324                   <span class="blue">Max = 14.2&#949; (Mean = 0.927&#949;)</span><br> <br>
 325                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 5.92e+03&#949; (Mean = 477&#949;))
 326                 </p>
 327               </td>
 328 <td>
 329                 <p>
 330                   <span class="blue">Max = 2.97e+04&#949; (Mean = 1.9e+03&#949;)</span>
 331                 </p>
 332               </td>
 333 <td>
 334                 <p>
 335                   <span class="blue">Max = 2.97e+04&#949; (Mean = 1.9e+03&#949;)</span>
 336                 </p>
 337               </td>
 338 <td>
 339                 <p>
 340                   <span class="blue">Max = 3.27e+04&#949; (Mean = 1.93e+03&#949;)</span>
 341                 </p>
 342               </td>
 343 </tr>
 344 </tbody>
 345 </table></div>
 346 </div>
 347 <br class="table-break"><div class="table">
 348 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_dn"></a><p class="title"><b>Table&#160;8.71.&#160;Error rates for jacobi_dn</b></p>
 349 <div class="table-contents"><table class="table" summary="Error rates for jacobi_dn">
 350 <colgroup>
 351 <col>
 352 <col>
 353 <col>
 354 <col>
 355 <col>
 356 </colgroup>
 357 <thead><tr>
 358 <th>
 359               </th>
 360 <th>
 361                 <p>
 362                   GNU C++ version 7.1.0<br> linux<br> double
 363                 </p>
 364               </th>
 365 <th>
 366                 <p>
 367                   GNU C++ version 7.1.0<br> linux<br> long double
 368                 </p>
 369               </th>
 370 <th>
 371                 <p>
 372                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 373                 </p>
 374               </th>
 375 <th>
 376                 <p>
 377                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 378                 </p>
 379               </th>
 380 </tr></thead>
 381 <tbody>
 382 <tr>
 383 <td>
 384                 <p>
 385                   Jacobi Elliptic: Mathworld Data
 386                 </p>
 387               </td>
 388 <td>
 389                 <p>
 390                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 391                   2.1:</em></span> Max = 2.82&#949; (Mean = 1.18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
 392                   other failures.</a>)
 393                 </p>
 394               </td>
 395 <td>
 396                 <p>
 397                   <span class="blue">Max = 49&#949; (Mean = 14&#949;)</span>
 398                 </p>
 399               </td>
 400 <td>
 401                 <p>
 402                   <span class="blue">Max = 49&#949; (Mean = 14&#949;)</span>
 403                 </p>
 404               </td>
 405 <td>
 406                 <p>
 407                   <span class="blue">Max = 34.3&#949; (Mean = 8.71&#949;)</span>
 408                 </p>
 409               </td>
 410 </tr>
 411 <tr>
 412 <td>
 413                 <p>
 414                   Jacobi Elliptic: Random Data
 415                 </p>
 416               </td>
 417 <td>
 418                 <p>
 419                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 420                   2.1:</em></span> Max = 3&#949; (Mean = 0.61&#949;))
 421                 </p>
 422               </td>
 423 <td>
 424                 <p>
 425                   <span class="blue">Max = 1.53&#949; (Mean = 0.473&#949;)</span>
 426                 </p>
 427               </td>
 428 <td>
 429                 <p>
 430                   <span class="blue">Max = 1.53&#949; (Mean = 0.481&#949;)</span>
 431                 </p>
 432               </td>
 433 <td>
 434                 <p>
 435                   <span class="blue">Max = 1.52&#949; (Mean = 0.466&#949;)</span>
 436                 </p>
 437               </td>
 438 </tr>
 439 <tr>
 440 <td>
 441                 <p>
 442                   Jacobi Elliptic: Random Small Values
 443                 </p>
 444               </td>
 445 <td>
 446                 <p>
 447                   <span class="blue">Max = 0.5&#949; (Mean = 0.0122&#949;)</span><br> <br>
 448                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5&#949; (Mean = 0.391&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
 449                   other failures.</a>)
 450                 </p>
 451               </td>
 452 <td>
 453                 <p>
 454                   <span class="blue">Max = 22.4&#949; (Mean = 0.777&#949;)</span>
 455                 </p>
 456               </td>
 457 <td>
 458                 <p>
 459                   <span class="blue">Max = 22.4&#949; (Mean = 0.763&#949;)</span>
 460                 </p>
 461               </td>
 462 <td>
 463                 <p>
 464                   <span class="blue">Max = 16.1&#949; (Mean = 0.685&#949;)</span>
 465                 </p>
 466               </td>
 467 </tr>
 468 <tr>
 469 <td>
 470                 <p>
 471                   Jacobi Elliptic: Modulus near 1
 472                 </p>
 473               </td>
 474 <td>
 475                 <p>
 476                   <span class="blue">Max = 2.28&#949; (Mean = 0.194&#949;)</span><br> <br>
 477                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
 478                   other failures.</a>)
 479                 </p>
 480               </td>
 481 <td>
 482                 <p>
 483                   <span class="blue">Max = 3.75e+03&#949; (Mean = 293&#949;)</span>
 484                 </p>
 485               </td>
 486 <td>
 487                 <p>
 488                   <span class="blue">Max = 3.75e+03&#949; (Mean = 293&#949;)</span>
 489                 </p>
 490               </td>
 491 <td>
 492                 <p>
 493                   <span class="blue">Max = 6.24e+03&#949; (Mean = 482&#949;)</span>
 494                 </p>
 495               </td>
 496 </tr>
 497 <tr>
 498 <td>
 499                 <p>
 500                   Jacobi Elliptic: Large Phi
 501                 </p>
 502               </td>
 503 <td>
 504                 <p>
 505                   <span class="blue">Max = 14.1&#949; (Mean = 0.897&#949;)</span><br> <br>
 506                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 121&#949; (Mean = 22&#949;))
 507                 </p>
 508               </td>
 509 <td>
 510                 <p>
 511                   <span class="blue">Max = 2.82e+04&#949; (Mean = 1.79e+03&#949;)</span>
 512                 </p>
 513               </td>
 514 <td>
 515                 <p>
 516                   <span class="blue">Max = 2.82e+04&#949; (Mean = 1.79e+03&#949;)</span>
 517                 </p>
 518               </td>
 519 <td>
 520                 <p>
 521                   <span class="blue">Max = 1.67e+04&#949; (Mean = 1e+03&#949;)</span>
 522                 </p>
 523               </td>
 524 </tr>
 525 </tbody>
 526 </table></div>
 527 </div>
 528 <br class="table-break"><div class="table">
 529 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_sn"></a><p class="title"><b>Table&#160;8.72.&#160;Error rates for jacobi_sn</b></p>
 530 <div class="table-contents"><table class="table" summary="Error rates for jacobi_sn">
 531 <colgroup>
 532 <col>
 533 <col>
 534 <col>
 535 <col>
 536 <col>
 537 </colgroup>
 538 <thead><tr>
 539 <th>
 540               </th>
 541 <th>
 542                 <p>
 543                   GNU C++ version 7.1.0<br> linux<br> double
 544                 </p>
 545               </th>
 546 <th>
 547                 <p>
 548                   GNU C++ version 7.1.0<br> linux<br> long double
 549                 </p>
 550               </th>
 551 <th>
 552                 <p>
 553                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 554                 </p>
 555               </th>
 556 <th>
 557                 <p>
 558                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 559                 </p>
 560               </th>
 561 </tr></thead>
 562 <tbody>
 563 <tr>
 564 <td>
 565                 <p>
 566                   Jacobi Elliptic: Mathworld Data
 567                 </p>
 568               </td>
 569 <td>
 570                 <p>
 571                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 572                   2.1:</em></span> Max = 588&#949; (Mean = 146&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
 573                   other failures.</a>)
 574                 </p>
 575               </td>
 576 <td>
 577                 <p>
 578                   <span class="blue">Max = 341&#949; (Mean = 80.7&#949;)</span>
 579                 </p>
 580               </td>
 581 <td>
 582                 <p>
 583                   <span class="blue">Max = 341&#949; (Mean = 80.7&#949;)</span>
 584                 </p>
 585               </td>
 586 <td>
 587                 <p>
 588                   <span class="blue">Max = 481&#949; (Mean = 113&#949;)</span>
 589                 </p>
 590               </td>
 591 </tr>
 592 <tr>
 593 <td>
 594                 <p>
 595                   Jacobi Elliptic: Random Data
 596                 </p>
 597               </td>
 598 <td>
 599                 <p>
 600                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 601                   2.1:</em></span> Max = 4.02&#949; (Mean = 1.07&#949;))
 602                 </p>
 603               </td>
 604 <td>
 605                 <p>
 606                   <span class="blue">Max = 2.01&#949; (Mean = 0.584&#949;)</span>
 607                 </p>
 608               </td>
 609 <td>
 610                 <p>
 611                   <span class="blue">Max = 2.01&#949; (Mean = 0.593&#949;)</span>
 612                 </p>
 613               </td>
 614 <td>
 615                 <p>
 616                   <span class="blue">Max = 1.92&#949; (Mean = 0.567&#949;)</span>
 617                 </p>
 618               </td>
 619 </tr>
 620 <tr>
 621 <td>
 622                 <p>
 623                   Jacobi Elliptic: Random Small Values
 624                 </p>
 625               </td>
 626 <td>
 627                 <p>
 628                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 629                   2.1:</em></span> Max = 11.7&#949; (Mean = 1.65&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
 630                   other failures.</a>)
 631                 </p>
 632               </td>
 633 <td>
 634                 <p>
 635                   <span class="blue">Max = 1.99&#949; (Mean = 0.347&#949;)</span>
 636                 </p>
 637               </td>
 638 <td>
 639                 <p>
 640                   <span class="blue">Max = 1.99&#949; (Mean = 0.347&#949;)</span>
 641                 </p>
 642               </td>
 643 <td>
 644                 <p>
 645                   <span class="blue">Max = 2.11&#949; (Mean = 0.385&#949;)</span>
 646                 </p>
 647               </td>
 648 </tr>
 649 <tr>
 650 <td>
 651                 <p>
 652                   Jacobi Elliptic: Modulus near 1
 653                 </p>
 654               </td>
 655 <td>
 656                 <p>
 657                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 658                   2.1:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
 659                   other failures.</a>)
 660                 </p>
 661               </td>
 662 <td>
 663                 <p>
 664                   <span class="blue">Max = 109&#949; (Mean = 7.35&#949;)</span>
 665                 </p>
 666               </td>
 667 <td>
 668                 <p>
 669                   <span class="blue">Max = 109&#949; (Mean = 7.38&#949;)</span>
 670                 </p>
 671               </td>
 672 <td>
 673                 <p>
 674                   <span class="blue">Max = 23.2&#949; (Mean = 1.85&#949;)</span>
 675                 </p>
 676               </td>
 677 </tr>
 678 <tr>
 679 <td>
 680                 <p>
 681                   Jacobi Elliptic: Large Phi
 682                 </p>
 683               </td>
 684 <td>
 685                 <p>
 686                   <span class="blue">Max = 12&#949; (Mean = 0.771&#949;)</span><br> <br>
 687                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.54e+04&#949; (Mean = 2.63e+03&#949;))
 688                 </p>
 689               </td>
 690 <td>
 691                 <p>
 692                   <span class="blue">Max = 2.45e+04&#949; (Mean = 1.51e+03&#949;)</span>
 693                 </p>
 694               </td>
 695 <td>
 696                 <p>
 697                   <span class="blue">Max = 2.45e+04&#949; (Mean = 1.51e+03&#949;)</span>
 698                 </p>
 699               </td>
 700 <td>
 701                 <p>
 702                   <span class="blue">Max = 4.36e+04&#949; (Mean = 2.54e+03&#949;)</span>
 703                 </p>
 704               </td>
 705 </tr>
 706 </tbody>
 707 </table></div>
 708 </div>
 709 <br class="table-break"><h5>
 710 <a name="math_toolkit.jacobi.jacobi_elliptic.h3"></a>
 711         <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.testing"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.testing">Testing</a>
 712       </h5>
 713 <p>
 714         The tests use a mixture of spot test values calculated using the online calculator
 715         at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
 716         and random test data generated using MPFR at 1000-bit precision and this
 717         implementation.
 718       </p>
 719 <h5>
 720 <a name="math_toolkit.jacobi.jacobi_elliptic.h4"></a>
 721         <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.implementation"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.implementation">Implementation</a>
 722       </h5>
 723 <p>
 724         For <span class="emphasis"><em>k &gt; 1</em></span> we apply the relations:
 725       </p>
 726 <div class="blockquote"><blockquote class="blockquote"><p>
 727           <span class="inlinemediaobject"><img src="../../../equations/jacobi3.svg"></span>
 728
 729         </p></blockquote></div>
 730 <p>
 731         Then filter off the special cases:
 732       </p>
 733 <div class="blockquote"><blockquote class="blockquote"><p>
 734           <span class="serif_italic"><span class="emphasis"><em>sn(0, k) = 0</em></span> and <span class="emphasis"><em>cn(0,
 735           k) = dn(0, k) = 1</em></span></span>
 736         </p></blockquote></div>
 737 <div class="blockquote"><blockquote class="blockquote"><p>
 738           <span class="serif_italic"><span class="emphasis"><em>sn(u, 0) = sin(u), cn(u, 0) = cos(u)
 739           and dn(u, 0) = 1</em></span></span>
 740         </p></blockquote></div>
 741 <div class="blockquote"><blockquote class="blockquote"><p>
 742           <span class="serif_italic"><span class="emphasis"><em>sn(u, 1) = tanh(u), cn(u, 1) = dn(u,
 743           1) = 1 / cosh(u)</em></span></span>
 744         </p></blockquote></div>
 745 <p>
 746         And for <span class="emphasis"><em>k<sup>4</sup> &lt; &#949;</em></span> we have:
 747       </p>
 748 <div class="blockquote"><blockquote class="blockquote"><p>
 749           <span class="inlinemediaobject"><img src="../../../equations/jacobi4.svg"></span>
 750
 751         </p></blockquote></div>
 752 <p>
 753         Otherwise the values are calculated using the method of <a href="http://dlmf.nist.gov/22.20#SS2" target="_top">arithmetic
 754         geometric means</a>.
 755       </p>
 756 </div>
 757 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 758 <td align="left"></td>
 759 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 760       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 761       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 762       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 763       Daryle Walker and Xiaogang Zhang<p>
 764         Distributed under the Boost Software License, Version 1.0. (See accompanying
 765         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>)
 766       </p>
 767 </div></td>
 768 </tr></table>
 769 <hr>
 770 <div class="spirit-nav">
 771 <a accesskey="p" href="jac_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../jacobi.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="jacobi_cd.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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