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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, <span class="emphasis"><em>*pcn</em></span>
  55         is set to <span class="emphasis"><em>cn(u, k)</em></span>, and <span class="emphasis"><em>*pdn</em></span> is
  56         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 <p>
  62         <span class="inlinemediaobject"><img src="../../../equations/jacobi1.svg"></span>
  63       </p>
  64 <p>
  65         The the angle &#966; is called the <span class="emphasis"><em>amplitude</em></span> and:
  66       </p>
  67 <p>
  68         <span class="inlinemediaobject"><img src="../../../equations/jacobi2.svg"></span>
  69       </p>
  70 <div class="note"><table border="0" summary="Note">
  71 <tr>
  72 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
  73 <th align="left">Note</th>
  74 </tr>
  75 <tr><td align="left" valign="top">
  76 <p>
  77           <span class="emphasis"><em>&#966;</em></span> is called the amplitude.
  78         </p>
  79 <p>
  80           <span class="emphasis"><em>k</em></span> is called the modulus.
  81         </p>
  82 </td></tr>
  83 </table></div>
  84 <div class="caution"><table border="0" summary="Caution">
  85 <tr>
  86 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
  87 <th align="left">Caution</th>
  88 </tr>
  89 <tr><td align="left" valign="top">
  90 <p>
  91           Rather like other elliptic functions, the Jacobi functions are expressed
  92           in a variety of different ways. In particular, the parameter <span class="emphasis"><em>k</em></span>
  93           (the modulus) may also be expressed using a modular angle &#945;, or a parameter
  94           <span class="emphasis"><em>m</em></span>. These are related by:
  95         </p>
  96 <p>
  97           k = sin&#945;
  98         </p>
  99 <p>
 100           m = k<sup>2</sup> = sin<sup>2</sup>&#945;
 101         </p>
 102 <p>
 103           So that the function <span class="emphasis"><em>sn</em></span> (for example) may be expressed
 104           as either:
 105         </p>
 106 <p>
 107           sn(u, k)
 108         </p>
 109 <p>
 110           sn(u \ &#945;)
 111         </p>
 112 <p>
 113           sn(u| m)
 114         </p>
 115 <p>
 116           To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
 117           of the parameter m</em></span>, or 1 - m, where:
 118         </p>
 119 <p>
 120           1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>&#945;
 121         </p>
 122 <p>
 123           This implementation uses <span class="emphasis"><em>k</em></span> throughout, and makes this
 124           the first argument to the functions: this is for alignment with the elliptic
 125           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
 126           Report on C++ Library Extensions</a>. However, you should be extra
 127           careful when using these functions!
 128         </p>
 129 </td></tr>
 130 </table></div>
 131 <p>
 132         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
 133         be used to control the behaviour of the function: how it handles errors,
 134         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
 135         documentation for more details</a>.
 136       </p>
 137 <p>
 138         The following graphs illustrate how these functions change as <span class="emphasis"><em>k</em></span>
 139         changes: for small <span class="emphasis"><em>k</em></span> these are sine waves, while as
 140         <span class="emphasis"><em>k</em></span> tends to 1 they become hyperbolic functions:
 141       </p>
 142 <p>
 143         <span class="inlinemediaobject"><img src="../../../graphs/jacobi_sn.svg" align="middle"></span>
 144       </p>
 145 <p>
 146         <span class="inlinemediaobject"><img src="../../../graphs/jacobi_cn.svg" align="middle"></span>
 147       </p>
 148 <p>
 149         <span class="inlinemediaobject"><img src="../../../graphs/jacobi_dn.svg" align="middle"></span>
 150       </p>
 151 <h5>
 152 <a name="math_toolkit.jacobi.jacobi_elliptic.h2"></a>
 153         <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>
 154       </h5>
 155 <p>
 156         These functions are computed using only basic arithmetic operations and trigomometric
 157         functions, so there isn't much variation in accuracy over differing platforms.
 158         Typically errors are trivially small for small angles, and as is typical
 159         for cyclic functions, grow as the angle increases. Note that only results
 160         for the widest floating point type on the system are given as narrower types
 161         have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
 162         error</a>. All values are relative errors in units of epsilon.
 163       </p>
 164 <div class="table">
 165 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_cn"></a><p class="title"><b>Table&#160;6.70.&#160;Error rates for jacobi_cn</b></p>
 166 <div class="table-contents"><table class="table" summary="Error rates for jacobi_cn">
 167 <colgroup>
 168 <col>
 169 <col>
 170 <col>
 171 <col>
 172 <col>
 173 </colgroup>
 174 <thead><tr>
 175 <th>
 176               </th>
 177 <th>
 178                 <p>
 179                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 180                 </p>
 181               </th>
 182 <th>
 183                 <p>
 184                   GNU C++ version 5.1.0<br> linux<br> double
 185                 </p>
 186               </th>
 187 <th>
 188                 <p>
 189                   GNU C++ version 5.1.0<br> linux<br> long double
 190                 </p>
 191               </th>
 192 <th>
 193                 <p>
 194                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 195                 </p>
 196               </th>
 197 </tr></thead>
 198 <tbody>
 199 <tr>
 200 <td>
 201                 <p>
 202                   Jacobi Elliptic: Mathworld Data
 203                 </p>
 204               </td>
 205 <td>
 206                 <p>
 207                   <span class="blue">Max = 45.8&#949; (Mean = 11.4&#949;)</span>
 208                 </p>
 209               </td>
 210 <td>
 211                 <p>
 212                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 213                   1.16:</em></span> Max = 17.3&#949; (Mean = 4.29&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_GSL_1_16_Jacobi_Elliptic_Mathworld_Data">And
 214                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_Cephes_Jacobi_Elliptic_Mathworld_Data">And
 215                   other failures.</a>)</span>
 216                 </p>
 217               </td>
 218 <td>
 219                 <p>
 220                   <span class="blue">Max = 71.6&#949; (Mean = 19.3&#949;)</span>
 221                 </p>
 222               </td>
 223 <td>
 224                 <p>
 225                   <span class="blue">Max = 71.6&#949; (Mean = 19.4&#949;)</span>
 226                 </p>
 227               </td>
 228 </tr>
 229 <tr>
 230 <td>
 231                 <p>
 232                   Jacobi Elliptic: Random Data
 233                 </p>
 234               </td>
 235 <td>
 236                 <p>
 237                   <span class="blue">Max = 1.83&#949; (Mean = 0.455&#949;)</span>
 238                 </p>
 239               </td>
 240 <td>
 241                 <p>
 242                   <span class="blue">Max = 0.816&#949; (Mean = 0.0563&#949;)</span><br>
 243                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.43&#949; (Mean = 0.803&#949;))<br>
 244                   (<span class="emphasis"><em>Cephes:</em></span> Max = 1.83&#949; (Mean = 0.437&#949;))
 245                 </p>
 246               </td>
 247 <td>
 248                 <p>
 249                   <span class="blue">Max = 1.68&#949; (Mean = 0.443&#949;)</span>
 250                 </p>
 251               </td>
 252 <td>
 253                 <p>
 254                   <span class="blue">Max = 1.68&#949; (Mean = 0.454&#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 = 26.2&#949; (Mean = 1.17&#949;)</span>
 267                 </p>
 268               </td>
 269 <td>
 270                 <p>
 271                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 272                   1.16:</em></span> Max = 55.2&#949; (Mean = 1.64&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_GSL_1_16_Jacobi_Elliptic_Random_Small_Values">And
 273                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_Cephes_Jacobi_Elliptic_Random_Small_Values">And
 274                   other failures.</a>)</span>
 275                 </p>
 276               </td>
 277 <td>
 278                 <p>
 279                   <span class="blue">Max = 10.4&#949; (Mean = 0.594&#949;)</span>
 280                 </p>
 281               </td>
 282 <td>
 283                 <p>
 284                   <span class="blue">Max = 10.4&#949; (Mean = 0.602&#949;)</span>
 285                 </p>
 286               </td>
 287 </tr>
 288 <tr>
 289 <td>
 290                 <p>
 291                   Jacobi Elliptic: Modulus near 1
 292                 </p>
 293               </td>
 294 <td>
 295                 <p>
 296                   <span class="blue">Max = 513&#949; (Mean = 126&#949;)</span>
 297                 </p>
 298               </td>
 299 <td>
 300                 <p>
 301                   <span class="blue">Max = 0.919&#949; (Mean = 0.127&#949;)</span><br> <br>
 302                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_GSL_1_16_Jacobi_Elliptic_Modulus_near_1">And
 303                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_cn_Cephes_Jacobi_Elliptic_Modulus_near_1">And
 304                   other failures.</a>)</span>
 305                 </p>
 306               </td>
 307 <td>
 308                 <p>
 309                   <span class="blue">Max = 675&#949; (Mean = 87.1&#949;)</span>
 310                 </p>
 311               </td>
 312 <td>
 313                 <p>
 314                   <span class="blue">Max = 675&#949; (Mean = 86.8&#949;)</span>
 315                 </p>
 316               </td>
 317 </tr>
 318 <tr>
 319 <td>
 320                 <p>
 321                   Jacobi Elliptic: Large Phi
 322                 </p>
 323               </td>
 324 <td>
 325                 <p>
 326                   <span class="blue">Max = 3.27e+004&#949; (Mean = 1.93e+003&#949;)</span>
 327                 </p>
 328               </td>
 329 <td>
 330                 <p>
 331                   <span class="blue">Max = 14.2&#949; (Mean = 0.927&#949;)</span><br> <br>
 332                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 5.92e+03&#949; (Mean = 477&#949;))<br>
 333                   (<span class="emphasis"><em>Cephes:</em></span> Max = 3.27e+04&#949; (Mean = 1.93e+03&#949;))
 334                 </p>
 335               </td>
 336 <td>
 337                 <p>
 338                   <span class="blue">Max = 2.97e+04&#949; (Mean = 1.9e+03&#949;)</span>
 339                 </p>
 340               </td>
 341 <td>
 342                 <p>
 343                   <span class="blue">Max = 2.97e+04&#949; (Mean = 1.9e+03&#949;)</span>
 344                 </p>
 345               </td>
 346 </tr>
 347 </tbody>
 348 </table></div>
 349 </div>
 350 <br class="table-break"><div class="table">
 351 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_dn"></a><p class="title"><b>Table&#160;6.71.&#160;Error rates for jacobi_dn</b></p>
 352 <div class="table-contents"><table class="table" summary="Error rates for jacobi_dn">
 353 <colgroup>
 354 <col>
 355 <col>
 356 <col>
 357 <col>
 358 <col>
 359 </colgroup>
 360 <thead><tr>
 361 <th>
 362               </th>
 363 <th>
 364                 <p>
 365                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 366                 </p>
 367               </th>
 368 <th>
 369                 <p>
 370                   GNU C++ version 5.1.0<br> linux<br> double
 371                 </p>
 372               </th>
 373 <th>
 374                 <p>
 375                   GNU C++ version 5.1.0<br> linux<br> long double
 376                 </p>
 377               </th>
 378 <th>
 379                 <p>
 380                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 381                 </p>
 382               </th>
 383 </tr></thead>
 384 <tbody>
 385 <tr>
 386 <td>
 387                 <p>
 388                   Jacobi Elliptic: Mathworld Data
 389                 </p>
 390               </td>
 391 <td>
 392                 <p>
 393                   <span class="blue">Max = 34.3&#949; (Mean = 8.71&#949;)</span>
 394                 </p>
 395               </td>
 396 <td>
 397                 <p>
 398                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 399                   1.16:</em></span> Max = 2.82&#949; (Mean = 1.18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_GSL_1_16_Jacobi_Elliptic_Mathworld_Data">And
 400                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_Cephes_Jacobi_Elliptic_Mathworld_Data">And
 401                   other failures.</a>)</span>
 402                 </p>
 403               </td>
 404 <td>
 405                 <p>
 406                   <span class="blue">Max = 49&#949; (Mean = 14&#949;)</span>
 407                 </p>
 408               </td>
 409 <td>
 410                 <p>
 411                   <span class="blue">Max = 49&#949; (Mean = 14&#949;)</span>
 412                 </p>
 413               </td>
 414 </tr>
 415 <tr>
 416 <td>
 417                 <p>
 418                   Jacobi Elliptic: Random Data
 419                 </p>
 420               </td>
 421 <td>
 422                 <p>
 423                   <span class="blue">Max = 1.52&#949; (Mean = 0.466&#949;)</span>
 424                 </p>
 425               </td>
 426 <td>
 427                 <p>
 428                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 429                   1.16:</em></span> Max = 3&#949; (Mean = 0.61&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 430                   Max = 1.49&#949; (Mean = 0.321&#949;))
 431                 </p>
 432               </td>
 433 <td>
 434                 <p>
 435                   <span class="blue">Max = 1.53&#949; (Mean = 0.473&#949;)</span>
 436                 </p>
 437               </td>
 438 <td>
 439                 <p>
 440                   <span class="blue">Max = 1.53&#949; (Mean = 0.481&#949;)</span>
 441                 </p>
 442               </td>
 443 </tr>
 444 <tr>
 445 <td>
 446                 <p>
 447                   Jacobi Elliptic: Random Small Values
 448                 </p>
 449               </td>
 450 <td>
 451                 <p>
 452                   <span class="blue">Max = 16.1&#949; (Mean = 0.685&#949;)</span>
 453                 </p>
 454               </td>
 455 <td>
 456                 <p>
 457                   <span class="blue">Max = 0.5&#949; (Mean = 0.0122&#949;)</span><br> <br>
 458                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.5&#949; (Mean = 0.391&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_GSL_1_16_Jacobi_Elliptic_Random_Small_Values">And
 459                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_Cephes_Jacobi_Elliptic_Random_Small_Values">And
 460                   other failures.</a>)</span>
 461                 </p>
 462               </td>
 463 <td>
 464                 <p>
 465                   <span class="blue">Max = 22.4&#949; (Mean = 0.777&#949;)</span>
 466                 </p>
 467               </td>
 468 <td>
 469                 <p>
 470                   <span class="blue">Max = 22.4&#949; (Mean = 0.763&#949;)</span>
 471                 </p>
 472               </td>
 473 </tr>
 474 <tr>
 475 <td>
 476                 <p>
 477                   Jacobi Elliptic: Modulus near 1
 478                 </p>
 479               </td>
 480 <td>
 481                 <p>
 482                   <span class="blue">Max = 6.24e+003&#949; (Mean = 482&#949;)</span>
 483                 </p>
 484               </td>
 485 <td>
 486                 <p>
 487                   <span class="blue">Max = 2.28&#949; (Mean = 0.194&#949;)</span><br> <br>
 488                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_GSL_1_16_Jacobi_Elliptic_Modulus_near_1">And
 489                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_dn_Cephes_Jacobi_Elliptic_Modulus_near_1">And
 490                   other failures.</a>)</span>
 491                 </p>
 492               </td>
 493 <td>
 494                 <p>
 495                   <span class="blue">Max = 3.75e+03&#949; (Mean = 293&#949;)</span>
 496                 </p>
 497               </td>
 498 <td>
 499                 <p>
 500                   <span class="blue">Max = 3.75e+03&#949; (Mean = 293&#949;)</span>
 501                 </p>
 502               </td>
 503 </tr>
 504 <tr>
 505 <td>
 506                 <p>
 507                   Jacobi Elliptic: Large Phi
 508                 </p>
 509               </td>
 510 <td>
 511                 <p>
 512                   <span class="blue">Max = 1.67e+004&#949; (Mean = 1e+003&#949;)</span>
 513                 </p>
 514               </td>
 515 <td>
 516                 <p>
 517                   <span class="blue">Max = 14.1&#949; (Mean = 0.897&#949;)</span><br> <br>
 518                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 121&#949; (Mean = 22&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 519                   Max = 196&#949; (Mean = 30.2&#949;))
 520                 </p>
 521               </td>
 522 <td>
 523                 <p>
 524                   <span class="blue">Max = 2.82e+04&#949; (Mean = 1.79e+03&#949;)</span>
 525                 </p>
 526               </td>
 527 <td>
 528                 <p>
 529                   <span class="blue">Max = 2.82e+04&#949; (Mean = 1.79e+03&#949;)</span>
 530                 </p>
 531               </td>
 532 </tr>
 533 </tbody>
 534 </table></div>
 535 </div>
 536 <br class="table-break"><div class="table">
 537 <a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_sn"></a><p class="title"><b>Table&#160;6.72.&#160;Error rates for jacobi_sn</b></p>
 538 <div class="table-contents"><table class="table" summary="Error rates for jacobi_sn">
 539 <colgroup>
 540 <col>
 541 <col>
 542 <col>
 543 <col>
 544 <col>
 545 </colgroup>
 546 <thead><tr>
 547 <th>
 548               </th>
 549 <th>
 550                 <p>
 551                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 552                 </p>
 553               </th>
 554 <th>
 555                 <p>
 556                   GNU C++ version 5.1.0<br> linux<br> double
 557                 </p>
 558               </th>
 559 <th>
 560                 <p>
 561                   GNU C++ version 5.1.0<br> linux<br> long double
 562                 </p>
 563               </th>
 564 <th>
 565                 <p>
 566                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 567                 </p>
 568               </th>
 569 </tr></thead>
 570 <tbody>
 571 <tr>
 572 <td>
 573                 <p>
 574                   Jacobi Elliptic: Mathworld Data
 575                 </p>
 576               </td>
 577 <td>
 578                 <p>
 579                   <span class="blue">Max = 481&#949; (Mean = 113&#949;)</span>
 580                 </p>
 581               </td>
 582 <td>
 583                 <p>
 584                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 585                   1.16:</em></span> Max = 588&#949; (Mean = 146&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_GSL_1_16_Jacobi_Elliptic_Mathworld_Data">And
 586                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_Cephes_Jacobi_Elliptic_Mathworld_Data">And
 587                   other failures.</a>)</span>
 588                 </p>
 589               </td>
 590 <td>
 591                 <p>
 592                   <span class="blue">Max = 341&#949; (Mean = 80.7&#949;)</span>
 593                 </p>
 594               </td>
 595 <td>
 596                 <p>
 597                   <span class="blue">Max = 341&#949; (Mean = 80.7&#949;)</span>
 598                 </p>
 599               </td>
 600 </tr>
 601 <tr>
 602 <td>
 603                 <p>
 604                   Jacobi Elliptic: Random Data
 605                 </p>
 606               </td>
 607 <td>
 608                 <p>
 609                   <span class="blue">Max = 1.92&#949; (Mean = 0.567&#949;)</span>
 610                 </p>
 611               </td>
 612 <td>
 613                 <p>
 614                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 615                   1.16:</em></span> Max = 4.02&#949; (Mean = 1.07&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 616                   Max = 1.92&#949; (Mean = 0.567&#949;))
 617                 </p>
 618               </td>
 619 <td>
 620                 <p>
 621                   <span class="blue">Max = 2.01&#949; (Mean = 0.584&#949;)</span>
 622                 </p>
 623               </td>
 624 <td>
 625                 <p>
 626                   <span class="blue">Max = 2.01&#949; (Mean = 0.593&#949;)</span>
 627                 </p>
 628               </td>
 629 </tr>
 630 <tr>
 631 <td>
 632                 <p>
 633                   Jacobi Elliptic: Random Small Values
 634                 </p>
 635               </td>
 636 <td>
 637                 <p>
 638                   <span class="blue">Max = 2.11&#949; (Mean = 0.385&#949;)</span>
 639                 </p>
 640               </td>
 641 <td>
 642                 <p>
 643                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 644                   1.16:</em></span> Max = 11.7&#949; (Mean = 1.65&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_GSL_1_16_Jacobi_Elliptic_Random_Small_Values">And
 645                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_Cephes_Jacobi_Elliptic_Random_Small_Values">And
 646                   other failures.</a>)</span>
 647                 </p>
 648               </td>
 649 <td>
 650                 <p>
 651                   <span class="blue">Max = 1.99&#949; (Mean = 0.347&#949;)</span>
 652                 </p>
 653               </td>
 654 <td>
 655                 <p>
 656                   <span class="blue">Max = 1.99&#949; (Mean = 0.347&#949;)</span>
 657                 </p>
 658               </td>
 659 </tr>
 660 <tr>
 661 <td>
 662                 <p>
 663                   Jacobi Elliptic: Modulus near 1
 664                 </p>
 665               </td>
 666 <td>
 667                 <p>
 668                   <span class="blue">Max = 23.2&#949; (Mean = 1.85&#949;)</span>
 669                 </p>
 670               </td>
 671 <td>
 672                 <p>
 673                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 674                   1.16:</em></span> Max = 0&#949; (Mean = 0&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_GSL_1_16_Jacobi_Elliptic_Modulus_near_1">And
 675                   other failures.</a>)<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_jacobi_sn_Cephes_Jacobi_Elliptic_Modulus_near_1">And
 676                   other failures.</a>)</span>
 677                 </p>
 678               </td>
 679 <td>
 680                 <p>
 681                   <span class="blue">Max = 109&#949; (Mean = 7.35&#949;)</span>
 682                 </p>
 683               </td>
 684 <td>
 685                 <p>
 686                   <span class="blue">Max = 109&#949; (Mean = 7.38&#949;)</span>
 687                 </p>
 688               </td>
 689 </tr>
 690 <tr>
 691 <td>
 692                 <p>
 693                   Jacobi Elliptic: Large Phi
 694                 </p>
 695               </td>
 696 <td>
 697                 <p>
 698                   <span class="blue">Max = 4.36e+004&#949; (Mean = 2.54e+003&#949;)</span>
 699                 </p>
 700               </td>
 701 <td>
 702                 <p>
 703                   <span class="blue">Max = 12&#949; (Mean = 0.771&#949;)</span><br> <br>
 704                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 4.54e+04&#949; (Mean = 2.63e+03&#949;))<br>
 705                   (<span class="emphasis"><em>Cephes:</em></span> Max = 4.36e+04&#949; (Mean = 2.54e+03&#949;))
 706                 </p>
 707               </td>
 708 <td>
 709                 <p>
 710                   <span class="blue">Max = 2.45e+04&#949; (Mean = 1.51e+03&#949;)</span>
 711                 </p>
 712               </td>
 713 <td>
 714                 <p>
 715                   <span class="blue">Max = 2.45e+04&#949; (Mean = 1.51e+03&#949;)</span>
 716                 </p>
 717               </td>
 718 </tr>
 719 </tbody>
 720 </table></div>
 721 </div>
 722 <br class="table-break"><h5>
 723 <a name="math_toolkit.jacobi.jacobi_elliptic.h3"></a>
 724         <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>
 725       </h5>
 726 <p>
 727         The tests use a mixture of spot test values calculated using the online calculator
 728         at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
 729         and random test data generated using MPFR at 1000-bit precision and this
 730         implementation.
 731       </p>
 732 <h5>
 733 <a name="math_toolkit.jacobi.jacobi_elliptic.h4"></a>
 734         <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>
 735       </h5>
 736 <p>
 737         For <span class="emphasis"><em>k &gt; 1</em></span> we apply the relations:
 738       </p>
 739 <p>
 740         <span class="inlinemediaobject"><img src="../../../equations/jacobi3.svg"></span>
 741       </p>
 742 <p>
 743         Then filter off the special cases:
 744       </p>
 745 <p>
 746         <span class="emphasis"><em>sn(0, k) = 0</em></span> and <span class="emphasis"><em>cn(0, k) = dn(0, k) = 1</em></span>.
 747       </p>
 748 <p>
 749         <span class="emphasis"><em>sn(u, 0) = sin(u), cn(u, 0) = cos(u) and dn(u, 0) = 1</em></span>.
 750       </p>
 751 <p>
 752         <span class="emphasis"><em>sn(u, 1) = tanh(u), cn(u, 1) = dn(u, 1) = 1 / cosh(u)</em></span>.
 753       </p>
 754 <p>
 755         And for <span class="emphasis"><em>k<sup>4</sup> &lt; &#949;</em></span> we have:
 756       </p>
 757 <p>
 758         <span class="inlinemediaobject"><img src="../../../equations/jacobi4.svg"></span>
 759       </p>
 760 <p>
 761         Otherwise the values are calculated using the method of <a href="http://dlmf.nist.gov/22.20#SS2" target="_top">arithmetic
 762         geometric means</a>.
 763       </p>
 764 </div>
 765 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 766 <td align="left"></td>
 767 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 768       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 769       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 770       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 771         Distributed under the Boost Software License, Version 1.0. (See accompanying
 772         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>)
 773       </p>
 774 </div></td>
 775 </tr></table>
 776 <hr>
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