1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Jacobi Elliptic SN, CN and DN</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../jacobi.html" title="Jacobi Elliptic Functions"> 9<link rel="prev" href="jac_over.html" title="Overview of the Jacobi Elliptic Functions"> 10<link rel="next" href="jacobi_cd.html" title="Jacobi Elliptic Function cd"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td 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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"><</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">></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"><</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> 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"><</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">></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">&);</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>φ</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>φ</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 α, 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 α</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>α</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 \ α)</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>α</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 21. 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 21. 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 trigonometric 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 8.70. 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 210 2.1:</em></span> Max = 17.3ε (Mean = 4.29ε) <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 = 71.6ε (Mean = 19.3ε)</span> 217 </p> 218 </td> 219<td> 220 <p> 221 <span class="blue">Max = 71.6ε (Mean = 19.4ε)</span> 222 </p> 223 </td> 224<td> 225 <p> 226 <span class="blue">Max = 45.8ε (Mean = 11.4ε)</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 = 0.816ε (Mean = 0.0563ε)</span><br> 239 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.43ε (Mean = 0.803ε)) 240 </p> 241 </td> 242<td> 243 <p> 244 <span class="blue">Max = 1.68ε (Mean = 0.443ε)</span> 245 </p> 246 </td> 247<td> 248 <p> 249 <span class="blue">Max = 1.68ε (Mean = 0.454ε)</span> 250 </p> 251 </td> 252<td> 253 <p> 254 <span class="blue">Max = 1.83ε (Mean = 0.455ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 267 2.1:</em></span> Max = 55.2ε (Mean = 1.64ε) <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 = 10.4ε (Mean = 0.594ε)</span> 274 </p> 275 </td> 276<td> 277 <p> 278 <span class="blue">Max = 10.4ε (Mean = 0.602ε)</span> 279 </p> 280 </td> 281<td> 282 <p> 283 <span class="blue">Max = 26.2ε (Mean = 1.17ε)</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 = 0.919ε (Mean = 0.127ε)</span><br> <br> 296 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0ε (Mean = 0ε) <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 = 675ε (Mean = 87.1ε)</span> 303 </p> 304 </td> 305<td> 306 <p> 307 <span class="blue">Max = 675ε (Mean = 86.8ε)</span> 308 </p> 309 </td> 310<td> 311 <p> 312 <span class="blue">Max = 513ε (Mean = 126ε)</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 = 14.2ε (Mean = 0.927ε)</span><br> <br> 325 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 5.92e+03ε (Mean = 477ε)) 326 </p> 327 </td> 328<td> 329 <p> 330 <span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</span> 331 </p> 332 </td> 333<td> 334 <p> 335 <span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</span> 336 </p> 337 </td> 338<td> 339 <p> 340 <span class="blue">Max = 3.27e+04ε (Mean = 1.93e+03ε)</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 8.71. 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 391 2.1:</em></span> Max = 2.82ε (Mean = 1.18ε) <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 = 49ε (Mean = 14ε)</span> 398 </p> 399 </td> 400<td> 401 <p> 402 <span class="blue">Max = 49ε (Mean = 14ε)</span> 403 </p> 404 </td> 405<td> 406 <p> 407 <span class="blue">Max = 34.3ε (Mean = 8.71ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 420 2.1:</em></span> Max = 3ε (Mean = 0.61ε)) 421 </p> 422 </td> 423<td> 424 <p> 425 <span class="blue">Max = 1.53ε (Mean = 0.473ε)</span> 426 </p> 427 </td> 428<td> 429 <p> 430 <span class="blue">Max = 1.53ε (Mean = 0.481ε)</span> 431 </p> 432 </td> 433<td> 434 <p> 435 <span class="blue">Max = 1.52ε (Mean = 0.466ε)</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 = 0.5ε (Mean = 0.0122ε)</span><br> <br> 448 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5ε (Mean = 0.391ε) <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 = 22.4ε (Mean = 0.777ε)</span> 455 </p> 456 </td> 457<td> 458 <p> 459 <span class="blue">Max = 22.4ε (Mean = 0.763ε)</span> 460 </p> 461 </td> 462<td> 463 <p> 464 <span class="blue">Max = 16.1ε (Mean = 0.685ε)</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 = 2.28ε (Mean = 0.194ε)</span><br> <br> 477 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0ε (Mean = 0ε) <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+03ε (Mean = 293ε)</span> 484 </p> 485 </td> 486<td> 487 <p> 488 <span class="blue">Max = 3.75e+03ε (Mean = 293ε)</span> 489 </p> 490 </td> 491<td> 492 <p> 493 <span class="blue">Max = 6.24e+03ε (Mean = 482ε)</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 = 14.1ε (Mean = 0.897ε)</span><br> <br> 506 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 121ε (Mean = 22ε)) 507 </p> 508 </td> 509<td> 510 <p> 511 <span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</span> 512 </p> 513 </td> 514<td> 515 <p> 516 <span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</span> 517 </p> 518 </td> 519<td> 520 <p> 521 <span class="blue">Max = 1.67e+04ε (Mean = 1e+03ε)</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 8.72. 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 572 2.1:</em></span> Max = 588ε (Mean = 146ε) <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 = 341ε (Mean = 80.7ε)</span> 579 </p> 580 </td> 581<td> 582 <p> 583 <span class="blue">Max = 341ε (Mean = 80.7ε)</span> 584 </p> 585 </td> 586<td> 587 <p> 588 <span class="blue">Max = 481ε (Mean = 113ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 601 2.1:</em></span> Max = 4.02ε (Mean = 1.07ε)) 602 </p> 603 </td> 604<td> 605 <p> 606 <span class="blue">Max = 2.01ε (Mean = 0.584ε)</span> 607 </p> 608 </td> 609<td> 610 <p> 611 <span class="blue">Max = 2.01ε (Mean = 0.593ε)</span> 612 </p> 613 </td> 614<td> 615 <p> 616 <span class="blue">Max = 1.92ε (Mean = 0.567ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 629 2.1:</em></span> Max = 11.7ε (Mean = 1.65ε) <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 = 1.99ε (Mean = 0.347ε)</span> 636 </p> 637 </td> 638<td> 639 <p> 640 <span class="blue">Max = 1.99ε (Mean = 0.347ε)</span> 641 </p> 642 </td> 643<td> 644 <p> 645 <span class="blue">Max = 2.11ε (Mean = 0.385ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 658 2.1:</em></span> Max = 0ε (Mean = 0ε) <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 = 109ε (Mean = 7.35ε)</span> 665 </p> 666 </td> 667<td> 668 <p> 669 <span class="blue">Max = 109ε (Mean = 7.38ε)</span> 670 </p> 671 </td> 672<td> 673 <p> 674 <span class="blue">Max = 23.2ε (Mean = 1.85ε)</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 = 12ε (Mean = 0.771ε)</span><br> <br> 687 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.54e+04ε (Mean = 2.63e+03ε)) 688 </p> 689 </td> 690<td> 691 <p> 692 <span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</span> 693 </p> 694 </td> 695<td> 696 <p> 697 <span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</span> 698 </p> 699 </td> 700<td> 701 <p> 702 <span class="blue">Max = 4.36e+04ε (Mean = 2.54e+03ε)</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 > 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> < ε</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 © 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 Råde, 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> 772</div> 773</body> 774</html> 775
