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24 | </div> | |
25 | <div class="section"> | |
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"><</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, <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 φ 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>φ</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 α, or a parameter | |
94 | <span class="emphasis"><em>m</em></span>. These are related by: | |
95 | </p> | |
96 | <p> | |
97 | k = sinα | |
98 | </p> | |
99 | <p> | |
100 | m = k<sup>2</sup> = sin<sup>2</sup>α | |
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 \ α) | |
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>α | |
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 15. 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 15. 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 6.70. 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 = 45.8ε (Mean = 11.4ε)</span> | |
208 | </p> | |
209 | </td> | |
210 | <td> | |
211 | <p> | |
212 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
213 | 1.16:</em></span> Max = 17.3ε (Mean = 4.29ε) <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 = +INFε (Mean = +INFε) <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 = 71.6ε (Mean = 19.3ε)</span> | |
221 | </p> | |
222 | </td> | |
223 | <td> | |
224 | <p> | |
225 | <span class="blue">Max = 71.6ε (Mean = 19.4ε)</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 = 1.83ε (Mean = 0.455ε)</span> | |
238 | </p> | |
239 | </td> | |
240 | <td> | |
241 | <p> | |
242 | <span class="blue">Max = 0.816ε (Mean = 0.0563ε)</span><br> | |
243 | <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.43ε (Mean = 0.803ε))<br> | |
244 | (<span class="emphasis"><em>Cephes:</em></span> Max = 1.83ε (Mean = 0.437ε)) | |
245 | </p> | |
246 | </td> | |
247 | <td> | |
248 | <p> | |
249 | <span class="blue">Max = 1.68ε (Mean = 0.443ε)</span> | |
250 | </p> | |
251 | </td> | |
252 | <td> | |
253 | <p> | |
254 | <span class="blue">Max = 1.68ε (Mean = 0.454ε)</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 = 26.2ε (Mean = 1.17ε)</span> | |
267 | </p> | |
268 | </td> | |
269 | <td> | |
270 | <p> | |
271 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
272 | 1.16:</em></span> Max = 55.2ε (Mean = 1.64ε) <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 = +INFε (Mean = +INFε) <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 = 10.4ε (Mean = 0.594ε)</span> | |
280 | </p> | |
281 | </td> | |
282 | <td> | |
283 | <p> | |
284 | <span class="blue">Max = 10.4ε (Mean = 0.602ε)</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 = 513ε (Mean = 126ε)</span> | |
297 | </p> | |
298 | </td> | |
299 | <td> | |
300 | <p> | |
301 | <span class="blue">Max = 0.919ε (Mean = 0.127ε)</span><br> <br> | |
302 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0ε (Mean = 0ε) <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 = +INFε (Mean = +INFε) <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 = 675ε (Mean = 87.1ε)</span> | |
310 | </p> | |
311 | </td> | |
312 | <td> | |
313 | <p> | |
314 | <span class="blue">Max = 675ε (Mean = 86.8ε)</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+004ε (Mean = 1.93e+003ε)</span> | |
327 | </p> | |
328 | </td> | |
329 | <td> | |
330 | <p> | |
331 | <span class="blue">Max = 14.2ε (Mean = 0.927ε)</span><br> <br> | |
332 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 5.92e+03ε (Mean = 477ε))<br> | |
333 | (<span class="emphasis"><em>Cephes:</em></span> Max = 3.27e+04ε (Mean = 1.93e+03ε)) | |
334 | </p> | |
335 | </td> | |
336 | <td> | |
337 | <p> | |
338 | <span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</span> | |
339 | </p> | |
340 | </td> | |
341 | <td> | |
342 | <p> | |
343 | <span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</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 6.71. 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 = 34.3ε (Mean = 8.71ε)</span> | |
394 | </p> | |
395 | </td> | |
396 | <td> | |
397 | <p> | |
398 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
399 | 1.16:</em></span> Max = 2.82ε (Mean = 1.18ε) <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 = +INFε (Mean = +INFε) <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 = 49ε (Mean = 14ε)</span> | |
407 | </p> | |
408 | </td> | |
409 | <td> | |
410 | <p> | |
411 | <span class="blue">Max = 49ε (Mean = 14ε)</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 = 1.52ε (Mean = 0.466ε)</span> | |
424 | </p> | |
425 | </td> | |
426 | <td> | |
427 | <p> | |
428 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
429 | 1.16:</em></span> Max = 3ε (Mean = 0.61ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
430 | Max = 1.49ε (Mean = 0.321ε)) | |
431 | </p> | |
432 | </td> | |
433 | <td> | |
434 | <p> | |
435 | <span class="blue">Max = 1.53ε (Mean = 0.473ε)</span> | |
436 | </p> | |
437 | </td> | |
438 | <td> | |
439 | <p> | |
440 | <span class="blue">Max = 1.53ε (Mean = 0.481ε)</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 = 16.1ε (Mean = 0.685ε)</span> | |
453 | </p> | |
454 | </td> | |
455 | <td> | |
456 | <p> | |
457 | <span class="blue">Max = 0.5ε (Mean = 0.0122ε)</span><br> <br> | |
458 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.5ε (Mean = 0.391ε) <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 = +INFε (Mean = +INFε) <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 = 22.4ε (Mean = 0.777ε)</span> | |
466 | </p> | |
467 | </td> | |
468 | <td> | |
469 | <p> | |
470 | <span class="blue">Max = 22.4ε (Mean = 0.763ε)</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+003ε (Mean = 482ε)</span> | |
483 | </p> | |
484 | </td> | |
485 | <td> | |
486 | <p> | |
487 | <span class="blue">Max = 2.28ε (Mean = 0.194ε)</span><br> <br> | |
488 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0ε (Mean = 0ε) <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 = +INFε (Mean = +INFε) <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+03ε (Mean = 293ε)</span> | |
496 | </p> | |
497 | </td> | |
498 | <td> | |
499 | <p> | |
500 | <span class="blue">Max = 3.75e+03ε (Mean = 293ε)</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+004ε (Mean = 1e+003ε)</span> | |
513 | </p> | |
514 | </td> | |
515 | <td> | |
516 | <p> | |
517 | <span class="blue">Max = 14.1ε (Mean = 0.897ε)</span><br> <br> | |
518 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 121ε (Mean = 22ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
519 | Max = 196ε (Mean = 30.2ε)) | |
520 | </p> | |
521 | </td> | |
522 | <td> | |
523 | <p> | |
524 | <span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</span> | |
525 | </p> | |
526 | </td> | |
527 | <td> | |
528 | <p> | |
529 | <span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</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 6.72. 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 = 481ε (Mean = 113ε)</span> | |
580 | </p> | |
581 | </td> | |
582 | <td> | |
583 | <p> | |
584 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
585 | 1.16:</em></span> Max = 588ε (Mean = 146ε) <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 = +INFε (Mean = +INFε) <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 = 341ε (Mean = 80.7ε)</span> | |
593 | </p> | |
594 | </td> | |
595 | <td> | |
596 | <p> | |
597 | <span class="blue">Max = 341ε (Mean = 80.7ε)</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 = 1.92ε (Mean = 0.567ε)</span> | |
610 | </p> | |
611 | </td> | |
612 | <td> | |
613 | <p> | |
614 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
615 | 1.16:</em></span> Max = 4.02ε (Mean = 1.07ε))<br> (<span class="emphasis"><em>Cephes:</em></span> | |
616 | Max = 1.92ε (Mean = 0.567ε)) | |
617 | </p> | |
618 | </td> | |
619 | <td> | |
620 | <p> | |
621 | <span class="blue">Max = 2.01ε (Mean = 0.584ε)</span> | |
622 | </p> | |
623 | </td> | |
624 | <td> | |
625 | <p> | |
626 | <span class="blue">Max = 2.01ε (Mean = 0.593ε)</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 = 2.11ε (Mean = 0.385ε)</span> | |
639 | </p> | |
640 | </td> | |
641 | <td> | |
642 | <p> | |
643 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
644 | 1.16:</em></span> Max = 11.7ε (Mean = 1.65ε) <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 = +INFε (Mean = +INFε) <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 = 1.99ε (Mean = 0.347ε)</span> | |
652 | </p> | |
653 | </td> | |
654 | <td> | |
655 | <p> | |
656 | <span class="blue">Max = 1.99ε (Mean = 0.347ε)</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 = 23.2ε (Mean = 1.85ε)</span> | |
669 | </p> | |
670 | </td> | |
671 | <td> | |
672 | <p> | |
673 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
674 | 1.16:</em></span> Max = 0ε (Mean = 0ε) <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 = +INFε (Mean = +INFε) <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 = 109ε (Mean = 7.35ε)</span> | |
682 | </p> | |
683 | </td> | |
684 | <td> | |
685 | <p> | |
686 | <span class="blue">Max = 109ε (Mean = 7.38ε)</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+004ε (Mean = 2.54e+003ε)</span> | |
699 | </p> | |
700 | </td> | |
701 | <td> | |
702 | <p> | |
703 | <span class="blue">Max = 12ε (Mean = 0.771ε)</span><br> <br> | |
704 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 4.54e+04ε (Mean = 2.63e+03ε))<br> | |
705 | (<span class="emphasis"><em>Cephes:</em></span> Max = 4.36e+04ε (Mean = 2.54e+03ε)) | |
706 | </p> | |
707 | </td> | |
708 | <td> | |
709 | <p> | |
710 | <span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</span> | |
711 | </p> | |
712 | </td> | |
713 | <td> | |
714 | <p> | |
715 | <span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</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 > 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> < ε</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 © 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 Råde, 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> | |
777 | <div class="spirit-nav"> | |
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