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1 | // Special functions -*- C++ -*- |
2 | ||
7adcbafe | 3 | // Copyright (C) 2006-2022 Free Software Foundation, Inc. |
7c62b943 BK |
4 | // |
5 | // This file is part of the GNU ISO C++ Library. This library is free | |
6 | // software; you can redistribute it and/or modify it under the | |
7 | // terms of the GNU General Public License as published by the | |
748086b7 | 8 | // Free Software Foundation; either version 3, or (at your option) |
7c62b943 BK |
9 | // any later version. |
10 | // | |
11 | // This library is distributed in the hope that it will be useful, | |
12 | // but WITHOUT ANY WARRANTY; without even the implied warranty of | |
13 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
14 | // GNU General Public License for more details. | |
15 | // | |
748086b7 JJ |
16 | // Under Section 7 of GPL version 3, you are granted additional |
17 | // permissions described in the GCC Runtime Library Exception, version | |
18 | // 3.1, as published by the Free Software Foundation. | |
19 | ||
20 | // You should have received a copy of the GNU General Public License and | |
21 | // a copy of the GCC Runtime Library Exception along with this program; | |
22 | // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see | |
23 | // <http://www.gnu.org/licenses/>. | |
7c62b943 BK |
24 | |
25 | /** @file tr1/ell_integral.tcc | |
26 | * This is an internal header file, included by other library headers. | |
f910786b | 27 | * Do not attempt to use it directly. @headername{tr1/cmath} |
7c62b943 BK |
28 | */ |
29 | ||
30 | // | |
31 | // ISO C++ 14882 TR1: 5.2 Special functions | |
32 | // | |
33 | ||
34 | // Written by Edward Smith-Rowland based on: | |
35 | // (1) B. C. Carlson Numer. Math. 33, 1 (1979) | |
36 | // (2) B. C. Carlson, Special Functions of Applied Mathematics (1977) | |
37 | // (3) The Gnu Scientific Library, http://www.gnu.org/software/gsl | |
38 | // (4) Numerical Recipes in C, 2nd ed, by W. H. Press, S. A. Teukolsky, | |
39 | // W. T. Vetterling, B. P. Flannery, Cambridge University Press | |
40 | // (1992), pp. 261-269 | |
41 | ||
e133ace8 PC |
42 | #ifndef _GLIBCXX_TR1_ELL_INTEGRAL_TCC |
43 | #define _GLIBCXX_TR1_ELL_INTEGRAL_TCC 1 | |
7c62b943 | 44 | |
12ffa228 | 45 | namespace std _GLIBCXX_VISIBILITY(default) |
7c62b943 | 46 | { |
4a15d842 FD |
47 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
48 | ||
f8571e51 | 49 | #if _GLIBCXX_USE_STD_SPEC_FUNCS |
2be75957 | 50 | #elif defined(_GLIBCXX_TR1_CMATH) |
e133ace8 PC |
51 | namespace tr1 |
52 | { | |
2be75957 ESR |
53 | #else |
54 | # error do not include this header directly, use <cmath> or <tr1/cmath> | |
55 | #endif | |
7c62b943 BK |
56 | // [5.2] Special functions |
57 | ||
7c62b943 | 58 | // Implementation-space details. |
7c62b943 BK |
59 | namespace __detail |
60 | { | |
7c62b943 BK |
61 | /** |
62 | * @brief Return the Carlson elliptic function @f$ R_F(x,y,z) @f$ | |
63 | * of the first kind. | |
64 | * | |
65 | * The Carlson elliptic function of the first kind is defined by: | |
66 | * @f[ | |
67 | * R_F(x,y,z) = \frac{1}{2} \int_0^\infty | |
68 | * \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} | |
69 | * @f] | |
70 | * | |
71 | * @param __x The first of three symmetric arguments. | |
72 | * @param __y The second of three symmetric arguments. | |
73 | * @param __z The third of three symmetric arguments. | |
74 | * @return The Carlson elliptic function of the first kind. | |
75 | */ | |
76 | template<typename _Tp> | |
77 | _Tp | |
be59c932 | 78 | __ellint_rf(_Tp __x, _Tp __y, _Tp __z) |
7c62b943 BK |
79 | { |
80 | const _Tp __min = std::numeric_limits<_Tp>::min(); | |
7c62b943 | 81 | const _Tp __lolim = _Tp(5) * __min; |
7c62b943 BK |
82 | |
83 | if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0)) | |
84 | std::__throw_domain_error(__N("Argument less than zero " | |
85 | "in __ellint_rf.")); | |
86 | else if (__x + __y < __lolim || __x + __z < __lolim | |
87 | || __y + __z < __lolim) | |
88 | std::__throw_domain_error(__N("Argument too small in __ellint_rf")); | |
89 | else | |
90 | { | |
91 | const _Tp __c0 = _Tp(1) / _Tp(4); | |
92 | const _Tp __c1 = _Tp(1) / _Tp(24); | |
93 | const _Tp __c2 = _Tp(1) / _Tp(10); | |
94 | const _Tp __c3 = _Tp(3) / _Tp(44); | |
95 | const _Tp __c4 = _Tp(1) / _Tp(14); | |
96 | ||
97 | _Tp __xn = __x; | |
98 | _Tp __yn = __y; | |
99 | _Tp __zn = __z; | |
100 | ||
101 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
102 | const _Tp __errtol = std::pow(__eps, _Tp(1) / _Tp(6)); | |
103 | _Tp __mu; | |
104 | _Tp __xndev, __yndev, __zndev; | |
105 | ||
106 | const unsigned int __max_iter = 100; | |
107 | for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) | |
108 | { | |
109 | __mu = (__xn + __yn + __zn) / _Tp(3); | |
110 | __xndev = 2 - (__mu + __xn) / __mu; | |
111 | __yndev = 2 - (__mu + __yn) / __mu; | |
112 | __zndev = 2 - (__mu + __zn) / __mu; | |
113 | _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); | |
114 | __epsilon = std::max(__epsilon, std::abs(__zndev)); | |
115 | if (__epsilon < __errtol) | |
116 | break; | |
117 | const _Tp __xnroot = std::sqrt(__xn); | |
118 | const _Tp __ynroot = std::sqrt(__yn); | |
119 | const _Tp __znroot = std::sqrt(__zn); | |
120 | const _Tp __lambda = __xnroot * (__ynroot + __znroot) | |
121 | + __ynroot * __znroot; | |
122 | __xn = __c0 * (__xn + __lambda); | |
123 | __yn = __c0 * (__yn + __lambda); | |
124 | __zn = __c0 * (__zn + __lambda); | |
125 | } | |
126 | ||
127 | const _Tp __e2 = __xndev * __yndev - __zndev * __zndev; | |
128 | const _Tp __e3 = __xndev * __yndev * __zndev; | |
129 | const _Tp __s = _Tp(1) + (__c1 * __e2 - __c2 - __c3 * __e3) * __e2 | |
130 | + __c4 * __e3; | |
131 | ||
132 | return __s / std::sqrt(__mu); | |
133 | } | |
134 | } | |
135 | ||
136 | ||
137 | /** | |
138 | * @brief Return the complete elliptic integral of the first kind | |
139 | * @f$ K(k) @f$ by series expansion. | |
140 | * | |
141 | * The complete elliptic integral of the first kind is defined as | |
142 | * @f[ | |
143 | * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} | |
144 | * {\sqrt{1 - k^2sin^2\theta}} | |
145 | * @f] | |
146 | * | |
147 | * This routine is not bad as long as |k| is somewhat smaller than 1 | |
148 | * but is not is good as the Carlson elliptic integral formulation. | |
149 | * | |
150 | * @param __k The argument of the complete elliptic function. | |
151 | * @return The complete elliptic function of the first kind. | |
152 | */ | |
153 | template<typename _Tp> | |
154 | _Tp | |
be59c932 | 155 | __comp_ellint_1_series(_Tp __k) |
7c62b943 BK |
156 | { |
157 | ||
158 | const _Tp __kk = __k * __k; | |
159 | ||
160 | _Tp __term = __kk / _Tp(4); | |
161 | _Tp __sum = _Tp(1) + __term; | |
162 | ||
163 | const unsigned int __max_iter = 1000; | |
164 | for (unsigned int __i = 2; __i < __max_iter; ++__i) | |
165 | { | |
166 | __term *= (2 * __i - 1) * __kk / (2 * __i); | |
167 | if (__term < std::numeric_limits<_Tp>::epsilon()) | |
168 | break; | |
169 | __sum += __term; | |
170 | } | |
171 | ||
172 | return __numeric_constants<_Tp>::__pi_2() * __sum; | |
173 | } | |
174 | ||
175 | ||
176 | /** | |
177 | * @brief Return the complete elliptic integral of the first kind | |
178 | * @f$ K(k) @f$ using the Carlson formulation. | |
179 | * | |
180 | * The complete elliptic integral of the first kind is defined as | |
181 | * @f[ | |
182 | * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} | |
183 | * {\sqrt{1 - k^2 sin^2\theta}} | |
184 | * @f] | |
185 | * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the | |
186 | * first kind. | |
187 | * | |
188 | * @param __k The argument of the complete elliptic function. | |
189 | * @return The complete elliptic function of the first kind. | |
190 | */ | |
191 | template<typename _Tp> | |
192 | _Tp | |
be59c932 | 193 | __comp_ellint_1(_Tp __k) |
7c62b943 BK |
194 | { |
195 | ||
196 | if (__isnan(__k)) | |
197 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
198 | else if (std::abs(__k) >= _Tp(1)) | |
199 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
200 | else | |
201 | return __ellint_rf(_Tp(0), _Tp(1) - __k * __k, _Tp(1)); | |
202 | } | |
203 | ||
204 | ||
205 | /** | |
206 | * @brief Return the incomplete elliptic integral of the first kind | |
207 | * @f$ F(k,\phi) @f$ using the Carlson formulation. | |
208 | * | |
209 | * The incomplete elliptic integral of the first kind is defined as | |
210 | * @f[ | |
211 | * F(k,\phi) = \int_0^{\phi}\frac{d\theta} | |
212 | * {\sqrt{1 - k^2 sin^2\theta}} | |
213 | * @f] | |
214 | * | |
215 | * @param __k The argument of the elliptic function. | |
216 | * @param __phi The integral limit argument of the elliptic function. | |
217 | * @return The elliptic function of the first kind. | |
218 | */ | |
219 | template<typename _Tp> | |
220 | _Tp | |
be59c932 | 221 | __ellint_1(_Tp __k, _Tp __phi) |
7c62b943 BK |
222 | { |
223 | ||
224 | if (__isnan(__k) || __isnan(__phi)) | |
225 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
226 | else if (std::abs(__k) > _Tp(1)) | |
227 | std::__throw_domain_error(__N("Bad argument in __ellint_1.")); | |
228 | else | |
229 | { | |
230 | // Reduce phi to -pi/2 < phi < +pi/2. | |
231 | const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() | |
232 | + _Tp(0.5L)); | |
233 | const _Tp __phi_red = __phi | |
234 | - __n * __numeric_constants<_Tp>::__pi(); | |
235 | ||
236 | const _Tp __s = std::sin(__phi_red); | |
237 | const _Tp __c = std::cos(__phi_red); | |
238 | ||
239 | const _Tp __F = __s | |
240 | * __ellint_rf(__c * __c, | |
241 | _Tp(1) - __k * __k * __s * __s, _Tp(1)); | |
242 | ||
243 | if (__n == 0) | |
244 | return __F; | |
245 | else | |
246 | return __F + _Tp(2) * __n * __comp_ellint_1(__k); | |
247 | } | |
248 | } | |
249 | ||
250 | ||
251 | /** | |
252 | * @brief Return the complete elliptic integral of the second kind | |
253 | * @f$ E(k) @f$ by series expansion. | |
254 | * | |
255 | * The complete elliptic integral of the second kind is defined as | |
256 | * @f[ | |
257 | * E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} | |
258 | * @f] | |
259 | * | |
260 | * This routine is not bad as long as |k| is somewhat smaller than 1 | |
261 | * but is not is good as the Carlson elliptic integral formulation. | |
262 | * | |
263 | * @param __k The argument of the complete elliptic function. | |
264 | * @return The complete elliptic function of the second kind. | |
265 | */ | |
266 | template<typename _Tp> | |
267 | _Tp | |
be59c932 | 268 | __comp_ellint_2_series(_Tp __k) |
7c62b943 BK |
269 | { |
270 | ||
271 | const _Tp __kk = __k * __k; | |
272 | ||
273 | _Tp __term = __kk; | |
274 | _Tp __sum = __term; | |
275 | ||
276 | const unsigned int __max_iter = 1000; | |
277 | for (unsigned int __i = 2; __i < __max_iter; ++__i) | |
278 | { | |
279 | const _Tp __i2m = 2 * __i - 1; | |
280 | const _Tp __i2 = 2 * __i; | |
281 | __term *= __i2m * __i2m * __kk / (__i2 * __i2); | |
282 | if (__term < std::numeric_limits<_Tp>::epsilon()) | |
283 | break; | |
284 | __sum += __term / __i2m; | |
285 | } | |
286 | ||
287 | return __numeric_constants<_Tp>::__pi_2() * (_Tp(1) - __sum); | |
288 | } | |
289 | ||
290 | ||
291 | /** | |
292 | * @brief Return the Carlson elliptic function of the second kind | |
293 | * @f$ R_D(x,y,z) = R_J(x,y,z,z) @f$ where | |
294 | * @f$ R_J(x,y,z,p) @f$ is the Carlson elliptic function | |
295 | * of the third kind. | |
296 | * | |
297 | * The Carlson elliptic function of the second kind is defined by: | |
298 | * @f[ | |
299 | * R_D(x,y,z) = \frac{3}{2} \int_0^\infty | |
300 | * \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} | |
301 | * @f] | |
302 | * | |
303 | * Based on Carlson's algorithms: | |
304 | * - B. C. Carlson Numer. Math. 33, 1 (1979) | |
305 | * - B. C. Carlson, Special Functions of Applied Mathematics (1977) | |
28dac70a | 306 | * - Numerical Recipes in C, 2nd ed, pp. 261-269, |
7c62b943 BK |
307 | * by Press, Teukolsky, Vetterling, Flannery (1992) |
308 | * | |
309 | * @param __x The first of two symmetric arguments. | |
310 | * @param __y The second of two symmetric arguments. | |
311 | * @param __z The third argument. | |
312 | * @return The Carlson elliptic function of the second kind. | |
313 | */ | |
314 | template<typename _Tp> | |
315 | _Tp | |
be59c932 | 316 | __ellint_rd(_Tp __x, _Tp __y, _Tp __z) |
7c62b943 BK |
317 | { |
318 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
319 | const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6)); | |
7c62b943 BK |
320 | const _Tp __max = std::numeric_limits<_Tp>::max(); |
321 | const _Tp __lolim = _Tp(2) / std::pow(__max, _Tp(2) / _Tp(3)); | |
7c62b943 BK |
322 | |
323 | if (__x < _Tp(0) || __y < _Tp(0)) | |
324 | std::__throw_domain_error(__N("Argument less than zero " | |
325 | "in __ellint_rd.")); | |
326 | else if (__x + __y < __lolim || __z < __lolim) | |
327 | std::__throw_domain_error(__N("Argument too small " | |
328 | "in __ellint_rd.")); | |
329 | else | |
330 | { | |
331 | const _Tp __c0 = _Tp(1) / _Tp(4); | |
332 | const _Tp __c1 = _Tp(3) / _Tp(14); | |
333 | const _Tp __c2 = _Tp(1) / _Tp(6); | |
334 | const _Tp __c3 = _Tp(9) / _Tp(22); | |
335 | const _Tp __c4 = _Tp(3) / _Tp(26); | |
336 | ||
337 | _Tp __xn = __x; | |
338 | _Tp __yn = __y; | |
339 | _Tp __zn = __z; | |
340 | _Tp __sigma = _Tp(0); | |
341 | _Tp __power4 = _Tp(1); | |
342 | ||
343 | _Tp __mu; | |
344 | _Tp __xndev, __yndev, __zndev; | |
345 | ||
346 | const unsigned int __max_iter = 100; | |
347 | for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) | |
348 | { | |
349 | __mu = (__xn + __yn + _Tp(3) * __zn) / _Tp(5); | |
350 | __xndev = (__mu - __xn) / __mu; | |
351 | __yndev = (__mu - __yn) / __mu; | |
352 | __zndev = (__mu - __zn) / __mu; | |
353 | _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); | |
354 | __epsilon = std::max(__epsilon, std::abs(__zndev)); | |
355 | if (__epsilon < __errtol) | |
356 | break; | |
357 | _Tp __xnroot = std::sqrt(__xn); | |
358 | _Tp __ynroot = std::sqrt(__yn); | |
359 | _Tp __znroot = std::sqrt(__zn); | |
360 | _Tp __lambda = __xnroot * (__ynroot + __znroot) | |
361 | + __ynroot * __znroot; | |
362 | __sigma += __power4 / (__znroot * (__zn + __lambda)); | |
363 | __power4 *= __c0; | |
364 | __xn = __c0 * (__xn + __lambda); | |
365 | __yn = __c0 * (__yn + __lambda); | |
366 | __zn = __c0 * (__zn + __lambda); | |
367 | } | |
368 | ||
2f2aeda9 | 369 | _Tp __ea = __xndev * __yndev; |
7c62b943 | 370 | _Tp __eb = __zndev * __zndev; |
2f2aeda9 UW |
371 | _Tp __ec = __ea - __eb; |
372 | _Tp __ed = __ea - _Tp(6) * __eb; | |
7c62b943 BK |
373 | _Tp __ef = __ed + __ec + __ec; |
374 | _Tp __s1 = __ed * (-__c1 + __c3 * __ed | |
375 | / _Tp(3) - _Tp(3) * __c4 * __zndev * __ef | |
376 | / _Tp(2)); | |
377 | _Tp __s2 = __zndev | |
378 | * (__c2 * __ef | |
2f2aeda9 | 379 | + __zndev * (-__c3 * __ec - __zndev * __c4 - __ea)); |
7c62b943 BK |
380 | |
381 | return _Tp(3) * __sigma + __power4 * (_Tp(1) + __s1 + __s2) | |
382 | / (__mu * std::sqrt(__mu)); | |
383 | } | |
384 | } | |
385 | ||
386 | ||
387 | /** | |
388 | * @brief Return the complete elliptic integral of the second kind | |
389 | * @f$ E(k) @f$ using the Carlson formulation. | |
390 | * | |
391 | * The complete elliptic integral of the second kind is defined as | |
392 | * @f[ | |
393 | * E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} | |
394 | * @f] | |
395 | * | |
396 | * @param __k The argument of the complete elliptic function. | |
397 | * @return The complete elliptic function of the second kind. | |
398 | */ | |
399 | template<typename _Tp> | |
400 | _Tp | |
be59c932 | 401 | __comp_ellint_2(_Tp __k) |
7c62b943 BK |
402 | { |
403 | ||
404 | if (__isnan(__k)) | |
405 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
406 | else if (std::abs(__k) == 1) | |
407 | return _Tp(1); | |
408 | else if (std::abs(__k) > _Tp(1)) | |
409 | std::__throw_domain_error(__N("Bad argument in __comp_ellint_2.")); | |
410 | else | |
411 | { | |
412 | const _Tp __kk = __k * __k; | |
413 | ||
414 | return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1)) | |
415 | - __kk * __ellint_rd(_Tp(0), _Tp(1) - __kk, _Tp(1)) / _Tp(3); | |
416 | } | |
417 | } | |
418 | ||
419 | ||
420 | /** | |
421 | * @brief Return the incomplete elliptic integral of the second kind | |
422 | * @f$ E(k,\phi) @f$ using the Carlson formulation. | |
423 | * | |
424 | * The incomplete elliptic integral of the second kind is defined as | |
425 | * @f[ | |
426 | * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} | |
427 | * @f] | |
428 | * | |
429 | * @param __k The argument of the elliptic function. | |
430 | * @param __phi The integral limit argument of the elliptic function. | |
431 | * @return The elliptic function of the second kind. | |
432 | */ | |
433 | template<typename _Tp> | |
434 | _Tp | |
be59c932 | 435 | __ellint_2(_Tp __k, _Tp __phi) |
7c62b943 BK |
436 | { |
437 | ||
438 | if (__isnan(__k) || __isnan(__phi)) | |
439 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
440 | else if (std::abs(__k) > _Tp(1)) | |
441 | std::__throw_domain_error(__N("Bad argument in __ellint_2.")); | |
442 | else | |
443 | { | |
444 | // Reduce phi to -pi/2 < phi < +pi/2. | |
445 | const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() | |
446 | + _Tp(0.5L)); | |
447 | const _Tp __phi_red = __phi | |
448 | - __n * __numeric_constants<_Tp>::__pi(); | |
449 | ||
450 | const _Tp __kk = __k * __k; | |
451 | const _Tp __s = std::sin(__phi_red); | |
452 | const _Tp __ss = __s * __s; | |
453 | const _Tp __sss = __ss * __s; | |
454 | const _Tp __c = std::cos(__phi_red); | |
455 | const _Tp __cc = __c * __c; | |
456 | ||
457 | const _Tp __E = __s | |
458 | * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1)) | |
459 | - __kk * __sss | |
460 | * __ellint_rd(__cc, _Tp(1) - __kk * __ss, _Tp(1)) | |
461 | / _Tp(3); | |
462 | ||
463 | if (__n == 0) | |
464 | return __E; | |
465 | else | |
466 | return __E + _Tp(2) * __n * __comp_ellint_2(__k); | |
467 | } | |
468 | } | |
469 | ||
470 | ||
471 | /** | |
472 | * @brief Return the Carlson elliptic function | |
473 | * @f$ R_C(x,y) = R_F(x,y,y) @f$ where @f$ R_F(x,y,z) @f$ | |
474 | * is the Carlson elliptic function of the first kind. | |
475 | * | |
476 | * The Carlson elliptic function is defined by: | |
477 | * @f[ | |
478 | * R_C(x,y) = \frac{1}{2} \int_0^\infty | |
479 | * \frac{dt}{(t + x)^{1/2}(t + y)} | |
480 | * @f] | |
481 | * | |
482 | * Based on Carlson's algorithms: | |
483 | * - B. C. Carlson Numer. Math. 33, 1 (1979) | |
484 | * - B. C. Carlson, Special Functions of Applied Mathematics (1977) | |
28dac70a | 485 | * - Numerical Recipes in C, 2nd ed, pp. 261-269, |
7c62b943 BK |
486 | * by Press, Teukolsky, Vetterling, Flannery (1992) |
487 | * | |
488 | * @param __x The first argument. | |
489 | * @param __y The second argument. | |
490 | * @return The Carlson elliptic function. | |
491 | */ | |
492 | template<typename _Tp> | |
493 | _Tp | |
be59c932 | 494 | __ellint_rc(_Tp __x, _Tp __y) |
7c62b943 BK |
495 | { |
496 | const _Tp __min = std::numeric_limits<_Tp>::min(); | |
7c62b943 | 497 | const _Tp __lolim = _Tp(5) * __min; |
7c62b943 BK |
498 | |
499 | if (__x < _Tp(0) || __y < _Tp(0) || __x + __y < __lolim) | |
500 | std::__throw_domain_error(__N("Argument less than zero " | |
501 | "in __ellint_rc.")); | |
502 | else | |
503 | { | |
504 | const _Tp __c0 = _Tp(1) / _Tp(4); | |
505 | const _Tp __c1 = _Tp(1) / _Tp(7); | |
506 | const _Tp __c2 = _Tp(9) / _Tp(22); | |
507 | const _Tp __c3 = _Tp(3) / _Tp(10); | |
508 | const _Tp __c4 = _Tp(3) / _Tp(8); | |
509 | ||
510 | _Tp __xn = __x; | |
511 | _Tp __yn = __y; | |
512 | ||
513 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
514 | const _Tp __errtol = std::pow(__eps / _Tp(30), _Tp(1) / _Tp(6)); | |
515 | _Tp __mu; | |
516 | _Tp __sn; | |
517 | ||
518 | const unsigned int __max_iter = 100; | |
519 | for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) | |
520 | { | |
521 | __mu = (__xn + _Tp(2) * __yn) / _Tp(3); | |
522 | __sn = (__yn + __mu) / __mu - _Tp(2); | |
523 | if (std::abs(__sn) < __errtol) | |
524 | break; | |
525 | const _Tp __lambda = _Tp(2) * std::sqrt(__xn) * std::sqrt(__yn) | |
526 | + __yn; | |
527 | __xn = __c0 * (__xn + __lambda); | |
528 | __yn = __c0 * (__yn + __lambda); | |
529 | } | |
530 | ||
531 | _Tp __s = __sn * __sn | |
532 | * (__c3 + __sn*(__c1 + __sn * (__c4 + __sn * __c2))); | |
533 | ||
534 | return (_Tp(1) + __s) / std::sqrt(__mu); | |
535 | } | |
536 | } | |
537 | ||
538 | ||
539 | /** | |
540 | * @brief Return the Carlson elliptic function @f$ R_J(x,y,z,p) @f$ | |
541 | * of the third kind. | |
542 | * | |
543 | * The Carlson elliptic function of the third kind is defined by: | |
544 | * @f[ | |
545 | * R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty | |
546 | * \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} | |
547 | * @f] | |
548 | * | |
549 | * Based on Carlson's algorithms: | |
550 | * - B. C. Carlson Numer. Math. 33, 1 (1979) | |
551 | * - B. C. Carlson, Special Functions of Applied Mathematics (1977) | |
28dac70a | 552 | * - Numerical Recipes in C, 2nd ed, pp. 261-269, |
7c62b943 BK |
553 | * by Press, Teukolsky, Vetterling, Flannery (1992) |
554 | * | |
555 | * @param __x The first of three symmetric arguments. | |
556 | * @param __y The second of three symmetric arguments. | |
557 | * @param __z The third of three symmetric arguments. | |
558 | * @param __p The fourth argument. | |
559 | * @return The Carlson elliptic function of the fourth kind. | |
560 | */ | |
561 | template<typename _Tp> | |
562 | _Tp | |
be59c932 | 563 | __ellint_rj(_Tp __x, _Tp __y, _Tp __z, _Tp __p) |
7c62b943 BK |
564 | { |
565 | const _Tp __min = std::numeric_limits<_Tp>::min(); | |
7c62b943 | 566 | const _Tp __lolim = std::pow(_Tp(5) * __min, _Tp(1)/_Tp(3)); |
7c62b943 BK |
567 | |
568 | if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0)) | |
569 | std::__throw_domain_error(__N("Argument less than zero " | |
570 | "in __ellint_rj.")); | |
571 | else if (__x + __y < __lolim || __x + __z < __lolim | |
572 | || __y + __z < __lolim || __p < __lolim) | |
573 | std::__throw_domain_error(__N("Argument too small " | |
574 | "in __ellint_rj")); | |
575 | else | |
576 | { | |
577 | const _Tp __c0 = _Tp(1) / _Tp(4); | |
578 | const _Tp __c1 = _Tp(3) / _Tp(14); | |
579 | const _Tp __c2 = _Tp(1) / _Tp(3); | |
580 | const _Tp __c3 = _Tp(3) / _Tp(22); | |
581 | const _Tp __c4 = _Tp(3) / _Tp(26); | |
582 | ||
583 | _Tp __xn = __x; | |
584 | _Tp __yn = __y; | |
585 | _Tp __zn = __z; | |
586 | _Tp __pn = __p; | |
587 | _Tp __sigma = _Tp(0); | |
588 | _Tp __power4 = _Tp(1); | |
589 | ||
590 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
591 | const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6)); | |
592 | ||
86558afc | 593 | _Tp __mu; |
7c62b943 BK |
594 | _Tp __xndev, __yndev, __zndev, __pndev; |
595 | ||
596 | const unsigned int __max_iter = 100; | |
597 | for (unsigned int __iter = 0; __iter < __max_iter; ++__iter) | |
598 | { | |
599 | __mu = (__xn + __yn + __zn + _Tp(2) * __pn) / _Tp(5); | |
600 | __xndev = (__mu - __xn) / __mu; | |
601 | __yndev = (__mu - __yn) / __mu; | |
602 | __zndev = (__mu - __zn) / __mu; | |
603 | __pndev = (__mu - __pn) / __mu; | |
604 | _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev)); | |
605 | __epsilon = std::max(__epsilon, std::abs(__zndev)); | |
606 | __epsilon = std::max(__epsilon, std::abs(__pndev)); | |
607 | if (__epsilon < __errtol) | |
608 | break; | |
609 | const _Tp __xnroot = std::sqrt(__xn); | |
610 | const _Tp __ynroot = std::sqrt(__yn); | |
611 | const _Tp __znroot = std::sqrt(__zn); | |
612 | const _Tp __lambda = __xnroot * (__ynroot + __znroot) | |
613 | + __ynroot * __znroot; | |
f070285a | 614 | const _Tp __alpha1 = __pn * (__xnroot + __ynroot + __znroot) |
7c62b943 | 615 | + __xnroot * __ynroot * __znroot; |
f070285a | 616 | const _Tp __alpha2 = __alpha1 * __alpha1; |
7c62b943 BK |
617 | const _Tp __beta = __pn * (__pn + __lambda) |
618 | * (__pn + __lambda); | |
619 | __sigma += __power4 * __ellint_rc(__alpha2, __beta); | |
620 | __power4 *= __c0; | |
621 | __xn = __c0 * (__xn + __lambda); | |
622 | __yn = __c0 * (__yn + __lambda); | |
623 | __zn = __c0 * (__zn + __lambda); | |
624 | __pn = __c0 * (__pn + __lambda); | |
625 | } | |
626 | ||
2f2aeda9 | 627 | _Tp __ea = __xndev * (__yndev + __zndev) + __yndev * __zndev; |
7c62b943 BK |
628 | _Tp __eb = __xndev * __yndev * __zndev; |
629 | _Tp __ec = __pndev * __pndev; | |
2f2aeda9 UW |
630 | _Tp __e2 = __ea - _Tp(3) * __ec; |
631 | _Tp __e3 = __eb + _Tp(2) * __pndev * (__ea - __ec); | |
7c62b943 BK |
632 | _Tp __s1 = _Tp(1) + __e2 * (-__c1 + _Tp(3) * __c3 * __e2 / _Tp(4) |
633 | - _Tp(3) * __c4 * __e3 / _Tp(2)); | |
634 | _Tp __s2 = __eb * (__c2 / _Tp(2) | |
635 | + __pndev * (-__c3 - __c3 + __pndev * __c4)); | |
2f2aeda9 | 636 | _Tp __s3 = __pndev * __ea * (__c2 - __pndev * __c3) |
7c62b943 BK |
637 | - __c2 * __pndev * __ec; |
638 | ||
639 | return _Tp(3) * __sigma + __power4 * (__s1 + __s2 + __s3) | |
640 | / (__mu * std::sqrt(__mu)); | |
641 | } | |
642 | } | |
643 | ||
644 | ||
645 | /** | |
646 | * @brief Return the complete elliptic integral of the third kind | |
647 | * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ using the | |
648 | * Carlson formulation. | |
649 | * | |
650 | * The complete elliptic integral of the third kind is defined as | |
651 | * @f[ | |
652 | * \Pi(k,\nu) = \int_0^{\pi/2} | |
653 | * \frac{d\theta} | |
654 | * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} | |
655 | * @f] | |
656 | * | |
657 | * @param __k The argument of the elliptic function. | |
658 | * @param __nu The second argument of the elliptic function. | |
659 | * @return The complete elliptic function of the third kind. | |
660 | */ | |
661 | template<typename _Tp> | |
662 | _Tp | |
be59c932 | 663 | __comp_ellint_3(_Tp __k, _Tp __nu) |
7c62b943 BK |
664 | { |
665 | ||
666 | if (__isnan(__k) || __isnan(__nu)) | |
667 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
668 | else if (__nu == _Tp(1)) | |
669 | return std::numeric_limits<_Tp>::infinity(); | |
670 | else if (std::abs(__k) > _Tp(1)) | |
671 | std::__throw_domain_error(__N("Bad argument in __comp_ellint_3.")); | |
672 | else | |
673 | { | |
674 | const _Tp __kk = __k * __k; | |
675 | ||
676 | return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1)) | |
b4688136 ESR |
677 | + __nu |
678 | * __ellint_rj(_Tp(0), _Tp(1) - __kk, _Tp(1), _Tp(1) - __nu) | |
7c62b943 BK |
679 | / _Tp(3); |
680 | } | |
681 | } | |
682 | ||
683 | ||
684 | /** | |
685 | * @brief Return the incomplete elliptic integral of the third kind | |
686 | * @f$ \Pi(k,\nu,\phi) @f$ using the Carlson formulation. | |
687 | * | |
688 | * The incomplete elliptic integral of the third kind is defined as | |
689 | * @f[ | |
690 | * \Pi(k,\nu,\phi) = \int_0^{\phi} | |
691 | * \frac{d\theta} | |
692 | * {(1 - \nu \sin^2\theta) | |
693 | * \sqrt{1 - k^2 \sin^2\theta}} | |
694 | * @f] | |
695 | * | |
696 | * @param __k The argument of the elliptic function. | |
697 | * @param __nu The second argument of the elliptic function. | |
698 | * @param __phi The integral limit argument of the elliptic function. | |
699 | * @return The elliptic function of the third kind. | |
700 | */ | |
701 | template<typename _Tp> | |
702 | _Tp | |
be59c932 | 703 | __ellint_3(_Tp __k, _Tp __nu, _Tp __phi) |
7c62b943 BK |
704 | { |
705 | ||
706 | if (__isnan(__k) || __isnan(__nu) || __isnan(__phi)) | |
707 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
708 | else if (std::abs(__k) > _Tp(1)) | |
709 | std::__throw_domain_error(__N("Bad argument in __ellint_3.")); | |
710 | else | |
711 | { | |
712 | // Reduce phi to -pi/2 < phi < +pi/2. | |
713 | const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi() | |
714 | + _Tp(0.5L)); | |
715 | const _Tp __phi_red = __phi | |
716 | - __n * __numeric_constants<_Tp>::__pi(); | |
717 | ||
718 | const _Tp __kk = __k * __k; | |
719 | const _Tp __s = std::sin(__phi_red); | |
720 | const _Tp __ss = __s * __s; | |
721 | const _Tp __sss = __ss * __s; | |
722 | const _Tp __c = std::cos(__phi_red); | |
723 | const _Tp __cc = __c * __c; | |
724 | ||
725 | const _Tp __Pi = __s | |
726 | * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1)) | |
b4688136 | 727 | + __nu * __sss |
7c62b943 | 728 | * __ellint_rj(__cc, _Tp(1) - __kk * __ss, _Tp(1), |
b4688136 | 729 | _Tp(1) - __nu * __ss) / _Tp(3); |
7c62b943 BK |
730 | |
731 | if (__n == 0) | |
732 | return __Pi; | |
733 | else | |
734 | return __Pi + _Tp(2) * __n * __comp_ellint_3(__k, __nu); | |
735 | } | |
736 | } | |
2be75957 | 737 | } // namespace __detail |
f8571e51 | 738 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
2be75957 ESR |
739 | } // namespace tr1 |
740 | #endif | |
4a15d842 FD |
741 | |
742 | _GLIBCXX_END_NAMESPACE_VERSION | |
7c62b943 BK |
743 | } |
744 | ||
e133ace8 | 745 | #endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC |
7c62b943 | 746 |