1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006-2018 Free Software Foundation, Inc.
5 // This file is part of the GNU ISO C++ Library. This library is free
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7 // terms of the GNU General Public License as published by the
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25 /** @file tr1/ell_integral.tcc
26 * This is an internal header file, included by other library headers.
27 * Do not attempt to use it directly. @headername{tr1/cmath}
31 // ISO C++ 14882 TR1: 5.2 Special functions
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
42 #ifndef _GLIBCXX_TR1_ELL_INTEGRAL_TCC
43 #define _GLIBCXX_TR1_ELL_INTEGRAL_TCC 1
45 namespace std _GLIBCXX_VISIBILITY(default)
47 _GLIBCXX_BEGIN_NAMESPACE_VERSION
49 #if _GLIBCXX_USE_STD_SPEC_FUNCS
50 #elif defined(_GLIBCXX_TR1_CMATH)
54 # error do not include this header directly, use <cmath> or <tr1/cmath>
56 // [5.2] Special functions
58 // Implementation-space details.
62 * @brief Return the Carlson elliptic function @f$ R_F(x,y,z) @f$
65 * The Carlson elliptic function of the first kind is defined by:
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}}
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.
76 template<typename _Tp>
78 __ellint_rf(_Tp __x, _Tp __y, _Tp __z)
80 const _Tp __min = std::numeric_limits<_Tp>::min();
81 const _Tp __max = std::numeric_limits<_Tp>::max();
82 const _Tp __lolim = _Tp(5) * __min;
83 const _Tp __uplim = __max / _Tp(5);
85 if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
86 std::__throw_domain_error(__N("Argument less than zero "
88 else if (__x + __y < __lolim || __x + __z < __lolim
89 || __y + __z < __lolim)
90 std::__throw_domain_error(__N("Argument too small in __ellint_rf"));
93 const _Tp __c0 = _Tp(1) / _Tp(4);
94 const _Tp __c1 = _Tp(1) / _Tp(24);
95 const _Tp __c2 = _Tp(1) / _Tp(10);
96 const _Tp __c3 = _Tp(3) / _Tp(44);
97 const _Tp __c4 = _Tp(1) / _Tp(14);
103 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
104 const _Tp __errtol = std::pow(__eps, _Tp(1) / _Tp(6));
106 _Tp __xndev, __yndev, __zndev;
108 const unsigned int __max_iter = 100;
109 for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
111 __mu = (__xn + __yn + __zn) / _Tp(3);
112 __xndev = 2 - (__mu + __xn) / __mu;
113 __yndev = 2 - (__mu + __yn) / __mu;
114 __zndev = 2 - (__mu + __zn) / __mu;
115 _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
116 __epsilon = std::max(__epsilon, std::abs(__zndev));
117 if (__epsilon < __errtol)
119 const _Tp __xnroot = std::sqrt(__xn);
120 const _Tp __ynroot = std::sqrt(__yn);
121 const _Tp __znroot = std::sqrt(__zn);
122 const _Tp __lambda = __xnroot * (__ynroot + __znroot)
123 + __ynroot * __znroot;
124 __xn = __c0 * (__xn + __lambda);
125 __yn = __c0 * (__yn + __lambda);
126 __zn = __c0 * (__zn + __lambda);
129 const _Tp __e2 = __xndev * __yndev - __zndev * __zndev;
130 const _Tp __e3 = __xndev * __yndev * __zndev;
131 const _Tp __s = _Tp(1) + (__c1 * __e2 - __c2 - __c3 * __e3) * __e2
134 return __s / std::sqrt(__mu);
140 * @brief Return the complete elliptic integral of the first kind
141 * @f$ K(k) @f$ by series expansion.
143 * The complete elliptic integral of the first kind is defined as
145 * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
146 * {\sqrt{1 - k^2sin^2\theta}}
149 * This routine is not bad as long as |k| is somewhat smaller than 1
150 * but is not is good as the Carlson elliptic integral formulation.
152 * @param __k The argument of the complete elliptic function.
153 * @return The complete elliptic function of the first kind.
155 template<typename _Tp>
157 __comp_ellint_1_series(_Tp __k)
160 const _Tp __kk = __k * __k;
162 _Tp __term = __kk / _Tp(4);
163 _Tp __sum = _Tp(1) + __term;
165 const unsigned int __max_iter = 1000;
166 for (unsigned int __i = 2; __i < __max_iter; ++__i)
168 __term *= (2 * __i - 1) * __kk / (2 * __i);
169 if (__term < std::numeric_limits<_Tp>::epsilon())
174 return __numeric_constants<_Tp>::__pi_2() * __sum;
179 * @brief Return the complete elliptic integral of the first kind
180 * @f$ K(k) @f$ using the Carlson formulation.
182 * The complete elliptic integral of the first kind is defined as
184 * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
185 * {\sqrt{1 - k^2 sin^2\theta}}
187 * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the
190 * @param __k The argument of the complete elliptic function.
191 * @return The complete elliptic function of the first kind.
193 template<typename _Tp>
195 __comp_ellint_1(_Tp __k)
199 return std::numeric_limits<_Tp>::quiet_NaN();
200 else if (std::abs(__k) >= _Tp(1))
201 return std::numeric_limits<_Tp>::quiet_NaN();
203 return __ellint_rf(_Tp(0), _Tp(1) - __k * __k, _Tp(1));
208 * @brief Return the incomplete elliptic integral of the first kind
209 * @f$ F(k,\phi) @f$ using the Carlson formulation.
211 * The incomplete elliptic integral of the first kind is defined as
213 * F(k,\phi) = \int_0^{\phi}\frac{d\theta}
214 * {\sqrt{1 - k^2 sin^2\theta}}
217 * @param __k The argument of the elliptic function.
218 * @param __phi The integral limit argument of the elliptic function.
219 * @return The elliptic function of the first kind.
221 template<typename _Tp>
223 __ellint_1(_Tp __k, _Tp __phi)
226 if (__isnan(__k) || __isnan(__phi))
227 return std::numeric_limits<_Tp>::quiet_NaN();
228 else if (std::abs(__k) > _Tp(1))
229 std::__throw_domain_error(__N("Bad argument in __ellint_1."));
232 // Reduce phi to -pi/2 < phi < +pi/2.
233 const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
235 const _Tp __phi_red = __phi
236 - __n * __numeric_constants<_Tp>::__pi();
238 const _Tp __s = std::sin(__phi_red);
239 const _Tp __c = std::cos(__phi_red);
242 * __ellint_rf(__c * __c,
243 _Tp(1) - __k * __k * __s * __s, _Tp(1));
248 return __F + _Tp(2) * __n * __comp_ellint_1(__k);
254 * @brief Return the complete elliptic integral of the second kind
255 * @f$ E(k) @f$ by series expansion.
257 * The complete elliptic integral of the second kind is defined as
259 * E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
262 * This routine is not bad as long as |k| is somewhat smaller than 1
263 * but is not is good as the Carlson elliptic integral formulation.
265 * @param __k The argument of the complete elliptic function.
266 * @return The complete elliptic function of the second kind.
268 template<typename _Tp>
270 __comp_ellint_2_series(_Tp __k)
273 const _Tp __kk = __k * __k;
278 const unsigned int __max_iter = 1000;
279 for (unsigned int __i = 2; __i < __max_iter; ++__i)
281 const _Tp __i2m = 2 * __i - 1;
282 const _Tp __i2 = 2 * __i;
283 __term *= __i2m * __i2m * __kk / (__i2 * __i2);
284 if (__term < std::numeric_limits<_Tp>::epsilon())
286 __sum += __term / __i2m;
289 return __numeric_constants<_Tp>::__pi_2() * (_Tp(1) - __sum);
294 * @brief Return the Carlson elliptic function of the second kind
295 * @f$ R_D(x,y,z) = R_J(x,y,z,z) @f$ where
296 * @f$ R_J(x,y,z,p) @f$ is the Carlson elliptic function
299 * The Carlson elliptic function of the second kind is defined by:
301 * R_D(x,y,z) = \frac{3}{2} \int_0^\infty
302 * \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}}
305 * Based on Carlson's algorithms:
306 * - B. C. Carlson Numer. Math. 33, 1 (1979)
307 * - B. C. Carlson, Special Functions of Applied Mathematics (1977)
308 * - Numerical Recipes in C, 2nd ed, pp. 261-269,
309 * by Press, Teukolsky, Vetterling, Flannery (1992)
311 * @param __x The first of two symmetric arguments.
312 * @param __y The second of two symmetric arguments.
313 * @param __z The third argument.
314 * @return The Carlson elliptic function of the second kind.
316 template<typename _Tp>
318 __ellint_rd(_Tp __x, _Tp __y, _Tp __z)
320 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
321 const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));
322 const _Tp __min = std::numeric_limits<_Tp>::min();
323 const _Tp __max = std::numeric_limits<_Tp>::max();
324 const _Tp __lolim = _Tp(2) / std::pow(__max, _Tp(2) / _Tp(3));
325 const _Tp __uplim = std::pow(_Tp(0.1L) * __errtol / __min, _Tp(2) / _Tp(3));
327 if (__x < _Tp(0) || __y < _Tp(0))
328 std::__throw_domain_error(__N("Argument less than zero "
330 else if (__x + __y < __lolim || __z < __lolim)
331 std::__throw_domain_error(__N("Argument too small "
335 const _Tp __c0 = _Tp(1) / _Tp(4);
336 const _Tp __c1 = _Tp(3) / _Tp(14);
337 const _Tp __c2 = _Tp(1) / _Tp(6);
338 const _Tp __c3 = _Tp(9) / _Tp(22);
339 const _Tp __c4 = _Tp(3) / _Tp(26);
344 _Tp __sigma = _Tp(0);
345 _Tp __power4 = _Tp(1);
348 _Tp __xndev, __yndev, __zndev;
350 const unsigned int __max_iter = 100;
351 for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
353 __mu = (__xn + __yn + _Tp(3) * __zn) / _Tp(5);
354 __xndev = (__mu - __xn) / __mu;
355 __yndev = (__mu - __yn) / __mu;
356 __zndev = (__mu - __zn) / __mu;
357 _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
358 __epsilon = std::max(__epsilon, std::abs(__zndev));
359 if (__epsilon < __errtol)
361 _Tp __xnroot = std::sqrt(__xn);
362 _Tp __ynroot = std::sqrt(__yn);
363 _Tp __znroot = std::sqrt(__zn);
364 _Tp __lambda = __xnroot * (__ynroot + __znroot)
365 + __ynroot * __znroot;
366 __sigma += __power4 / (__znroot * (__zn + __lambda));
368 __xn = __c0 * (__xn + __lambda);
369 __yn = __c0 * (__yn + __lambda);
370 __zn = __c0 * (__zn + __lambda);
373 // Note: __ea is an SPU badname.
374 _Tp __eaa = __xndev * __yndev;
375 _Tp __eb = __zndev * __zndev;
376 _Tp __ec = __eaa - __eb;
377 _Tp __ed = __eaa - _Tp(6) * __eb;
378 _Tp __ef = __ed + __ec + __ec;
379 _Tp __s1 = __ed * (-__c1 + __c3 * __ed
380 / _Tp(3) - _Tp(3) * __c4 * __zndev * __ef
384 + __zndev * (-__c3 * __ec - __zndev * __c4 - __eaa));
386 return _Tp(3) * __sigma + __power4 * (_Tp(1) + __s1 + __s2)
387 / (__mu * std::sqrt(__mu));
393 * @brief Return the complete elliptic integral of the second kind
394 * @f$ E(k) @f$ using the Carlson formulation.
396 * The complete elliptic integral of the second kind is defined as
398 * E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
401 * @param __k The argument of the complete elliptic function.
402 * @return The complete elliptic function of the second kind.
404 template<typename _Tp>
406 __comp_ellint_2(_Tp __k)
410 return std::numeric_limits<_Tp>::quiet_NaN();
411 else if (std::abs(__k) == 1)
413 else if (std::abs(__k) > _Tp(1))
414 std::__throw_domain_error(__N("Bad argument in __comp_ellint_2."));
417 const _Tp __kk = __k * __k;
419 return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
420 - __kk * __ellint_rd(_Tp(0), _Tp(1) - __kk, _Tp(1)) / _Tp(3);
426 * @brief Return the incomplete elliptic integral of the second kind
427 * @f$ E(k,\phi) @f$ using the Carlson formulation.
429 * The incomplete elliptic integral of the second kind is defined as
431 * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}
434 * @param __k The argument of the elliptic function.
435 * @param __phi The integral limit argument of the elliptic function.
436 * @return The elliptic function of the second kind.
438 template<typename _Tp>
440 __ellint_2(_Tp __k, _Tp __phi)
443 if (__isnan(__k) || __isnan(__phi))
444 return std::numeric_limits<_Tp>::quiet_NaN();
445 else if (std::abs(__k) > _Tp(1))
446 std::__throw_domain_error(__N("Bad argument in __ellint_2."));
449 // Reduce phi to -pi/2 < phi < +pi/2.
450 const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
452 const _Tp __phi_red = __phi
453 - __n * __numeric_constants<_Tp>::__pi();
455 const _Tp __kk = __k * __k;
456 const _Tp __s = std::sin(__phi_red);
457 const _Tp __ss = __s * __s;
458 const _Tp __sss = __ss * __s;
459 const _Tp __c = std::cos(__phi_red);
460 const _Tp __cc = __c * __c;
463 * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
465 * __ellint_rd(__cc, _Tp(1) - __kk * __ss, _Tp(1))
471 return __E + _Tp(2) * __n * __comp_ellint_2(__k);
477 * @brief Return the Carlson elliptic function
478 * @f$ R_C(x,y) = R_F(x,y,y) @f$ where @f$ R_F(x,y,z) @f$
479 * is the Carlson elliptic function of the first kind.
481 * The Carlson elliptic function is defined by:
483 * R_C(x,y) = \frac{1}{2} \int_0^\infty
484 * \frac{dt}{(t + x)^{1/2}(t + y)}
487 * Based on Carlson's algorithms:
488 * - B. C. Carlson Numer. Math. 33, 1 (1979)
489 * - B. C. Carlson, Special Functions of Applied Mathematics (1977)
490 * - Numerical Recipes in C, 2nd ed, pp. 261-269,
491 * by Press, Teukolsky, Vetterling, Flannery (1992)
493 * @param __x The first argument.
494 * @param __y The second argument.
495 * @return The Carlson elliptic function.
497 template<typename _Tp>
499 __ellint_rc(_Tp __x, _Tp __y)
501 const _Tp __min = std::numeric_limits<_Tp>::min();
502 const _Tp __max = std::numeric_limits<_Tp>::max();
503 const _Tp __lolim = _Tp(5) * __min;
504 const _Tp __uplim = __max / _Tp(5);
506 if (__x < _Tp(0) || __y < _Tp(0) || __x + __y < __lolim)
507 std::__throw_domain_error(__N("Argument less than zero "
511 const _Tp __c0 = _Tp(1) / _Tp(4);
512 const _Tp __c1 = _Tp(1) / _Tp(7);
513 const _Tp __c2 = _Tp(9) / _Tp(22);
514 const _Tp __c3 = _Tp(3) / _Tp(10);
515 const _Tp __c4 = _Tp(3) / _Tp(8);
520 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
521 const _Tp __errtol = std::pow(__eps / _Tp(30), _Tp(1) / _Tp(6));
525 const unsigned int __max_iter = 100;
526 for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
528 __mu = (__xn + _Tp(2) * __yn) / _Tp(3);
529 __sn = (__yn + __mu) / __mu - _Tp(2);
530 if (std::abs(__sn) < __errtol)
532 const _Tp __lambda = _Tp(2) * std::sqrt(__xn) * std::sqrt(__yn)
534 __xn = __c0 * (__xn + __lambda);
535 __yn = __c0 * (__yn + __lambda);
538 _Tp __s = __sn * __sn
539 * (__c3 + __sn*(__c1 + __sn * (__c4 + __sn * __c2)));
541 return (_Tp(1) + __s) / std::sqrt(__mu);
547 * @brief Return the Carlson elliptic function @f$ R_J(x,y,z,p) @f$
550 * The Carlson elliptic function of the third kind is defined by:
552 * R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty
553 * \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)}
556 * Based on Carlson's algorithms:
557 * - B. C. Carlson Numer. Math. 33, 1 (1979)
558 * - B. C. Carlson, Special Functions of Applied Mathematics (1977)
559 * - Numerical Recipes in C, 2nd ed, pp. 261-269,
560 * by Press, Teukolsky, Vetterling, Flannery (1992)
562 * @param __x The first of three symmetric arguments.
563 * @param __y The second of three symmetric arguments.
564 * @param __z The third of three symmetric arguments.
565 * @param __p The fourth argument.
566 * @return The Carlson elliptic function of the fourth kind.
568 template<typename _Tp>
570 __ellint_rj(_Tp __x, _Tp __y, _Tp __z, _Tp __p)
572 const _Tp __min = std::numeric_limits<_Tp>::min();
573 const _Tp __max = std::numeric_limits<_Tp>::max();
574 const _Tp __lolim = std::pow(_Tp(5) * __min, _Tp(1)/_Tp(3));
575 const _Tp __uplim = _Tp(0.3L)
576 * std::pow(_Tp(0.2L) * __max, _Tp(1)/_Tp(3));
578 if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
579 std::__throw_domain_error(__N("Argument less than zero "
581 else if (__x + __y < __lolim || __x + __z < __lolim
582 || __y + __z < __lolim || __p < __lolim)
583 std::__throw_domain_error(__N("Argument too small "
587 const _Tp __c0 = _Tp(1) / _Tp(4);
588 const _Tp __c1 = _Tp(3) / _Tp(14);
589 const _Tp __c2 = _Tp(1) / _Tp(3);
590 const _Tp __c3 = _Tp(3) / _Tp(22);
591 const _Tp __c4 = _Tp(3) / _Tp(26);
597 _Tp __sigma = _Tp(0);
598 _Tp __power4 = _Tp(1);
600 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
601 const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));
604 _Tp __xndev, __yndev, __zndev, __pndev;
606 const unsigned int __max_iter = 100;
607 for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
609 __mu = (__xn + __yn + __zn + _Tp(2) * __pn) / _Tp(5);
610 __xndev = (__mu - __xn) / __mu;
611 __yndev = (__mu - __yn) / __mu;
612 __zndev = (__mu - __zn) / __mu;
613 __pndev = (__mu - __pn) / __mu;
614 _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
615 __epsilon = std::max(__epsilon, std::abs(__zndev));
616 __epsilon = std::max(__epsilon, std::abs(__pndev));
617 if (__epsilon < __errtol)
619 const _Tp __xnroot = std::sqrt(__xn);
620 const _Tp __ynroot = std::sqrt(__yn);
621 const _Tp __znroot = std::sqrt(__zn);
622 const _Tp __lambda = __xnroot * (__ynroot + __znroot)
623 + __ynroot * __znroot;
624 const _Tp __alpha1 = __pn * (__xnroot + __ynroot + __znroot)
625 + __xnroot * __ynroot * __znroot;
626 const _Tp __alpha2 = __alpha1 * __alpha1;
627 const _Tp __beta = __pn * (__pn + __lambda)
629 __sigma += __power4 * __ellint_rc(__alpha2, __beta);
631 __xn = __c0 * (__xn + __lambda);
632 __yn = __c0 * (__yn + __lambda);
633 __zn = __c0 * (__zn + __lambda);
634 __pn = __c0 * (__pn + __lambda);
637 // Note: __ea is an SPU badname.
638 _Tp __eaa = __xndev * (__yndev + __zndev) + __yndev * __zndev;
639 _Tp __eb = __xndev * __yndev * __zndev;
640 _Tp __ec = __pndev * __pndev;
641 _Tp __e2 = __eaa - _Tp(3) * __ec;
642 _Tp __e3 = __eb + _Tp(2) * __pndev * (__eaa - __ec);
643 _Tp __s1 = _Tp(1) + __e2 * (-__c1 + _Tp(3) * __c3 * __e2 / _Tp(4)
644 - _Tp(3) * __c4 * __e3 / _Tp(2));
645 _Tp __s2 = __eb * (__c2 / _Tp(2)
646 + __pndev * (-__c3 - __c3 + __pndev * __c4));
647 _Tp __s3 = __pndev * __eaa * (__c2 - __pndev * __c3)
648 - __c2 * __pndev * __ec;
650 return _Tp(3) * __sigma + __power4 * (__s1 + __s2 + __s3)
651 / (__mu * std::sqrt(__mu));
657 * @brief Return the complete elliptic integral of the third kind
658 * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ using the
659 * Carlson formulation.
661 * The complete elliptic integral of the third kind is defined as
663 * \Pi(k,\nu) = \int_0^{\pi/2}
665 * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}
668 * @param __k The argument of the elliptic function.
669 * @param __nu The second argument of the elliptic function.
670 * @return The complete elliptic function of the third kind.
672 template<typename _Tp>
674 __comp_ellint_3(_Tp __k, _Tp __nu)
677 if (__isnan(__k) || __isnan(__nu))
678 return std::numeric_limits<_Tp>::quiet_NaN();
679 else if (__nu == _Tp(1))
680 return std::numeric_limits<_Tp>::infinity();
681 else if (std::abs(__k) > _Tp(1))
682 std::__throw_domain_error(__N("Bad argument in __comp_ellint_3."));
685 const _Tp __kk = __k * __k;
687 return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
689 * __ellint_rj(_Tp(0), _Tp(1) - __kk, _Tp(1), _Tp(1) - __nu)
696 * @brief Return the incomplete elliptic integral of the third kind
697 * @f$ \Pi(k,\nu,\phi) @f$ using the Carlson formulation.
699 * The incomplete elliptic integral of the third kind is defined as
701 * \Pi(k,\nu,\phi) = \int_0^{\phi}
703 * {(1 - \nu \sin^2\theta)
704 * \sqrt{1 - k^2 \sin^2\theta}}
707 * @param __k The argument of the elliptic function.
708 * @param __nu The second argument of the elliptic function.
709 * @param __phi The integral limit argument of the elliptic function.
710 * @return The elliptic function of the third kind.
712 template<typename _Tp>
714 __ellint_3(_Tp __k, _Tp __nu, _Tp __phi)
717 if (__isnan(__k) || __isnan(__nu) || __isnan(__phi))
718 return std::numeric_limits<_Tp>::quiet_NaN();
719 else if (std::abs(__k) > _Tp(1))
720 std::__throw_domain_error(__N("Bad argument in __ellint_3."));
723 // Reduce phi to -pi/2 < phi < +pi/2.
724 const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
726 const _Tp __phi_red = __phi
727 - __n * __numeric_constants<_Tp>::__pi();
729 const _Tp __kk = __k * __k;
730 const _Tp __s = std::sin(__phi_red);
731 const _Tp __ss = __s * __s;
732 const _Tp __sss = __ss * __s;
733 const _Tp __c = std::cos(__phi_red);
734 const _Tp __cc = __c * __c;
737 * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
739 * __ellint_rj(__cc, _Tp(1) - __kk * __ss, _Tp(1),
740 _Tp(1) - __nu * __ss) / _Tp(3);
745 return __Pi + _Tp(2) * __n * __comp_ellint_3(__k, __nu);
748 } // namespace __detail
749 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
753 _GLIBCXX_END_NAMESPACE_VERSION
756 #endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC