150276Speter// Special functions -*- C++ -*- 2262685Sdelphij 350276Speter// Copyright (C) 2006-2020 Free Software Foundation, Inc. 450276Speter// 550276Speter// This file is part of the GNU ISO C++ Library. This library is free 650276Speter// software; you can redistribute it and/or modify it under the 750276Speter// terms of the GNU General Public License as published by the 850276Speter// Free Software Foundation; either version 3, or (at your option) 950276Speter// any later version. 1050276Speter// 1150276Speter// This library is distributed in the hope that it will be useful, 1250276Speter// but WITHOUT ANY WARRANTY; without even the implied warranty of 1350276Speter// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 1450276Speter// GNU General Public License for more details. 1550276Speter// 1650276Speter// Under Section 7 of GPL version 3, you are granted additional 1750276Speter// permissions described in the GCC Runtime Library Exception, version 1850276Speter// 3.1, as published by the Free Software Foundation. 1950276Speter 2050276Speter// You should have received a copy of the GNU General Public License and 2150276Speter// a copy of the GCC Runtime Library Exception along with this program; 2250276Speter// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 2350276Speter// <http://www.gnu.org/licenses/>. 2450276Speter 2550276Speter/** @file tr1/modified_bessel_func.tcc 2650276Speter * This is an internal header file, included by other library headers. 2750276Speter * Do not attempt to use it directly. @headername{tr1/cmath} 2850276Speter */ 2950276Speter 30166124Srafan// 3150276Speter// ISO C++ 14882 TR1: 5.2 Special functions 3250276Speter// 33262685Sdelphij 3450276Speter// Written by Edward Smith-Rowland. 3550276Speter// 3650276Speter// References: 3750276Speter// (1) Handbook of Mathematical Functions, 3850276Speter// Ed. Milton Abramowitz and Irene A. Stegun, 3950276Speter// Dover Publications, 4050276Speter// Section 9, pp. 355-434, Section 10 pp. 435-478 4150276Speter// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 4266963Speter// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 4350276Speter// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 4450276Speter// 2nd ed, pp. 246-249. 45262685Sdelphij 4650276Speter#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 4750276Speter#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 4850276Speter 4950276Speter#include <tr1/special_function_util.h> 5050276Speter 5150276Speternamespace std _GLIBCXX_VISIBILITY(default) 5250276Speter{ 5350276Speter_GLIBCXX_BEGIN_NAMESPACE_VERSION 5450276Speter 5550276Speter#if _GLIBCXX_USE_STD_SPEC_FUNCS 5650276Speter#elif defined(_GLIBCXX_TR1_CMATH) 5750276Speternamespace tr1 5850276Speter{ 5950276Speter#else 6050276Speter# error do not include this header directly, use <cmath> or <tr1/cmath> 6150276Speter#endif 6250276Speter // [5.2] Special functions 6350276Speter 6450276Speter // Implementation-space details. 6550276Speter namespace __detail 6650276Speter { 6750276Speter /** 6850276Speter * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and 6950276Speter * @f$ K_\nu(x) @f$ and their first derivatives 70166124Srafan * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. 71166124Srafan * These four functions are computed together for numerical 72166124Srafan * stability. 73166124Srafan * 74166124Srafan * @param __nu The order of the Bessel functions. 75166124Srafan * @param __x The argument of the Bessel functions. 76166124Srafan * @param __Inu The output regular modified Bessel function. 77166124Srafan * @param __Knu The output irregular modified Bessel function. 78166124Srafan * @param __Ipnu The output derivative of the regular 7950276Speter * modified Bessel function. 80166124Srafan * @param __Kpnu The output derivative of the irregular 8150276Speter * modified Bessel function. 8250276Speter */ 8350276Speter template <typename _Tp> 8450276Speter void 8550276Speter __bessel_ik(_Tp __nu, _Tp __x, 8650276Speter _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) 8750276Speter { 8850276Speter if (__x == _Tp(0)) 8950276Speter { 9050276Speter if (__nu == _Tp(0)) 9150276Speter { 9250276Speter __Inu = _Tp(1); 93262685Sdelphij __Ipnu = _Tp(0); 94262685Sdelphij } 95262685Sdelphij else if (__nu == _Tp(1)) 96262685Sdelphij { 97262685Sdelphij __Inu = _Tp(0); 98262685Sdelphij __Ipnu = _Tp(0.5L); 99262685Sdelphij } 100262685Sdelphij else 101184989Srafan { 10250276Speter __Inu = _Tp(0); 10350276Speter __Ipnu = _Tp(0); 10450276Speter } 10550276Speter __Knu = std::numeric_limits<_Tp>::infinity(); 10650276Speter __Kpnu = -std::numeric_limits<_Tp>::infinity(); 10750276Speter return; 10850276Speter } 10950276Speter 110262685Sdelphij const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 11150276Speter const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); 11250276Speter const int __max_iter = 15000; 11350276Speter const _Tp __x_min = _Tp(2); 11450276Speter 11550276Speter const int __nl = static_cast<int>(__nu + _Tp(0.5L)); 11650276Speter 11750276Speter const _Tp __mu = __nu - __nl; 11850276Speter const _Tp __mu2 = __mu * __mu; 11950276Speter const _Tp __xi = _Tp(1) / __x; 120262685Sdelphij const _Tp __xi2 = _Tp(2) * __xi; 12150276Speter _Tp __h = __nu * __xi; 122166124Srafan if ( __h < __fp_min ) 123174993Srafan __h = __fp_min; 124262685Sdelphij _Tp __b = __xi2 * __nu; 12550276Speter _Tp __d = _Tp(0); 126174993Srafan _Tp __c = __h; 127174993Srafan int __i; 128174993Srafan for ( __i = 1; __i <= __max_iter; ++__i ) 129174993Srafan { 130174993Srafan __b += __xi2; 131174993Srafan __d = _Tp(1) / (__b + __d); 132174993Srafan __c = __b + _Tp(1) / __c; 133174993Srafan const _Tp __del = __c * __d; 13450276Speter __h *= __del; 13550276Speter if (std::abs(__del - _Tp(1)) < __eps) 13650276Speter break; 13750276Speter } 13850276Speter if (__i > __max_iter) 13950276Speter std::__throw_runtime_error(__N("Argument x too large " 14050276Speter "in __bessel_ik; " 14150276Speter "try asymptotic expansion.")); 14250276Speter _Tp __Inul = __fp_min; 14350276Speter _Tp __Ipnul = __h * __Inul; 14450276Speter _Tp __Inul1 = __Inul; 14550276Speter _Tp __Ipnu1 = __Ipnul; 14650276Speter _Tp __fact = __nu * __xi; 14750276Speter for (int __l = __nl; __l >= 1; --__l) 14850276Speter { 14950276Speter const _Tp __Inutemp = __fact * __Inul + __Ipnul; 15050276Speter __fact -= __xi; 15150276Speter __Ipnul = __fact * __Inutemp + __Inul; 15262449Speter __Inul = __Inutemp; 15362449Speter } 15462449Speter _Tp __f = __Ipnul / __Inul; 15562449Speter _Tp __Kmu, __Knu1; 15662449Speter if (__x < __x_min) 15762449Speter { 15862449Speter const _Tp __x2 = __x / _Tp(2); 15962449Speter const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 16062449Speter const _Tp __fact = (std::abs(__pimu) < __eps 16162449Speter ? _Tp(1) : __pimu / std::sin(__pimu)); 16262449Speter _Tp __d = -std::log(__x2); 16362449Speter _Tp __e = __mu * __d; 16450276Speter const _Tp __fact2 = (std::abs(__e) < __eps 16550276Speter ? _Tp(1) : std::sinh(__e) / __e); 16650276Speter _Tp __gam1, __gam2, __gampl, __gammi; 16750276Speter __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 16850276Speter _Tp __ff = __fact 16950276Speter * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 17050276Speter _Tp __sum = __ff; 17150276Speter __e = std::exp(__e); 17250276Speter _Tp __p = __e / (_Tp(2) * __gampl); 17350276Speter _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); 17450276Speter _Tp __c = _Tp(1); 17550276Speter __d = __x2 * __x2; 17650276Speter _Tp __sum1 = __p; 17750276Speter int __i; 17850276Speter for (__i = 1; __i <= __max_iter; ++__i) 17950276Speter { 18050276Speter __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 18150276Speter __c *= __d / __i; 18250276Speter __p /= __i - __mu; 18350276Speter __q /= __i + __mu; 184262685Sdelphij const _Tp __del = __c * __ff; 18550276Speter __sum += __del; 18650276Speter const _Tp __del1 = __c * (__p - __i * __ff); 18750276Speter __sum1 += __del1; 18850276Speter if (std::abs(__del) < __eps * std::abs(__sum)) 18950276Speter break; 19050276Speter } 19166963Speter if (__i > __max_iter) 19250276Speter std::__throw_runtime_error(__N("Bessel k series failed to converge " 19350276Speter "in __bessel_ik.")); 19497049Speter __Kmu = __sum; 19550276Speter __Knu1 = __sum1 * __xi2; 196262685Sdelphij } 19750276Speter else 19850276Speter { 19966963Speter _Tp __b = _Tp(2) * (_Tp(1) + __x); 20097049Speter _Tp __d = _Tp(1) / __b; 20176726Speter _Tp __delh = __d; 20266963Speter _Tp __h = __delh; 203 _Tp __q1 = _Tp(0); 204 _Tp __q2 = _Tp(1); 205 _Tp __a1 = _Tp(0.25L) - __mu2; 206 _Tp __q = __c = __a1; 207 _Tp __a = -__a1; 208 _Tp __s = _Tp(1) + __q * __delh; 209 int __i; 210 for (__i = 2; __i <= __max_iter; ++__i) 211 { 212 __a -= 2 * (__i - 1); 213 __c = -__a * __c / __i; 214 const _Tp __qnew = (__q1 - __b * __q2) / __a; 215 __q1 = __q2; 216 __q2 = __qnew; 217 __q += __c * __qnew; 218 __b += _Tp(2); 219 __d = _Tp(1) / (__b + __a * __d); 220 __delh = (__b * __d - _Tp(1)) * __delh; 221 __h += __delh; 222 const _Tp __dels = __q * __delh; 223 __s += __dels; 224 if ( std::abs(__dels / __s) < __eps ) 225 break; 226 } 227 if (__i > __max_iter) 228 std::__throw_runtime_error(__N("Steed's method failed " 229 "in __bessel_ik.")); 230 __h = __a1 * __h; 231 __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) 232 * std::exp(-__x) / __s; 233 __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; 234 } 235 236 _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; 237 _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); 238 __Inu = __Inumu * __Inul1 / __Inul; 239 __Ipnu = __Inumu * __Ipnu1 / __Inul; 240 for ( __i = 1; __i <= __nl; ++__i ) 241 { 242 const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; 243 __Kmu = __Knu1; 244 __Knu1 = __Knutemp; 245 } 246 __Knu = __Kmu; 247 __Kpnu = __nu * __xi * __Kmu - __Knu1; 248 249 return; 250 } 251 252 253 /** 254 * @brief Return the regular modified Bessel function of order 255 * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. 256 * 257 * The regular modified cylindrical Bessel function is: 258 * @f[ 259 * I_{\nu}(x) = \sum_{k=0}^{\infty} 260 * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 261 * @f] 262 * 263 * @param __nu The order of the regular modified Bessel function. 264 * @param __x The argument of the regular modified Bessel function. 265 * @return The output regular modified Bessel function. 266 */ 267 template<typename _Tp> 268 _Tp 269 __cyl_bessel_i(_Tp __nu, _Tp __x) 270 { 271 if (__nu < _Tp(0) || __x < _Tp(0)) 272 std::__throw_domain_error(__N("Bad argument " 273 "in __cyl_bessel_i.")); 274 else if (__isnan(__nu) || __isnan(__x)) 275 return std::numeric_limits<_Tp>::quiet_NaN(); 276 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 277 return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); 278 else 279 { 280 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 281 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 282 return __I_nu; 283 } 284 } 285 286 287 /** 288 * @brief Return the irregular modified Bessel function 289 * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. 290 * 291 * The irregular modified Bessel function is defined by: 292 * @f[ 293 * K_{\nu}(x) = \frac{\pi}{2} 294 * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} 295 * @f] 296 * where for integral \f$ \nu = n \f$ a limit is taken: 297 * \f$ lim_{\nu \to n} \f$. 298 * 299 * @param __nu The order of the irregular modified Bessel function. 300 * @param __x The argument of the irregular modified Bessel function. 301 * @return The output irregular modified Bessel function. 302 */ 303 template<typename _Tp> 304 _Tp 305 __cyl_bessel_k(_Tp __nu, _Tp __x) 306 { 307 if (__nu < _Tp(0) || __x < _Tp(0)) 308 std::__throw_domain_error(__N("Bad argument " 309 "in __cyl_bessel_k.")); 310 else if (__isnan(__nu) || __isnan(__x)) 311 return std::numeric_limits<_Tp>::quiet_NaN(); 312 else 313 { 314 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 315 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 316 return __K_nu; 317 } 318 } 319 320 321 /** 322 * @brief Compute the spherical modified Bessel functions 323 * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first 324 * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ 325 * respectively. 326 * 327 * @param __n The order of the modified spherical Bessel function. 328 * @param __x The argument of the modified spherical Bessel function. 329 * @param __i_n The output regular modified spherical Bessel function. 330 * @param __k_n The output irregular modified spherical 331 * Bessel function. 332 * @param __ip_n The output derivative of the regular modified 333 * spherical Bessel function. 334 * @param __kp_n The output derivative of the irregular modified 335 * spherical Bessel function. 336 */ 337 template <typename _Tp> 338 void 339 __sph_bessel_ik(unsigned int __n, _Tp __x, 340 _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) 341 { 342 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 343 344 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 345 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 346 347 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 348 / std::sqrt(__x); 349 350 __i_n = __factor * __I_nu; 351 __k_n = __factor * __K_nu; 352 __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); 353 __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); 354 355 return; 356 } 357 358 359 /** 360 * @brief Compute the Airy functions 361 * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first 362 * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ 363 * respectively. 364 * 365 * @param __x The argument of the Airy functions. 366 * @param __Ai The output Airy function of the first kind. 367 * @param __Bi The output Airy function of the second kind. 368 * @param __Aip The output derivative of the Airy function 369 * of the first kind. 370 * @param __Bip The output derivative of the Airy function 371 * of the second kind. 372 */ 373 template <typename _Tp> 374 void 375 __airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) 376 { 377 const _Tp __absx = std::abs(__x); 378 const _Tp __rootx = std::sqrt(__absx); 379 const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); 380 const _Tp _S_NaN = std::numeric_limits<_Tp>::quiet_NaN(); 381 const _Tp _S_inf = std::numeric_limits<_Tp>::infinity(); 382 383 if (__isnan(__x)) 384 __Bip = __Aip = __Bi = __Ai = std::numeric_limits<_Tp>::quiet_NaN(); 385 else if (__z == _S_inf) 386 { 387 __Aip = __Ai = _Tp(0); 388 __Bip = __Bi = _S_inf; 389 } 390 else if (__z == -_S_inf) 391 __Bip = __Aip = __Bi = __Ai = _Tp(0); 392 else if (__x > _Tp(0)) 393 { 394 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 395 396 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 397 __Ai = __rootx * __K_nu 398 / (__numeric_constants<_Tp>::__sqrt3() 399 * __numeric_constants<_Tp>::__pi()); 400 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 401 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 402 403 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 404 __Aip = -__x * __K_nu 405 / (__numeric_constants<_Tp>::__sqrt3() 406 * __numeric_constants<_Tp>::__pi()); 407 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 408 + _Tp(2) * __I_nu 409 / __numeric_constants<_Tp>::__sqrt3()); 410 } 411 else if (__x < _Tp(0)) 412 { 413 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 414 415 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 416 __Ai = __rootx * (__J_nu 417 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 418 __Bi = -__rootx * (__N_nu 419 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 420 421 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 422 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 423 + __J_nu) / _Tp(2); 424 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 425 - __N_nu) / _Tp(2); 426 } 427 else 428 { 429 // Reference: 430 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 431 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 432 __Ai = _Tp(0.35502805388781723926L); 433 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 434 435 // Reference: 436 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 437 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 438 __Aip = -_Tp(0.25881940379280679840L); 439 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 440 } 441 442 return; 443 } 444 } // namespace __detail 445#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 446} // namespace tr1 447#endif 448 449_GLIBCXX_END_NAMESPACE_VERSION 450} 451 452#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 453