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1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006-2025 Free Software Foundation, Inc. 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 8 // Free Software Foundation; either version 3, or (at your option) 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 // 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/>. 24 25 /** @file tr1/modified_bessel_func.tcc 26 * This is an internal header file, included by other library headers. 27 * Do not attempt to use it directly. @headername{tr1/cmath} 28 */ 29 30 // 31 // ISO C++ 14882 TR1: 5.2 Special functions 32 // 33 34 // Written by Edward Smith-Rowland. 35 // 36 // References: 37 // (1) Handbook of Mathematical Functions, 38 // Ed. Milton Abramowitz and Irene A. Stegun, 39 // Dover Publications, 40 // Section 9, pp. 355-434, Section 10 pp. 435-478 41 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 42 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 43 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 44 // 2nd ed, pp. 246-249. 45 46 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 47 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 48 49 #include <tr1/special_function_util.h> 50 51 namespace std _GLIBCXX_VISIBILITY(default) 52 { 53 _GLIBCXX_BEGIN_NAMESPACE_VERSION 54 55 #if _GLIBCXX_USE_STD_SPEC_FUNCS 56 #elif defined(_GLIBCXX_TR1_CMATH) 57 namespace tr1 58 { 59 #else 60 # error do not include this header directly, use <cmath> or <tr1/cmath> 61 #endif 62 // [5.2] Special functions 63 64 // Implementation-space details. 65 namespace __detail 66 { 67 /** 68 * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and 69 * @f$ K_\nu(x) @f$ and their first derivatives 70 * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. 71 * These four functions are computed together for numerical 72 * stability. 73 * 74 * @param __nu The order of the Bessel functions. 75 * @param __x The argument of the Bessel functions. 76 * @param __Inu The output regular modified Bessel function. 77 * @param __Knu The output irregular modified Bessel function. 78 * @param __Ipnu The output derivative of the regular 79 * modified Bessel function. 80 * @param __Kpnu The output derivative of the irregular 81 * modified Bessel function. 82 */ 83 template <typename _Tp> 84 void 85 __bessel_ik(_Tp __nu, _Tp __x, 86 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) 87 { 88 if (__x == _Tp(0)) 89 { 90 if (__nu == _Tp(0)) 91 { 92 __Inu = _Tp(1); 93 __Ipnu = _Tp(0); 94 } 95 else if (__nu == _Tp(1)) 96 { 97 __Inu = _Tp(0); 98 __Ipnu = _Tp(0.5L); 99 } 100 else 101 { 102 __Inu = _Tp(0); 103 __Ipnu = _Tp(0); 104 } 105 __Knu = std::numeric_limits<_Tp>::infinity(); 106 __Kpnu = -std::numeric_limits<_Tp>::infinity(); 107 return; 108 } 109 110 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 111 const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); 112 const int __max_iter = 15000; 113 const _Tp __x_min = _Tp(2); 114 115 const int __nl = static_cast<int>(__nu + _Tp(0.5L)); 116 117 const _Tp __mu = __nu - __nl; 118 const _Tp __mu2 = __mu * __mu; 119 const _Tp __xi = _Tp(1) / __x; 120 const _Tp __xi2 = _Tp(2) * __xi; 121 _Tp __h = __nu * __xi; 122 if ( __h < __fp_min ) 123 __h = __fp_min; 124 _Tp __b = __xi2 * __nu; 125 _Tp __d = _Tp(0); 126 _Tp __c = __h; 127 int __i; 128 for ( __i = 1; __i <= __max_iter; ++__i ) 129 { 130 __b += __xi2; 131 __d = _Tp(1) / (__b + __d); 132 __c = __b + _Tp(1) / __c; 133 const _Tp __del = __c * __d; 134 __h *= __del; 135 if (std::abs(__del - _Tp(1)) < __eps) 136 break; 137 } 138 if (__i > __max_iter) 139 std::__throw_runtime_error(__N("Argument x too large " 140 "in __bessel_ik; " 141 "try asymptotic expansion.")); 142 _Tp __Inul = __fp_min; 143 _Tp __Ipnul = __h * __Inul; 144 _Tp __Inul1 = __Inul; 145 _Tp __Ipnu1 = __Ipnul; 146 _Tp __fact = __nu * __xi; 147 for (int __l = __nl; __l >= 1; --__l) 148 { 149 const _Tp __Inutemp = __fact * __Inul + __Ipnul; 150 __fact -= __xi; 151 __Ipnul = __fact * __Inutemp + __Inul; 152 __Inul = __Inutemp; 153 } 154 _Tp __f = __Ipnul / __Inul; 155 _Tp __Kmu, __Knu1; 156 if (__x < __x_min) 157 { 158 const _Tp __x2 = __x / _Tp(2); 159 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 160 const _Tp __fact = (std::abs(__pimu) < __eps 161 ? _Tp(1) : __pimu / std::sin(__pimu)); 162 _Tp __d = -std::log(__x2); 163 _Tp __e = __mu * __d; 164 const _Tp __fact2 = (std::abs(__e) < __eps 165 ? _Tp(1) : std::sinh(__e) / __e); 166 _Tp __gam1, __gam2, __gampl, __gammi; 167 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 168 _Tp __ff = __fact 169 * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 170 _Tp __sum = __ff; 171 __e = std::exp(__e); 172 _Tp __p = __e / (_Tp(2) * __gampl); 173 _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); 174 _Tp __c = _Tp(1); 175 __d = __x2 * __x2; 176 _Tp __sum1 = __p; 177 int __i; 178 for (__i = 1; __i <= __max_iter; ++__i) 179 { 180 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 181 __c *= __d / __i; 182 __p /= __i - __mu; 183 __q /= __i + __mu; 184 const _Tp __del = __c * __ff; 185 __sum += __del; 186 const _Tp __del1 = __c * (__p - __i * __ff); 187 __sum1 += __del1; 188 if (std::abs(__del) < __eps * std::abs(__sum)) 189 break; 190 } 191 if (__i > __max_iter) 192 std::__throw_runtime_error(__N("Bessel k series failed to converge " 193 "in __bessel_ik.")); 194 __Kmu = __sum; 195 __Knu1 = __sum1 * __xi2; 196 } 197 else 198 { 199 _Tp __b = _Tp(2) * (_Tp(1) + __x); 200 _Tp __d = _Tp(1) / __b; 201 _Tp __delh = __d; 202 _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_inf = std::numeric_limits<_Tp>::infinity(); 381 382 if (__isnan(__x)) 383 __Bip = __Aip = __Bi = __Ai = std::numeric_limits<_Tp>::quiet_NaN(); 384 else if (__z == _S_inf) 385 { 386 __Aip = __Ai = _Tp(0); 387 __Bip = __Bi = _S_inf; 388 } 389 else if (__z == -_S_inf) 390 __Bip = __Aip = __Bi = __Ai = _Tp(0); 391 else if (__x > _Tp(0)) 392 { 393 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 394 395 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 396 __Ai = __rootx * __K_nu 397 / (__numeric_constants<_Tp>::__sqrt3() 398 * __numeric_constants<_Tp>::__pi()); 399 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 400 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 401 402 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 403 __Aip = -__x * __K_nu 404 / (__numeric_constants<_Tp>::__sqrt3() 405 * __numeric_constants<_Tp>::__pi()); 406 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 407 + _Tp(2) * __I_nu 408 / __numeric_constants<_Tp>::__sqrt3()); 409 } 410 else if (__x < _Tp(0)) 411 { 412 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 413 414 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 415 __Ai = __rootx * (__J_nu 416 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 417 __Bi = -__rootx * (__N_nu 418 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 419 420 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 421 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 422 + __J_nu) / _Tp(2); 423 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 424 - __N_nu) / _Tp(2); 425 } 426 else 427 { 428 // Reference: 429 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 430 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 431 __Ai = _Tp(0.35502805388781723926L); 432 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 433 434 // Reference: 435 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 436 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 437 __Aip = -_Tp(0.25881940379280679840L); 438 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 439 } 440 441 return; 442 } 443 } // namespace __detail 444 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 445 } // namespace tr1 446 #endif 447 448 _GLIBCXX_END_NAMESPACE_VERSION 449 } 450 451 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
