1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006-2014 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 "special_function_util.h" 50 51 namespace std _GLIBCXX_VISIBILITY(default) 52 { 53 namespace tr1 54 { 55 // [5.2] Special functions 56 57 // Implementation-space details. 58 namespace __detail 59 { 60 _GLIBCXX_BEGIN_NAMESPACE_VERSION 61 62 /** 63 * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and 64 * @f$ K_\nu(x) @f$ and their first derivatives 65 * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. 66 * These four functions are computed together for numerical 67 * stability. 68 * 69 * @param __nu The order of the Bessel functions. 70 * @param __x The argument of the Bessel functions. 71 * @param __Inu The output regular modified Bessel function. 72 * @param __Knu The output irregular modified Bessel function. 73 * @param __Ipnu The output derivative of the regular 74 * modified Bessel function. 75 * @param __Kpnu The output derivative of the irregular 76 * modified Bessel function. 77 */ 78 template <typename _Tp> 79 void __bessel_ik(_Tp __nu,_Tp __x,_Tp & __Inu,_Tp & __Knu,_Tp & __Ipnu,_Tp & __Kpnu)80 __bessel_ik(_Tp __nu, _Tp __x, 81 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) 82 { 83 if (__x == _Tp(0)) 84 { 85 if (__nu == _Tp(0)) 86 { 87 __Inu = _Tp(1); 88 __Ipnu = _Tp(0); 89 } 90 else if (__nu == _Tp(1)) 91 { 92 __Inu = _Tp(0); 93 __Ipnu = _Tp(0.5L); 94 } 95 else 96 { 97 __Inu = _Tp(0); 98 __Ipnu = _Tp(0); 99 } 100 __Knu = std::numeric_limits<_Tp>::infinity(); 101 __Kpnu = -std::numeric_limits<_Tp>::infinity(); 102 return; 103 } 104 105 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 106 const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); 107 const int __max_iter = 15000; 108 const _Tp __x_min = _Tp(2); 109 110 const int __nl = static_cast<int>(__nu + _Tp(0.5L)); 111 112 const _Tp __mu = __nu - __nl; 113 const _Tp __mu2 = __mu * __mu; 114 const _Tp __xi = _Tp(1) / __x; 115 const _Tp __xi2 = _Tp(2) * __xi; 116 _Tp __h = __nu * __xi; 117 if ( __h < __fp_min ) 118 __h = __fp_min; 119 _Tp __b = __xi2 * __nu; 120 _Tp __d = _Tp(0); 121 _Tp __c = __h; 122 int __i; 123 for ( __i = 1; __i <= __max_iter; ++__i ) 124 { 125 __b += __xi2; 126 __d = _Tp(1) / (__b + __d); 127 __c = __b + _Tp(1) / __c; 128 const _Tp __del = __c * __d; 129 __h *= __del; 130 if (std::abs(__del - _Tp(1)) < __eps) 131 break; 132 } 133 if (__i > __max_iter) 134 std::__throw_runtime_error(__N("Argument x too large " 135 "in __bessel_ik; " 136 "try asymptotic expansion.")); 137 _Tp __Inul = __fp_min; 138 _Tp __Ipnul = __h * __Inul; 139 _Tp __Inul1 = __Inul; 140 _Tp __Ipnu1 = __Ipnul; 141 _Tp __fact = __nu * __xi; 142 for (int __l = __nl; __l >= 1; --__l) 143 { 144 const _Tp __Inutemp = __fact * __Inul + __Ipnul; 145 __fact -= __xi; 146 __Ipnul = __fact * __Inutemp + __Inul; 147 __Inul = __Inutemp; 148 } 149 _Tp __f = __Ipnul / __Inul; 150 _Tp __Kmu, __Knu1; 151 if (__x < __x_min) 152 { 153 const _Tp __x2 = __x / _Tp(2); 154 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 155 const _Tp __fact = (std::abs(__pimu) < __eps 156 ? _Tp(1) : __pimu / std::sin(__pimu)); 157 _Tp __d = -std::log(__x2); 158 _Tp __e = __mu * __d; 159 const _Tp __fact2 = (std::abs(__e) < __eps 160 ? _Tp(1) : std::sinh(__e) / __e); 161 _Tp __gam1, __gam2, __gampl, __gammi; 162 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 163 _Tp __ff = __fact 164 * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 165 _Tp __sum = __ff; 166 __e = std::exp(__e); 167 _Tp __p = __e / (_Tp(2) * __gampl); 168 _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); 169 _Tp __c = _Tp(1); 170 __d = __x2 * __x2; 171 _Tp __sum1 = __p; 172 int __i; 173 for (__i = 1; __i <= __max_iter; ++__i) 174 { 175 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 176 __c *= __d / __i; 177 __p /= __i - __mu; 178 __q /= __i + __mu; 179 const _Tp __del = __c * __ff; 180 __sum += __del; 181 const _Tp __del1 = __c * (__p - __i * __ff); 182 __sum1 += __del1; 183 if (std::abs(__del) < __eps * std::abs(__sum)) 184 break; 185 } 186 if (__i > __max_iter) 187 std::__throw_runtime_error(__N("Bessel k series failed to converge " 188 "in __bessel_ik.")); 189 __Kmu = __sum; 190 __Knu1 = __sum1 * __xi2; 191 } 192 else 193 { 194 _Tp __b = _Tp(2) * (_Tp(1) + __x); 195 _Tp __d = _Tp(1) / __b; 196 _Tp __delh = __d; 197 _Tp __h = __delh; 198 _Tp __q1 = _Tp(0); 199 _Tp __q2 = _Tp(1); 200 _Tp __a1 = _Tp(0.25L) - __mu2; 201 _Tp __q = __c = __a1; 202 _Tp __a = -__a1; 203 _Tp __s = _Tp(1) + __q * __delh; 204 int __i; 205 for (__i = 2; __i <= __max_iter; ++__i) 206 { 207 __a -= 2 * (__i - 1); 208 __c = -__a * __c / __i; 209 const _Tp __qnew = (__q1 - __b * __q2) / __a; 210 __q1 = __q2; 211 __q2 = __qnew; 212 __q += __c * __qnew; 213 __b += _Tp(2); 214 __d = _Tp(1) / (__b + __a * __d); 215 __delh = (__b * __d - _Tp(1)) * __delh; 216 __h += __delh; 217 const _Tp __dels = __q * __delh; 218 __s += __dels; 219 if ( std::abs(__dels / __s) < __eps ) 220 break; 221 } 222 if (__i > __max_iter) 223 std::__throw_runtime_error(__N("Steed's method failed " 224 "in __bessel_ik.")); 225 __h = __a1 * __h; 226 __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) 227 * std::exp(-__x) / __s; 228 __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; 229 } 230 231 _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; 232 _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); 233 __Inu = __Inumu * __Inul1 / __Inul; 234 __Ipnu = __Inumu * __Ipnu1 / __Inul; 235 for ( __i = 1; __i <= __nl; ++__i ) 236 { 237 const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; 238 __Kmu = __Knu1; 239 __Knu1 = __Knutemp; 240 } 241 __Knu = __Kmu; 242 __Kpnu = __nu * __xi * __Kmu - __Knu1; 243 244 return; 245 } 246 247 248 /** 249 * @brief Return the regular modified Bessel function of order 250 * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. 251 * 252 * The regular modified cylindrical Bessel function is: 253 * @f[ 254 * I_{\nu}(x) = \sum_{k=0}^{\infty} 255 * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 256 * @f] 257 * 258 * @param __nu The order of the regular modified Bessel function. 259 * @param __x The argument of the regular modified Bessel function. 260 * @return The output regular modified Bessel function. 261 */ 262 template<typename _Tp> 263 _Tp __cyl_bessel_i(_Tp __nu,_Tp __x)264 __cyl_bessel_i(_Tp __nu, _Tp __x) 265 { 266 if (__nu < _Tp(0) || __x < _Tp(0)) 267 std::__throw_domain_error(__N("Bad argument " 268 "in __cyl_bessel_i.")); 269 else if (__isnan(__nu) || __isnan(__x)) 270 return std::numeric_limits<_Tp>::quiet_NaN(); 271 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 272 return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); 273 else 274 { 275 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 276 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 277 return __I_nu; 278 } 279 } 280 281 282 /** 283 * @brief Return the irregular modified Bessel function 284 * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. 285 * 286 * The irregular modified Bessel function is defined by: 287 * @f[ 288 * K_{\nu}(x) = \frac{\pi}{2} 289 * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} 290 * @f] 291 * where for integral \f$ \nu = n \f$ a limit is taken: 292 * \f$ lim_{\nu \to n} \f$. 293 * 294 * @param __nu The order of the irregular modified Bessel function. 295 * @param __x The argument of the irregular modified Bessel function. 296 * @return The output irregular modified Bessel function. 297 */ 298 template<typename _Tp> 299 _Tp __cyl_bessel_k(_Tp __nu,_Tp __x)300 __cyl_bessel_k(_Tp __nu, _Tp __x) 301 { 302 if (__nu < _Tp(0) || __x < _Tp(0)) 303 std::__throw_domain_error(__N("Bad argument " 304 "in __cyl_bessel_k.")); 305 else if (__isnan(__nu) || __isnan(__x)) 306 return std::numeric_limits<_Tp>::quiet_NaN(); 307 else 308 { 309 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 310 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 311 return __K_nu; 312 } 313 } 314 315 316 /** 317 * @brief Compute the spherical modified Bessel functions 318 * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first 319 * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ 320 * respectively. 321 * 322 * @param __n The order of the modified spherical Bessel function. 323 * @param __x The argument of the modified spherical Bessel function. 324 * @param __i_n The output regular modified spherical Bessel function. 325 * @param __k_n The output irregular modified spherical 326 * Bessel function. 327 * @param __ip_n The output derivative of the regular modified 328 * spherical Bessel function. 329 * @param __kp_n The output derivative of the irregular modified 330 * spherical Bessel function. 331 */ 332 template <typename _Tp> 333 void __sph_bessel_ik(unsigned int __n,_Tp __x,_Tp & __i_n,_Tp & __k_n,_Tp & __ip_n,_Tp & __kp_n)334 __sph_bessel_ik(unsigned int __n, _Tp __x, 335 _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) 336 { 337 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 338 339 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 340 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 341 342 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 343 / std::sqrt(__x); 344 345 __i_n = __factor * __I_nu; 346 __k_n = __factor * __K_nu; 347 __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); 348 __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); 349 350 return; 351 } 352 353 354 /** 355 * @brief Compute the Airy functions 356 * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first 357 * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ 358 * respectively. 359 * 360 * @param __x The argument of the Airy functions. 361 * @param __Ai The output Airy function of the first kind. 362 * @param __Bi The output Airy function of the second kind. 363 * @param __Aip The output derivative of the Airy function 364 * of the first kind. 365 * @param __Bip The output derivative of the Airy function 366 * of the second kind. 367 */ 368 template <typename _Tp> 369 void __airy(_Tp __x,_Tp & __Ai,_Tp & __Bi,_Tp & __Aip,_Tp & __Bip)370 __airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) 371 { 372 const _Tp __absx = std::abs(__x); 373 const _Tp __rootx = std::sqrt(__absx); 374 const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); 375 376 if (__x > _Tp(0)) 377 { 378 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 379 380 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 381 __Ai = __rootx * __K_nu 382 / (__numeric_constants<_Tp>::__sqrt3() 383 * __numeric_constants<_Tp>::__pi()); 384 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 385 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 386 387 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 388 __Aip = -__x * __K_nu 389 / (__numeric_constants<_Tp>::__sqrt3() 390 * __numeric_constants<_Tp>::__pi()); 391 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 392 + _Tp(2) * __I_nu 393 / __numeric_constants<_Tp>::__sqrt3()); 394 } 395 else if (__x < _Tp(0)) 396 { 397 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 398 399 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 400 __Ai = __rootx * (__J_nu 401 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 402 __Bi = -__rootx * (__N_nu 403 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 404 405 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 406 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 407 + __J_nu) / _Tp(2); 408 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 409 - __N_nu) / _Tp(2); 410 } 411 else 412 { 413 // Reference: 414 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 415 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 416 __Ai = _Tp(0.35502805388781723926L); 417 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 418 419 // Reference: 420 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 421 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 422 __Aip = -_Tp(0.25881940379280679840L); 423 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 424 } 425 426 return; 427 } 428 429 _GLIBCXX_END_NAMESPACE_VERSION 430 } // namespace std::tr1::__detail 431 } 432 } 433 434 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 435