1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006, 2007, 2008, 2009 4 // Free Software Foundation, Inc. 5 // 6 // This file is part of the GNU ISO C++ Library. This library is free 7 // software; you can redistribute it and/or modify it under the 8 // terms of the GNU General Public License as published by the 9 // Free Software Foundation; either version 3, or (at your option) 10 // any later version. 11 // 12 // This library is distributed in the hope that it will be useful, 13 // but WITHOUT ANY WARRANTY; without even the implied warranty of 14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 // GNU General Public License for more details. 16 // 17 // Under Section 7 of GPL version 3, you are granted additional 18 // permissions described in the GCC Runtime Library Exception, version 19 // 3.1, as published by the Free Software Foundation. 20 21 // You should have received a copy of the GNU General Public License and 22 // a copy of the GCC Runtime Library Exception along with this program; 23 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 24 // <http://www.gnu.org/licenses/>. 25 26 /** @file tr1/modified_bessel_func.tcc 27 * This is an internal header file, included by other library headers. 28 * You should not attempt to use it directly. 29 */ 30 31 // 32 // ISO C++ 14882 TR1: 5.2 Special functions 33 // 34 35 // Written by Edward Smith-Rowland. 36 // 37 // References: 38 // (1) Handbook of Mathematical Functions, 39 // Ed. Milton Abramowitz and Irene A. Stegun, 40 // Dover Publications, 41 // Section 9, pp. 355-434, Section 10 pp. 435-478 42 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 43 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 44 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 45 // 2nd ed, pp. 246-249. 46 47 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 48 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 49 50 #include "special_function_util.h" 51 52 namespace std 53 { 54 namespace tr1 55 { 56 57 // [5.2] Special functions 58 59 // Implementation-space details. 60 namespace __detail 61 { 62 63 /** 64 * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and 65 * @f$ K_\nu(x) @f$ and their first derivatives 66 * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. 67 * These four functions are computed together for numerical 68 * stability. 69 * 70 * @param __nu The order of the Bessel functions. 71 * @param __x The argument of the Bessel functions. 72 * @param __Inu The output regular modified Bessel function. 73 * @param __Knu The output irregular modified Bessel function. 74 * @param __Ipnu The output derivative of the regular 75 * modified Bessel function. 76 * @param __Kpnu The output derivative of the irregular 77 * modified Bessel function. 78 */ 79 template <typename _Tp> 80 void 81 __bessel_ik(const _Tp __nu, const _Tp __x, 82 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) 83 { 84 if (__x == _Tp(0)) 85 { 86 if (__nu == _Tp(0)) 87 { 88 __Inu = _Tp(1); 89 __Ipnu = _Tp(0); 90 } 91 else if (__nu == _Tp(1)) 92 { 93 __Inu = _Tp(0); 94 __Ipnu = _Tp(0.5L); 95 } 96 else 97 { 98 __Inu = _Tp(0); 99 __Ipnu = _Tp(0); 100 } 101 __Knu = std::numeric_limits<_Tp>::infinity(); 102 __Kpnu = -std::numeric_limits<_Tp>::infinity(); 103 return; 104 } 105 106 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 107 const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); 108 const int __max_iter = 15000; 109 const _Tp __x_min = _Tp(2); 110 111 const int __nl = static_cast<int>(__nu + _Tp(0.5L)); 112 113 const _Tp __mu = __nu - __nl; 114 const _Tp __mu2 = __mu * __mu; 115 const _Tp __xi = _Tp(1) / __x; 116 const _Tp __xi2 = _Tp(2) * __xi; 117 _Tp __h = __nu * __xi; 118 if ( __h < __fp_min ) 119 __h = __fp_min; 120 _Tp __b = __xi2 * __nu; 121 _Tp __d = _Tp(0); 122 _Tp __c = __h; 123 int __i; 124 for ( __i = 1; __i <= __max_iter; ++__i ) 125 { 126 __b += __xi2; 127 __d = _Tp(1) / (__b + __d); 128 __c = __b + _Tp(1) / __c; 129 const _Tp __del = __c * __d; 130 __h *= __del; 131 if (std::abs(__del - _Tp(1)) < __eps) 132 break; 133 } 134 if (__i > __max_iter) 135 std::__throw_runtime_error(__N("Argument x too large " 136 "in __bessel_jn; " 137 "try asymptotic expansion.")); 138 _Tp __Inul = __fp_min; 139 _Tp __Ipnul = __h * __Inul; 140 _Tp __Inul1 = __Inul; 141 _Tp __Ipnu1 = __Ipnul; 142 _Tp __fact = __nu * __xi; 143 for (int __l = __nl; __l >= 1; --__l) 144 { 145 const _Tp __Inutemp = __fact * __Inul + __Ipnul; 146 __fact -= __xi; 147 __Ipnul = __fact * __Inutemp + __Inul; 148 __Inul = __Inutemp; 149 } 150 _Tp __f = __Ipnul / __Inul; 151 _Tp __Kmu, __Knu1; 152 if (__x < __x_min) 153 { 154 const _Tp __x2 = __x / _Tp(2); 155 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 156 const _Tp __fact = (std::abs(__pimu) < __eps 157 ? _Tp(1) : __pimu / std::sin(__pimu)); 158 _Tp __d = -std::log(__x2); 159 _Tp __e = __mu * __d; 160 const _Tp __fact2 = (std::abs(__e) < __eps 161 ? _Tp(1) : std::sinh(__e) / __e); 162 _Tp __gam1, __gam2, __gampl, __gammi; 163 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 164 _Tp __ff = __fact 165 * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 166 _Tp __sum = __ff; 167 __e = std::exp(__e); 168 _Tp __p = __e / (_Tp(2) * __gampl); 169 _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); 170 _Tp __c = _Tp(1); 171 __d = __x2 * __x2; 172 _Tp __sum1 = __p; 173 int __i; 174 for (__i = 1; __i <= __max_iter; ++__i) 175 { 176 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 177 __c *= __d / __i; 178 __p /= __i - __mu; 179 __q /= __i + __mu; 180 const _Tp __del = __c * __ff; 181 __sum += __del; 182 const _Tp __del1 = __c * (__p - __i * __ff); 183 __sum1 += __del1; 184 if (std::abs(__del) < __eps * std::abs(__sum)) 185 break; 186 } 187 if (__i > __max_iter) 188 std::__throw_runtime_error(__N("Bessel k series failed to converge " 189 "in __bessel_jn.")); 190 __Kmu = __sum; 191 __Knu1 = __sum1 * __xi2; 192 } 193 else 194 { 195 _Tp __b = _Tp(2) * (_Tp(1) + __x); 196 _Tp __d = _Tp(1) / __b; 197 _Tp __delh = __d; 198 _Tp __h = __delh; 199 _Tp __q1 = _Tp(0); 200 _Tp __q2 = _Tp(1); 201 _Tp __a1 = _Tp(0.25L) - __mu2; 202 _Tp __q = __c = __a1; 203 _Tp __a = -__a1; 204 _Tp __s = _Tp(1) + __q * __delh; 205 int __i; 206 for (__i = 2; __i <= __max_iter; ++__i) 207 { 208 __a -= 2 * (__i - 1); 209 __c = -__a * __c / __i; 210 const _Tp __qnew = (__q1 - __b * __q2) / __a; 211 __q1 = __q2; 212 __q2 = __qnew; 213 __q += __c * __qnew; 214 __b += _Tp(2); 215 __d = _Tp(1) / (__b + __a * __d); 216 __delh = (__b * __d - _Tp(1)) * __delh; 217 __h += __delh; 218 const _Tp __dels = __q * __delh; 219 __s += __dels; 220 if ( std::abs(__dels / __s) < __eps ) 221 break; 222 } 223 if (__i > __max_iter) 224 std::__throw_runtime_error(__N("Steed's method failed " 225 "in __bessel_jn.")); 226 __h = __a1 * __h; 227 __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) 228 * std::exp(-__x) / __s; 229 __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; 230 } 231 232 _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; 233 _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); 234 __Inu = __Inumu * __Inul1 / __Inul; 235 __Ipnu = __Inumu * __Ipnu1 / __Inul; 236 for ( __i = 1; __i <= __nl; ++__i ) 237 { 238 const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; 239 __Kmu = __Knu1; 240 __Knu1 = __Knutemp; 241 } 242 __Knu = __Kmu; 243 __Kpnu = __nu * __xi * __Kmu - __Knu1; 244 245 return; 246 } 247 248 249 /** 250 * @brief Return the regular modified Bessel function of order 251 * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. 252 * 253 * The regular modified cylindrical Bessel function is: 254 * @f[ 255 * I_{\nu}(x) = \sum_{k=0}^{\infty} 256 * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 257 * @f] 258 * 259 * @param __nu The order of the regular modified Bessel function. 260 * @param __x The argument of the regular modified Bessel function. 261 * @return The output regular modified Bessel function. 262 */ 263 template<typename _Tp> 264 _Tp 265 __cyl_bessel_i(const _Tp __nu, const _Tp __x) 266 { 267 if (__nu < _Tp(0) || __x < _Tp(0)) 268 std::__throw_domain_error(__N("Bad argument " 269 "in __cyl_bessel_i.")); 270 else if (__isnan(__nu) || __isnan(__x)) 271 return std::numeric_limits<_Tp>::quiet_NaN(); 272 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 273 return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); 274 else 275 { 276 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 277 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 278 return __I_nu; 279 } 280 } 281 282 283 /** 284 * @brief Return the irregular modified Bessel function 285 * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. 286 * 287 * The irregular modified Bessel function is defined by: 288 * @f[ 289 * K_{\nu}(x) = \frac{\pi}{2} 290 * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} 291 * @f] 292 * where for integral \f$ \nu = n \f$ a limit is taken: 293 * \f$ lim_{\nu \to n} \f$. 294 * 295 * @param __nu The order of the irregular modified Bessel function. 296 * @param __x The argument of the irregular modified Bessel function. 297 * @return The output irregular modified Bessel function. 298 */ 299 template<typename _Tp> 300 _Tp 301 __cyl_bessel_k(const _Tp __nu, const _Tp __x) 302 { 303 if (__nu < _Tp(0) || __x < _Tp(0)) 304 std::__throw_domain_error(__N("Bad argument " 305 "in __cyl_bessel_k.")); 306 else if (__isnan(__nu) || __isnan(__x)) 307 return std::numeric_limits<_Tp>::quiet_NaN(); 308 else 309 { 310 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 311 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 312 return __K_nu; 313 } 314 } 315 316 317 /** 318 * @brief Compute the spherical modified Bessel functions 319 * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first 320 * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ 321 * respectively. 322 * 323 * @param __n The order of the modified spherical Bessel function. 324 * @param __x The argument of the modified spherical Bessel function. 325 * @param __i_n The output regular modified spherical Bessel function. 326 * @param __k_n The output irregular modified spherical 327 * Bessel function. 328 * @param __ip_n The output derivative of the regular modified 329 * spherical Bessel function. 330 * @param __kp_n The output derivative of the irregular modified 331 * spherical Bessel function. 332 */ 333 template <typename _Tp> 334 void 335 __sph_bessel_ik(const unsigned int __n, const _Tp __x, 336 _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) 337 { 338 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 339 340 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 341 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 342 343 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 344 / std::sqrt(__x); 345 346 __i_n = __factor * __I_nu; 347 __k_n = __factor * __K_nu; 348 __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); 349 __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); 350 351 return; 352 } 353 354 355 /** 356 * @brief Compute the Airy functions 357 * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first 358 * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ 359 * respectively. 360 * 361 * @param __n The order of the Airy functions. 362 * @param __x The argument of the Airy functions. 363 * @param __i_n The output Airy function. 364 * @param __k_n The output Airy function. 365 * @param __ip_n The output derivative of the Airy function. 366 * @param __kp_n The output derivative of the Airy function. 367 */ 368 template <typename _Tp> 369 void 370 __airy(const _Tp __x, 371 _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) 372 { 373 const _Tp __absx = std::abs(__x); 374 const _Tp __rootx = std::sqrt(__absx); 375 const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); 376 377 if (__isnan(__x)) 378 return std::numeric_limits<_Tp>::quiet_NaN(); 379 else if (__x > _Tp(0)) 380 { 381 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 382 383 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 384 __Ai = __rootx * __K_nu 385 / (__numeric_constants<_Tp>::__sqrt3() 386 * __numeric_constants<_Tp>::__pi()); 387 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 388 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 389 390 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 391 __Aip = -__x * __K_nu 392 / (__numeric_constants<_Tp>::__sqrt3() 393 * __numeric_constants<_Tp>::__pi()); 394 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 395 + _Tp(2) * __I_nu 396 / __numeric_constants<_Tp>::__sqrt3()); 397 } 398 else if (__x < _Tp(0)) 399 { 400 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 401 402 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 403 __Ai = __rootx * (__J_nu 404 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 405 __Bi = -__rootx * (__N_nu 406 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 407 408 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 409 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 410 + __J_nu) / _Tp(2); 411 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 412 - __N_nu) / _Tp(2); 413 } 414 else 415 { 416 // Reference: 417 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 418 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 419 __Ai = _Tp(0.35502805388781723926L); 420 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 421 422 // Reference: 423 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 424 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 425 __Aip = -_Tp(0.25881940379280679840L); 426 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 427 } 428 429 return; 430 } 431 432 } // namespace std::tr1::__detail 433 } 434 } 435 436 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 437
