libstdc++
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00001 // Special functions -*- C++ -*- 00002 00003 // Copyright (C) 2006, 2007, 2008, 2009 00004 // Free Software Foundation, Inc. 00005 // 00006 // This file is part of the GNU ISO C++ Library. This library is free 00007 // software; you can redistribute it and/or modify it under the 00008 // terms of the GNU General Public License as published by the 00009 // Free Software Foundation; either version 3, or (at your option) 00010 // any later version. 00011 // 00012 // This library is distributed in the hope that it will be useful, 00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of 00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00015 // GNU General Public License for more details. 00016 // 00017 // Under Section 7 of GPL version 3, you are granted additional 00018 // permissions described in the GCC Runtime Library Exception, version 00019 // 3.1, as published by the Free Software Foundation. 00020 00021 // You should have received a copy of the GNU General Public License and 00022 // a copy of the GCC Runtime Library Exception along with this program; 00023 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 00024 // <http://www.gnu.org/licenses/>. 00025 00026 /** @file tr1/modified_bessel_func.tcc 00027 * This is an internal header file, included by other library headers. 00028 * You should not attempt to use it directly. 00029 */ 00030 00031 // 00032 // ISO C++ 14882 TR1: 5.2 Special functions 00033 // 00034 00035 // Written by Edward Smith-Rowland. 00036 // 00037 // References: 00038 // (1) Handbook of Mathematical Functions, 00039 // Ed. Milton Abramowitz and Irene A. Stegun, 00040 // Dover Publications, 00041 // Section 9, pp. 355-434, Section 10 pp. 435-478 00042 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 00043 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 00044 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 00045 // 2nd ed, pp. 246-249. 00046 00047 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 00048 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 00049 00050 #include "special_function_util.h" 00051 00052 namespace std 00053 { 00054 namespace tr1 00055 { 00056 00057 // [5.2] Special functions 00058 00059 // Implementation-space details. 00060 namespace __detail 00061 { 00062 00063 /** 00064 * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and 00065 * @f$ K_\nu(x) @f$ and their first derivatives 00066 * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. 00067 * These four functions are computed together for numerical 00068 * stability. 00069 * 00070 * @param __nu The order of the Bessel functions. 00071 * @param __x The argument of the Bessel functions. 00072 * @param __Inu The output regular modified Bessel function. 00073 * @param __Knu The output irregular modified Bessel function. 00074 * @param __Ipnu The output derivative of the regular 00075 * modified Bessel function. 00076 * @param __Kpnu The output derivative of the irregular 00077 * modified Bessel function. 00078 */ 00079 template <typename _Tp> 00080 void 00081 __bessel_ik(const _Tp __nu, const _Tp __x, 00082 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) 00083 { 00084 if (__x == _Tp(0)) 00085 { 00086 if (__nu == _Tp(0)) 00087 { 00088 __Inu = _Tp(1); 00089 __Ipnu = _Tp(0); 00090 } 00091 else if (__nu == _Tp(1)) 00092 { 00093 __Inu = _Tp(0); 00094 __Ipnu = _Tp(0.5L); 00095 } 00096 else 00097 { 00098 __Inu = _Tp(0); 00099 __Ipnu = _Tp(0); 00100 } 00101 __Knu = std::numeric_limits<_Tp>::infinity(); 00102 __Kpnu = -std::numeric_limits<_Tp>::infinity(); 00103 return; 00104 } 00105 00106 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00107 const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); 00108 const int __max_iter = 15000; 00109 const _Tp __x_min = _Tp(2); 00110 00111 const int __nl = static_cast<int>(__nu + _Tp(0.5L)); 00112 00113 const _Tp __mu = __nu - __nl; 00114 const _Tp __mu2 = __mu * __mu; 00115 const _Tp __xi = _Tp(1) / __x; 00116 const _Tp __xi2 = _Tp(2) * __xi; 00117 _Tp __h = __nu * __xi; 00118 if ( __h < __fp_min ) 00119 __h = __fp_min; 00120 _Tp __b = __xi2 * __nu; 00121 _Tp __d = _Tp(0); 00122 _Tp __c = __h; 00123 int __i; 00124 for ( __i = 1; __i <= __max_iter; ++__i ) 00125 { 00126 __b += __xi2; 00127 __d = _Tp(1) / (__b + __d); 00128 __c = __b + _Tp(1) / __c; 00129 const _Tp __del = __c * __d; 00130 __h *= __del; 00131 if (std::abs(__del - _Tp(1)) < __eps) 00132 break; 00133 } 00134 if (__i > __max_iter) 00135 std::__throw_runtime_error(__N("Argument x too large " 00136 "in __bessel_jn; " 00137 "try asymptotic expansion.")); 00138 _Tp __Inul = __fp_min; 00139 _Tp __Ipnul = __h * __Inul; 00140 _Tp __Inul1 = __Inul; 00141 _Tp __Ipnu1 = __Ipnul; 00142 _Tp __fact = __nu * __xi; 00143 for (int __l = __nl; __l >= 1; --__l) 00144 { 00145 const _Tp __Inutemp = __fact * __Inul + __Ipnul; 00146 __fact -= __xi; 00147 __Ipnul = __fact * __Inutemp + __Inul; 00148 __Inul = __Inutemp; 00149 } 00150 _Tp __f = __Ipnul / __Inul; 00151 _Tp __Kmu, __Knu1; 00152 if (__x < __x_min) 00153 { 00154 const _Tp __x2 = __x / _Tp(2); 00155 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 00156 const _Tp __fact = (std::abs(__pimu) < __eps 00157 ? _Tp(1) : __pimu / std::sin(__pimu)); 00158 _Tp __d = -std::log(__x2); 00159 _Tp __e = __mu * __d; 00160 const _Tp __fact2 = (std::abs(__e) < __eps 00161 ? _Tp(1) : std::sinh(__e) / __e); 00162 _Tp __gam1, __gam2, __gampl, __gammi; 00163 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 00164 _Tp __ff = __fact 00165 * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 00166 _Tp __sum = __ff; 00167 __e = std::exp(__e); 00168 _Tp __p = __e / (_Tp(2) * __gampl); 00169 _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); 00170 _Tp __c = _Tp(1); 00171 __d = __x2 * __x2; 00172 _Tp __sum1 = __p; 00173 int __i; 00174 for (__i = 1; __i <= __max_iter; ++__i) 00175 { 00176 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 00177 __c *= __d / __i; 00178 __p /= __i - __mu; 00179 __q /= __i + __mu; 00180 const _Tp __del = __c * __ff; 00181 __sum += __del; 00182 const _Tp __del1 = __c * (__p - __i * __ff); 00183 __sum1 += __del1; 00184 if (std::abs(__del) < __eps * std::abs(__sum)) 00185 break; 00186 } 00187 if (__i > __max_iter) 00188 std::__throw_runtime_error(__N("Bessel k series failed to converge " 00189 "in __bessel_jn.")); 00190 __Kmu = __sum; 00191 __Knu1 = __sum1 * __xi2; 00192 } 00193 else 00194 { 00195 _Tp __b = _Tp(2) * (_Tp(1) + __x); 00196 _Tp __d = _Tp(1) / __b; 00197 _Tp __delh = __d; 00198 _Tp __h = __delh; 00199 _Tp __q1 = _Tp(0); 00200 _Tp __q2 = _Tp(1); 00201 _Tp __a1 = _Tp(0.25L) - __mu2; 00202 _Tp __q = __c = __a1; 00203 _Tp __a = -__a1; 00204 _Tp __s = _Tp(1) + __q * __delh; 00205 int __i; 00206 for (__i = 2; __i <= __max_iter; ++__i) 00207 { 00208 __a -= 2 * (__i - 1); 00209 __c = -__a * __c / __i; 00210 const _Tp __qnew = (__q1 - __b * __q2) / __a; 00211 __q1 = __q2; 00212 __q2 = __qnew; 00213 __q += __c * __qnew; 00214 __b += _Tp(2); 00215 __d = _Tp(1) / (__b + __a * __d); 00216 __delh = (__b * __d - _Tp(1)) * __delh; 00217 __h += __delh; 00218 const _Tp __dels = __q * __delh; 00219 __s += __dels; 00220 if ( std::abs(__dels / __s) < __eps ) 00221 break; 00222 } 00223 if (__i > __max_iter) 00224 std::__throw_runtime_error(__N("Steed's method failed " 00225 "in __bessel_jn.")); 00226 __h = __a1 * __h; 00227 __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) 00228 * std::exp(-__x) / __s; 00229 __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; 00230 } 00231 00232 _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; 00233 _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); 00234 __Inu = __Inumu * __Inul1 / __Inul; 00235 __Ipnu = __Inumu * __Ipnu1 / __Inul; 00236 for ( __i = 1; __i <= __nl; ++__i ) 00237 { 00238 const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; 00239 __Kmu = __Knu1; 00240 __Knu1 = __Knutemp; 00241 } 00242 __Knu = __Kmu; 00243 __Kpnu = __nu * __xi * __Kmu - __Knu1; 00244 00245 return; 00246 } 00247 00248 00249 /** 00250 * @brief Return the regular modified Bessel function of order 00251 * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. 00252 * 00253 * The regular modified cylindrical Bessel function is: 00254 * @f[ 00255 * I_{\nu}(x) = \sum_{k=0}^{\infty} 00256 * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 00257 * @f] 00258 * 00259 * @param __nu The order of the regular modified Bessel function. 00260 * @param __x The argument of the regular modified Bessel function. 00261 * @return The output regular modified Bessel function. 00262 */ 00263 template<typename _Tp> 00264 _Tp 00265 __cyl_bessel_i(const _Tp __nu, const _Tp __x) 00266 { 00267 if (__nu < _Tp(0) || __x < _Tp(0)) 00268 std::__throw_domain_error(__N("Bad argument " 00269 "in __cyl_bessel_i.")); 00270 else if (__isnan(__nu) || __isnan(__x)) 00271 return std::numeric_limits<_Tp>::quiet_NaN(); 00272 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 00273 return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); 00274 else 00275 { 00276 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 00277 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 00278 return __I_nu; 00279 } 00280 } 00281 00282 00283 /** 00284 * @brief Return the irregular modified Bessel function 00285 * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. 00286 * 00287 * The irregular modified Bessel function is defined by: 00288 * @f[ 00289 * K_{\nu}(x) = \frac{\pi}{2} 00290 * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} 00291 * @f] 00292 * where for integral \f$ \nu = n \f$ a limit is taken: 00293 * \f$ lim_{\nu \to n} \f$. 00294 * 00295 * @param __nu The order of the irregular modified Bessel function. 00296 * @param __x The argument of the irregular modified Bessel function. 00297 * @return The output irregular modified Bessel function. 00298 */ 00299 template<typename _Tp> 00300 _Tp 00301 __cyl_bessel_k(const _Tp __nu, const _Tp __x) 00302 { 00303 if (__nu < _Tp(0) || __x < _Tp(0)) 00304 std::__throw_domain_error(__N("Bad argument " 00305 "in __cyl_bessel_k.")); 00306 else if (__isnan(__nu) || __isnan(__x)) 00307 return std::numeric_limits<_Tp>::quiet_NaN(); 00308 else 00309 { 00310 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; 00311 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 00312 return __K_nu; 00313 } 00314 } 00315 00316 00317 /** 00318 * @brief Compute the spherical modified Bessel functions 00319 * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first 00320 * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ 00321 * respectively. 00322 * 00323 * @param __n The order of the modified spherical Bessel function. 00324 * @param __x The argument of the modified spherical Bessel function. 00325 * @param __i_n The output regular modified spherical Bessel function. 00326 * @param __k_n The output irregular modified spherical 00327 * Bessel function. 00328 * @param __ip_n The output derivative of the regular modified 00329 * spherical Bessel function. 00330 * @param __kp_n The output derivative of the irregular modified 00331 * spherical Bessel function. 00332 */ 00333 template <typename _Tp> 00334 void 00335 __sph_bessel_ik(const unsigned int __n, const _Tp __x, 00336 _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) 00337 { 00338 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 00339 00340 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 00341 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 00342 00343 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 00344 / std::sqrt(__x); 00345 00346 __i_n = __factor * __I_nu; 00347 __k_n = __factor * __K_nu; 00348 __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); 00349 __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); 00350 00351 return; 00352 } 00353 00354 00355 /** 00356 * @brief Compute the Airy functions 00357 * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first 00358 * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ 00359 * respectively. 00360 * 00361 * @param __n The order of the Airy functions. 00362 * @param __x The argument of the Airy functions. 00363 * @param __i_n The output Airy function. 00364 * @param __k_n The output Airy function. 00365 * @param __ip_n The output derivative of the Airy function. 00366 * @param __kp_n The output derivative of the Airy function. 00367 */ 00368 template <typename _Tp> 00369 void 00370 __airy(const _Tp __x, 00371 _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) 00372 { 00373 const _Tp __absx = std::abs(__x); 00374 const _Tp __rootx = std::sqrt(__absx); 00375 const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); 00376 00377 if (__isnan(__x)) 00378 return std::numeric_limits<_Tp>::quiet_NaN(); 00379 else if (__x > _Tp(0)) 00380 { 00381 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 00382 00383 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 00384 __Ai = __rootx * __K_nu 00385 / (__numeric_constants<_Tp>::__sqrt3() 00386 * __numeric_constants<_Tp>::__pi()); 00387 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 00388 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 00389 00390 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 00391 __Aip = -__x * __K_nu 00392 / (__numeric_constants<_Tp>::__sqrt3() 00393 * __numeric_constants<_Tp>::__pi()); 00394 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 00395 + _Tp(2) * __I_nu 00396 / __numeric_constants<_Tp>::__sqrt3()); 00397 } 00398 else if (__x < _Tp(0)) 00399 { 00400 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 00401 00402 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 00403 __Ai = __rootx * (__J_nu 00404 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 00405 __Bi = -__rootx * (__N_nu 00406 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 00407 00408 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 00409 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 00410 + __J_nu) / _Tp(2); 00411 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 00412 - __N_nu) / _Tp(2); 00413 } 00414 else 00415 { 00416 // Reference: 00417 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 00418 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 00419 __Ai = _Tp(0.35502805388781723926L); 00420 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 00421 00422 // Reference: 00423 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 00424 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 00425 __Aip = -_Tp(0.25881940379280679840L); 00426 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 00427 } 00428 00429 return; 00430 } 00431 00432 } // namespace std::tr1::__detail 00433 } 00434 } 00435 00436 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC