Edit File: modified_bessel_func.tcc
// Special functions -*- C++ -*- // Copyright (C) 2006-2013 Free Software Foundation, Inc. // // This file is part of the GNU ISO C++ Library. This library is free // software; you can redistribute it and/or modify it under the // terms of the GNU General Public License as published by the // Free Software Foundation; either version 3, or (at your option) // any later version. // // This library is distributed in the hope that it will be useful, // but WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // // Under Section 7 of GPL version 3, you are granted additional // permissions described in the GCC Runtime Library Exception, version // 3.1, as published by the Free Software Foundation. // You should have received a copy of the GNU General Public License and // a copy of the GCC Runtime Library Exception along with this program; // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see // <http://www.gnu.org/licenses/>. /** @file tr1/modified_bessel_func.tcc * This is an internal header file, included by other library headers. * Do not attempt to use it directly. @headername{tr1/cmath} */ // // ISO C++ 14882 TR1: 5.2 Special functions // // Written by Edward Smith-Rowland. // // References: // (1) Handbook of Mathematical Functions, // Ed. Milton Abramowitz and Irene A. Stegun, // Dover Publications, // Section 9, pp. 355-434, Section 10 pp. 435-478 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), // 2nd ed, pp. 246-249. #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 #include "special_function_util.h" namespace std _GLIBCXX_VISIBILITY(default) { namespace tr1 { // [5.2] Special functions // Implementation-space details. namespace __detail { _GLIBCXX_BEGIN_NAMESPACE_VERSION /** * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and * @f$ K_\nu(x) @f$ and their first derivatives * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. * These four functions are computed together for numerical * stability. * * @param __nu The order of the Bessel functions. * @param __x The argument of the Bessel functions. * @param __Inu The output regular modified Bessel function. * @param __Knu The output irregular modified Bessel function. * @param __Ipnu The output derivative of the regular * modified Bessel function. * @param __Kpnu The output derivative of the irregular * modified Bessel function. */ template <typename _Tp> void __bessel_ik(_Tp __nu, _Tp __x, _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) { if (__x == _Tp(0)) { if (__nu == _Tp(0)) { __Inu = _Tp(1); __Ipnu = _Tp(0); } else if (__nu == _Tp(1)) { __Inu = _Tp(0); __Ipnu = _Tp(0.5L); } else { __Inu = _Tp(0); __Ipnu = _Tp(0); } __Knu = std::numeric_limits<_Tp>::infinity(); __Kpnu = -std::numeric_limits<_Tp>::infinity(); return; } const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); const int __max_iter = 15000; const _Tp __x_min = _Tp(2); const int __nl = static_cast<int>(__nu + _Tp(0.5L)); const _Tp __mu = __nu - __nl; const _Tp __mu2 = __mu * __mu; const _Tp __xi = _Tp(1) / __x; const _Tp __xi2 = _Tp(2) * __xi; _Tp __h = __nu * __xi; if ( __h < __fp_min ) __h = __fp_min; _Tp __b = __xi2 * __nu; _Tp __d = _Tp(0); _Tp __c = __h; int __i; for ( __i = 1; __i <= __max_iter; ++__i ) { __b += __xi2; __d = _Tp(1) / (__b + __d); __c = __b + _Tp(1) / __c; const _Tp __del = __c * __d; __h *= __del; if (std::abs(__del - _Tp(1)) < __eps) break; } if (__i > __max_iter) std::__throw_runtime_error(__N("Argument x too large " "in __bessel_ik; " "try asymptotic expansion.")); _Tp __Inul = __fp_min; _Tp __Ipnul = __h * __Inul; _Tp __Inul1 = __Inul; _Tp __Ipnu1 = __Ipnul; _Tp __fact = __nu * __xi; for (int __l = __nl; __l >= 1; --__l) { const _Tp __Inutemp = __fact * __Inul + __Ipnul; __fact -= __xi; __Ipnul = __fact * __Inutemp + __Inul; __Inul = __Inutemp; } _Tp __f = __Ipnul / __Inul; _Tp __Kmu, __Knu1; if (__x < __x_min) { const _Tp __x2 = __x / _Tp(2); const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; const _Tp __fact = (std::abs(__pimu) < __eps ? _Tp(1) : __pimu / std::sin(__pimu)); _Tp __d = -std::log(__x2); _Tp __e = __mu * __d; const _Tp __fact2 = (std::abs(__e) < __eps ? _Tp(1) : std::sinh(__e) / __e); _Tp __gam1, __gam2, __gampl, __gammi; __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); _Tp __ff = __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); _Tp __sum = __ff; __e = std::exp(__e); _Tp __p = __e / (_Tp(2) * __gampl); _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); _Tp __c = _Tp(1); __d = __x2 * __x2; _Tp __sum1 = __p; int __i; for (__i = 1; __i <= __max_iter; ++__i) { __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); __c *= __d / __i; __p /= __i - __mu; __q /= __i + __mu; const _Tp __del = __c * __ff; __sum += __del; const _Tp __del1 = __c * (__p - __i * __ff); __sum1 += __del1; if (std::abs(__del) < __eps * std::abs(__sum)) break; } if (__i > __max_iter) std::__throw_runtime_error(__N("Bessel k series failed to converge " "in __bessel_ik.")); __Kmu = __sum; __Knu1 = __sum1 * __xi2; } else { _Tp __b = _Tp(2) * (_Tp(1) + __x); _Tp __d = _Tp(1) / __b; _Tp __delh = __d; _Tp __h = __delh; _Tp __q1 = _Tp(0); _Tp __q2 = _Tp(1); _Tp __a1 = _Tp(0.25L) - __mu2; _Tp __q = __c = __a1; _Tp __a = -__a1; _Tp __s = _Tp(1) + __q * __delh; int __i; for (__i = 2; __i <= __max_iter; ++__i) { __a -= 2 * (__i - 1); __c = -__a * __c / __i; const _Tp __qnew = (__q1 - __b * __q2) / __a; __q1 = __q2; __q2 = __qnew; __q += __c * __qnew; __b += _Tp(2); __d = _Tp(1) / (__b + __a * __d); __delh = (__b * __d - _Tp(1)) * __delh; __h += __delh; const _Tp __dels = __q * __delh; __s += __dels; if ( std::abs(__dels / __s) < __eps ) break; } if (__i > __max_iter) std::__throw_runtime_error(__N("Steed's method failed " "in __bessel_ik.")); __h = __a1 * __h; __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) * std::exp(-__x) / __s; __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; } _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); __Inu = __Inumu * __Inul1 / __Inul; __Ipnu = __Inumu * __Ipnu1 / __Inul; for ( __i = 1; __i <= __nl; ++__i ) { const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; __Kmu = __Knu1; __Knu1 = __Knutemp; } __Knu = __Kmu; __Kpnu = __nu * __xi * __Kmu - __Knu1; return; } /** * @brief Return the regular modified Bessel function of order * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. * * The regular modified cylindrical Bessel function is: * @f[ * I_{\nu}(x) = \sum_{k=0}^{\infty} * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} * @f] * * @param __nu The order of the regular modified Bessel function. * @param __x The argument of the regular modified Bessel function. * @return The output regular modified Bessel function. */ template<typename _Tp> _Tp __cyl_bessel_i(_Tp __nu, _Tp __x) { if (__nu < _Tp(0) || __x < _Tp(0)) std::__throw_domain_error(__N("Bad argument " "in __cyl_bessel_i.")); else if (__isnan(__nu) || __isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); else { _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); return __I_nu; } } /** * @brief Return the irregular modified Bessel function * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. * * The irregular modified Bessel function is defined by: * @f[ * K_{\nu}(x) = \frac{\pi}{2} * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} * @f] * where for integral \f$ \nu = n \f$ a limit is taken: * \f$ lim_{\nu \to n} \f$. * * @param __nu The order of the irregular modified Bessel function. * @param __x The argument of the irregular modified Bessel function. * @return The output irregular modified Bessel function. */ template<typename _Tp> _Tp __cyl_bessel_k(_Tp __nu, _Tp __x) { if (__nu < _Tp(0) || __x < _Tp(0)) std::__throw_domain_error(__N("Bad argument " "in __cyl_bessel_k.")); else if (__isnan(__nu) || __isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else { _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); return __K_nu; } } /** * @brief Compute the spherical modified Bessel functions * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ * respectively. * * @param __n The order of the modified spherical Bessel function. * @param __x The argument of the modified spherical Bessel function. * @param __i_n The output regular modified spherical Bessel function. * @param __k_n The output irregular modified spherical * Bessel function. * @param __ip_n The output derivative of the regular modified * spherical Bessel function. * @param __kp_n The output derivative of the irregular modified * spherical Bessel function. */ template <typename _Tp> void __sph_bessel_ik(unsigned int __n, _Tp __x, _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) { const _Tp __nu = _Tp(__n) + _Tp(0.5L); _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() / std::sqrt(__x); __i_n = __factor * __I_nu; __k_n = __factor * __K_nu; __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); return; } /** * @brief Compute the Airy functions * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ * respectively. * * @param __n The order of the Airy functions. * @param __x The argument of the Airy functions. * @param __i_n The output Airy function. * @param __k_n The output Airy function. * @param __ip_n The output derivative of the Airy function. * @param __kp_n The output derivative of the Airy function. */ template <typename _Tp> void __airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) { const _Tp __absx = std::abs(__x); const _Tp __rootx = std::sqrt(__absx); const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); if (__isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__x > _Tp(0)) { _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); __Ai = __rootx * __K_nu / (__numeric_constants<_Tp>::__sqrt3() * __numeric_constants<_Tp>::__pi()); __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); __Aip = -__x * __K_nu / (__numeric_constants<_Tp>::__sqrt3() * __numeric_constants<_Tp>::__pi()); __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); } else if (__x < _Tp(0)) { _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); __Ai = __rootx * (__J_nu - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); __Bi = -__rootx * (__N_nu + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() + __J_nu) / _Tp(2); __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() - __N_nu) / _Tp(2); } else { // Reference: // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). __Ai = _Tp(0.35502805388781723926L); __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); // Reference: // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). __Aip = -_Tp(0.25881940379280679840L); __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); } return; } _GLIBCXX_END_NAMESPACE_VERSION } // namespace std::tr1::__detail } } #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC