| // Special functions -*- C++ -*- | |
| // Copyright (C) 2006-2026 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/bessel_function.tcc | |
| * This is an internal header file, included by other library headers. | |
| * Do not attempt to use it directly. @headername{tr1/cmath} | |
| */ | |
| /* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c | |
| * Copyright (C) 1996-2003 Gerard Jungman | |
| */ | |
| // | |
| // 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. 240-245 | |
| #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC | |
| #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 | |
| #include <tr1/special_function_util.h> | |
| namespace std _GLIBCXX_VISIBILITY(default) | |
| { | |
| _GLIBCXX_BEGIN_NAMESPACE_VERSION | |
| #if _GLIBCXX_USE_STD_SPEC_FUNCS | |
| # define _GLIBCXX_MATH_NS ::std | |
| #elif defined(_GLIBCXX_TR1_CMATH) | |
| namespace tr1 | |
| { | |
| # define _GLIBCXX_MATH_NS ::std::tr1 | |
| #else | |
| # error do not include this header directly, use <cmath> or <tr1/cmath> | |
| #endif | |
| // [5.2] Special functions | |
| // Implementation-space details. | |
| namespace __detail | |
| { | |
| /** | |
| * @brief Compute the gamma functions required by the Temme series | |
| * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. | |
| * @f[ | |
| * \Gamma_1 = \frac{1}{2\mu} | |
| * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] | |
| * @f] | |
| * and | |
| * @f[ | |
| * \Gamma_2 = \frac{1}{2} | |
| * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] | |
| * @f] | |
| * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. | |
| * is the nearest integer to @f$ \nu @f$. | |
| * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ | |
| * are returned as well. | |
| * | |
| * The accuracy requirements on this are exquisite. | |
| * | |
| * @param __mu The input parameter of the gamma functions. | |
| * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ | |
| * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ | |
| * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ | |
| * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ | |
| */ | |
| template <typename _Tp> | |
| void | |
| __gamma_temme(_Tp __mu, | |
| _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) | |
| { | |
| #if _GLIBCXX_USE_C99_MATH_TR1 | |
| __gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu); | |
| __gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu); | |
| #else | |
| __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); | |
| __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); | |
| #endif | |
| if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) | |
| __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); | |
| else | |
| __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); | |
| __gam2 = (__gammi + __gampl) / (_Tp(2)); | |
| return; | |
| } | |
| /** | |
| * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann | |
| * @f$ N_\nu(x) @f$ functions and their first derivatives | |
| * @f$ J'_\nu(x) @f$ and @f$ N'_\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 __Jnu The output Bessel function of the first kind. | |
| * @param __Nnu The output Neumann function (Bessel function of the second kind). | |
| * @param __Jpnu The output derivative of the Bessel function of the first kind. | |
| * @param __Npnu The output derivative of the Neumann function. | |
| */ | |
| template <typename _Tp> | |
| void | |
| __bessel_jn(_Tp __nu, _Tp __x, | |
| _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) | |
| { | |
| if (__x == _Tp(0)) | |
| { | |
| if (__nu == _Tp(0)) | |
| { | |
| __Jnu = _Tp(1); | |
| __Jpnu = _Tp(0); | |
| } | |
| else if (__nu == _Tp(1)) | |
| { | |
| __Jnu = _Tp(0); | |
| __Jpnu = _Tp(0.5L); | |
| } | |
| else | |
| { | |
| __Jnu = _Tp(0); | |
| __Jpnu = _Tp(0); | |
| } | |
| __Nnu = -std::numeric_limits<_Tp>::infinity(); | |
| __Npnu = std::numeric_limits<_Tp>::infinity(); | |
| return; | |
| } | |
| const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
| // When the multiplier is N i.e. | |
| // fp_min = N * min() | |
| // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! | |
| //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); | |
| const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); | |
| const int __max_iter = 15000; | |
| const _Tp __x_min = _Tp(2); | |
| const int __nl = (__x < __x_min | |
| ? static_cast<int>(__nu + _Tp(0.5L)) | |
| : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); | |
| const _Tp __mu = __nu - __nl; | |
| const _Tp __mu2 = __mu * __mu; | |
| const _Tp __xi = _Tp(1) / __x; | |
| const _Tp __xi2 = _Tp(2) * __xi; | |
| _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); | |
| int __isign = 1; | |
| _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 = __b - __d; | |
| if (std::abs(__d) < __fp_min) | |
| __d = __fp_min; | |
| __c = __b - _Tp(1) / __c; | |
| if (std::abs(__c) < __fp_min) | |
| __c = __fp_min; | |
| __d = _Tp(1) / __d; | |
| const _Tp __del = __c * __d; | |
| __h *= __del; | |
| if (__d < _Tp(0)) | |
| __isign = -__isign; | |
| if (std::abs(__del - _Tp(1)) < __eps) | |
| break; | |
| } | |
| if (__i > __max_iter) | |
| std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " | |
| "try asymptotic expansion.")); | |
| _Tp __Jnul = __isign * __fp_min; | |
| _Tp __Jpnul = __h * __Jnul; | |
| _Tp __Jnul1 = __Jnul; | |
| _Tp __Jpnu1 = __Jpnul; | |
| _Tp __fact = __nu * __xi; | |
| for ( int __l = __nl; __l >= 1; --__l ) | |
| { | |
| const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; | |
| __fact -= __xi; | |
| __Jpnul = __fact * __Jnutemp - __Jnul; | |
| __Jnul = __Jnutemp; | |
| } | |
| if (__Jnul == _Tp(0)) | |
| __Jnul = __eps; | |
| _Tp __f= __Jpnul / __Jnul; | |
| _Tp __Nmu, __Nnu1, __Npmu, __Jmu; | |
| if (__x < __x_min) | |
| { | |
| const _Tp __x2 = __x / _Tp(2); | |
| const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; | |
| _Tp __fact = (std::abs(__pimu) < __eps | |
| ? _Tp(1) : __pimu / std::sin(__pimu)); | |
| _Tp __d = -std::log(__x2); | |
| _Tp __e = __mu * __d; | |
| _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 = (_Tp(2) / __numeric_constants<_Tp>::__pi()) | |
| * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); | |
| __e = std::exp(__e); | |
| _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); | |
| _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); | |
| const _Tp __pimu2 = __pimu / _Tp(2); | |
| _Tp __fact3 = (std::abs(__pimu2) < __eps | |
| ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); | |
| _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; | |
| _Tp __c = _Tp(1); | |
| __d = -__x2 * __x2; | |
| _Tp __sum = __ff + __r * __q; | |
| _Tp __sum1 = __p; | |
| for (__i = 1; __i <= __max_iter; ++__i) | |
| { | |
| __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); | |
| __c *= __d / _Tp(__i); | |
| __p /= _Tp(__i) - __mu; | |
| __q /= _Tp(__i) + __mu; | |
| const _Tp __del = __c * (__ff + __r * __q); | |
| __sum += __del; | |
| const _Tp __del1 = __c * __p - __i * __del; | |
| __sum1 += __del1; | |
| if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) | |
| break; | |
| } | |
| if ( __i > __max_iter ) | |
| std::__throw_runtime_error(__N("Bessel y series failed to converge " | |
| "in __bessel_jn.")); | |
| __Nmu = -__sum; | |
| __Nnu1 = -__sum1 * __xi2; | |
| __Npmu = __mu * __xi * __Nmu - __Nnu1; | |
| __Jmu = __w / (__Npmu - __f * __Nmu); | |
| } | |
| else | |
| { | |
| _Tp __a = _Tp(0.25L) - __mu2; | |
| _Tp __q = _Tp(1); | |
| _Tp __p = -__xi / _Tp(2); | |
| _Tp __br = _Tp(2) * __x; | |
| _Tp __bi = _Tp(2); | |
| _Tp __fact = __a * __xi / (__p * __p + __q * __q); | |
| _Tp __cr = __br + __q * __fact; | |
| _Tp __ci = __bi + __p * __fact; | |
| _Tp __den = __br * __br + __bi * __bi; | |
| _Tp __dr = __br / __den; | |
| _Tp __di = -__bi / __den; | |
| _Tp __dlr = __cr * __dr - __ci * __di; | |
| _Tp __dli = __cr * __di + __ci * __dr; | |
| _Tp __temp = __p * __dlr - __q * __dli; | |
| __q = __p * __dli + __q * __dlr; | |
| __p = __temp; | |
| int __i; | |
| for (__i = 2; __i <= __max_iter; ++__i) | |
| { | |
| __a += _Tp(2 * (__i - 1)); | |
| __bi += _Tp(2); | |
| __dr = __a * __dr + __br; | |
| __di = __a * __di + __bi; | |
| if (std::abs(__dr) + std::abs(__di) < __fp_min) | |
| __dr = __fp_min; | |
| __fact = __a / (__cr * __cr + __ci * __ci); | |
| __cr = __br + __cr * __fact; | |
| __ci = __bi - __ci * __fact; | |
| if (std::abs(__cr) + std::abs(__ci) < __fp_min) | |
| __cr = __fp_min; | |
| __den = __dr * __dr + __di * __di; | |
| __dr /= __den; | |
| __di /= -__den; | |
| __dlr = __cr * __dr - __ci * __di; | |
| __dli = __cr * __di + __ci * __dr; | |
| __temp = __p * __dlr - __q * __dli; | |
| __q = __p * __dli + __q * __dlr; | |
| __p = __temp; | |
| if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) | |
| break; | |
| } | |
| if (__i > __max_iter) | |
| std::__throw_runtime_error(__N("Lentz's method failed " | |
| "in __bessel_jn.")); | |
| const _Tp __gam = (__p - __f) / __q; | |
| __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); | |
| #if _GLIBCXX_USE_C99_MATH_TR1 | |
| __Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul); | |
| #else | |
| if (__Jmu * __Jnul < _Tp(0)) | |
| __Jmu = -__Jmu; | |
| #endif | |
| __Nmu = __gam * __Jmu; | |
| __Npmu = (__p + __q / __gam) * __Nmu; | |
| __Nnu1 = __mu * __xi * __Nmu - __Npmu; | |
| } | |
| __fact = __Jmu / __Jnul; | |
| __Jnu = __fact * __Jnul1; | |
| __Jpnu = __fact * __Jpnu1; | |
| for (__i = 1; __i <= __nl; ++__i) | |
| { | |
| const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; | |
| __Nmu = __Nnu1; | |
| __Nnu1 = __Nnutemp; | |
| } | |
| __Nnu = __Nmu; | |
| __Npnu = __nu * __xi * __Nmu - __Nnu1; | |
| return; | |
| } | |
| /** | |
| * @brief This routine computes the asymptotic cylindrical Bessel | |
| * and Neumann functions of order nu: \f$ J_{\nu} \f$, | |
| * \f$ N_{\nu} \f$. | |
| * | |
| * References: | |
| * (1) Handbook of Mathematical Functions, | |
| * ed. Milton Abramowitz and Irene A. Stegun, | |
| * Dover Publications, | |
| * Section 9 p. 364, Equations 9.2.5-9.2.10 | |
| * | |
| * @param __nu The order of the Bessel functions. | |
| * @param __x The argument of the Bessel functions. | |
| * @param __Jnu The output Bessel function of the first kind. | |
| * @param __Nnu The output Neumann function (Bessel function of the second kind). | |
| */ | |
| template <typename _Tp> | |
| void | |
| __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) | |
| { | |
| if (__builtin_expect(__builtin_isinf(__x), 0)) | |
| { | |
| __Jnu = __Nnu = _Tp(0); | |
| return; | |
| } | |
| const _Tp __mu = _Tp(4) * __nu * __nu; | |
| const _Tp __8x = _Tp(8) * __x; | |
| _Tp __P = _Tp(0); | |
| _Tp __Q = _Tp(0); | |
| _Tp __k = _Tp(0); | |
| _Tp __term = _Tp(1); | |
| int __epsP = 0; | |
| int __epsQ = 0; | |
| _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
| do | |
| { | |
| __term *= (__k == 0 | |
| ? _Tp(1) | |
| : -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x)); | |
| __epsP = std::abs(__term) < __eps * std::abs(__P); | |
| __P += __term; | |
| __k++; | |
| __term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x); | |
| __epsQ = std::abs(__term) < __eps * std::abs(__Q); | |
| __Q += __term; | |
| if (__epsP && __epsQ && __k > (__nu / 2.)) | |
| break; | |
| __k++; | |
| } | |
| while (__k < 1000); | |
| const _Tp __chi = __x - (__nu + _Tp(0.5L)) | |
| * __numeric_constants<_Tp>::__pi_2(); | |
| const _Tp __c = std::cos(__chi); | |
| const _Tp __s = std::sin(__chi); | |
| const _Tp __coef = std::sqrt(_Tp(2) | |
| / (__numeric_constants<_Tp>::__pi() * __x)); | |
| __Jnu = __coef * (__c * __P - __s * __Q); | |
| __Nnu = __coef * (__s * __P + __c * __Q); | |
| return; | |
| } | |
| /** | |
| * @brief This routine returns the cylindrical Bessel functions | |
| * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ | |
| * by series expansion. | |
| * | |
| * The modified cylindrical Bessel function is: | |
| * @f[ | |
| * Z_{\nu}(x) = \sum_{k=0}^{\infty} | |
| * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
| * @f] | |
| * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for | |
| * \f$ Z = I \f$ or \f$ J \f$ respectively. | |
| * | |
| * See Abramowitz & Stegun, 9.1.10 | |
| * Abramowitz & Stegun, 9.6.7 | |
| * (1) Handbook of Mathematical Functions, | |
| * ed. Milton Abramowitz and Irene A. Stegun, | |
| * Dover Publications, | |
| * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 | |
| * | |
| * @param __nu The order of the Bessel function. | |
| * @param __x The argument of the Bessel function. | |
| * @param __sgn The sign of the alternate terms | |
| * -1 for the Bessel function of the first kind. | |
| * +1 for the modified Bessel function of the first kind. | |
| * @return The output Bessel function. | |
| */ | |
| template <typename _Tp> | |
| _Tp | |
| __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, | |
| unsigned int __max_iter) | |
| { | |
| if (__x == _Tp(0)) | |
| return __nu == _Tp(0) ? _Tp(1) : _Tp(0); | |
| const _Tp __x2 = __x / _Tp(2); | |
| _Tp __fact = __nu * std::log(__x2); | |
| #if _GLIBCXX_USE_C99_MATH_TR1 | |
| __fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1)); | |
| #else | |
| __fact -= __log_gamma(__nu + _Tp(1)); | |
| #endif | |
| __fact = std::exp(__fact); | |
| const _Tp __xx4 = __sgn * __x2 * __x2; | |
| _Tp __Jn = _Tp(1); | |
| _Tp __term = _Tp(1); | |
| for (unsigned int __i = 1; __i < __max_iter; ++__i) | |
| { | |
| __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); | |
| __Jn += __term; | |
| if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) | |
| break; | |
| } | |
| return __fact * __Jn; | |
| } | |
| /** | |
| * @brief Return the Bessel function of order \f$ \nu \f$: | |
| * \f$ J_{\nu}(x) \f$. | |
| * | |
| * The cylindrical Bessel function is: | |
| * @f[ | |
| * J_{\nu}(x) = \sum_{k=0}^{\infty} | |
| * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
| * @f] | |
| * | |
| * @param __nu The order of the Bessel function. | |
| * @param __x The argument of the Bessel function. | |
| * @return The output Bessel function. | |
| */ | |
| template<typename _Tp> | |
| _Tp | |
| __cyl_bessel_j(_Tp __nu, _Tp __x) | |
| { | |
| if (__nu < _Tp(0) || __x < _Tp(0)) | |
| std::__throw_domain_error(__N("Bad argument " | |
| "in __cyl_bessel_j.")); | |
| else if (__isnan(__nu) || __isnan(__x)) | |
| return std::numeric_limits<_Tp>::quiet_NaN(); | |
| else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) | |
| return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); | |
| else if (__x > _Tp(1000)) | |
| { | |
| _Tp __J_nu, __N_nu; | |
| __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
| return __J_nu; | |
| } | |
| else | |
| { | |
| _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
| __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
| return __J_nu; | |
| } | |
| } | |
| /** | |
| * @brief Return the Neumann function of order \f$ \nu \f$: | |
| * \f$ N_{\nu}(x) \f$. | |
| * | |
| * The Neumann function is defined by: | |
| * @f[ | |
| * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} | |
| * {\sin \nu\pi} | |
| * @f] | |
| * where for integral \f$ \nu = n \f$ a limit is taken: | |
| * \f$ lim_{\nu \to n} \f$. | |
| * | |
| * @param __nu The order of the Neumann function. | |
| * @param __x The argument of the Neumann function. | |
| * @return The output Neumann function. | |
| */ | |
| template<typename _Tp> | |
| _Tp | |
| __cyl_neumann_n(_Tp __nu, _Tp __x) | |
| { | |
| if (__nu < _Tp(0) || __x < _Tp(0)) | |
| std::__throw_domain_error(__N("Bad argument " | |
| "in __cyl_neumann_n.")); | |
| else if (__isnan(__nu) || __isnan(__x)) | |
| return std::numeric_limits<_Tp>::quiet_NaN(); | |
| else if (__x > _Tp(1000)) | |
| { | |
| _Tp __J_nu, __N_nu; | |
| __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
| return __N_nu; | |
| } | |
| else | |
| { | |
| _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
| __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
| return __N_nu; | |
| } | |
| } | |
| /** | |
| * @brief Compute the spherical Bessel @f$ j_n(x) @f$ | |
| * and Neumann @f$ n_n(x) @f$ functions and their first | |
| * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ | |
| * respectively. | |
| * | |
| * @param __n The order of the spherical Bessel function. | |
| * @param __x The argument of the spherical Bessel function. | |
| * @param __j_n The output spherical Bessel function. | |
| * @param __n_n The output spherical Neumann function. | |
| * @param __jp_n The output derivative of the spherical Bessel function. | |
| * @param __np_n The output derivative of the spherical Neumann function. | |
| */ | |
| template <typename _Tp> | |
| void | |
| __sph_bessel_jn(unsigned int __n, _Tp __x, | |
| _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) | |
| { | |
| const _Tp __nu = _Tp(__n) + _Tp(0.5L); | |
| _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
| __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
| const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() | |
| / std::sqrt(__x); | |
| __j_n = __factor * __J_nu; | |
| __n_n = __factor * __N_nu; | |
| __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); | |
| __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); | |
| return; | |
| } | |
| /** | |
| * @brief Return the spherical Bessel function | |
| * @f$ j_n(x) @f$ of order n. | |
| * | |
| * The spherical Bessel function is defined by: | |
| * @f[ | |
| * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) | |
| * @f] | |
| * | |
| * @param __n The order of the spherical Bessel function. | |
| * @param __x The argument of the spherical Bessel function. | |
| * @return The output spherical Bessel function. | |
| */ | |
| template <typename _Tp> | |
| _Tp | |
| __sph_bessel(unsigned int __n, _Tp __x) | |
| { | |
| if (__x < _Tp(0)) | |
| std::__throw_domain_error(__N("Bad argument " | |
| "in __sph_bessel.")); | |
| else if (__isnan(__x)) | |
| return std::numeric_limits<_Tp>::quiet_NaN(); | |
| else if (__x == _Tp(0)) | |
| { | |
| if (__n == 0) | |
| return _Tp(1); | |
| else | |
| return _Tp(0); | |
| } | |
| else | |
| { | |
| _Tp __j_n, __n_n, __jp_n, __np_n; | |
| __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
| return __j_n; | |
| } | |
| } | |
| /** | |
| * @brief Return the spherical Neumann function | |
| * @f$ n_n(x) @f$. | |
| * | |
| * The spherical Neumann function is defined by: | |
| * @f[ | |
| * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) | |
| * @f] | |
| * | |
| * @param __n The order of the spherical Neumann function. | |
| * @param __x The argument of the spherical Neumann function. | |
| * @return The output spherical Neumann function. | |
| */ | |
| template <typename _Tp> | |
| _Tp | |
| __sph_neumann(unsigned int __n, _Tp __x) | |
| { | |
| if (__x < _Tp(0)) | |
| std::__throw_domain_error(__N("Bad argument " | |
| "in __sph_neumann.")); | |
| else if (__isnan(__x)) | |
| return std::numeric_limits<_Tp>::quiet_NaN(); | |
| else if (__x == _Tp(0)) | |
| return -std::numeric_limits<_Tp>::infinity(); | |
| else | |
| { | |
| _Tp __j_n, __n_n, __jp_n, __np_n; | |
| __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
| return __n_n; | |
| } | |
| } | |
| } // namespace __detail | |
| #undef _GLIBCXX_MATH_NS | |
| #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) | |
| } // namespace tr1 | |
| #endif | |
| _GLIBCXX_END_NAMESPACE_VERSION | |
| } | |
| #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |