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Diffstat (limited to 'gcc-4.8.1/libstdc++-v3/include/tr1/bessel_function.tcc')
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diff --git a/gcc-4.8.1/libstdc++-v3/include/tr1/bessel_function.tcc b/gcc-4.8.1/libstdc++-v3/include/tr1/bessel_function.tcc deleted file mode 100644 index 20481b8e0..000000000 --- a/gcc-4.8.1/libstdc++-v3/include/tr1/bessel_function.tcc +++ /dev/null @@ -1,628 +0,0 @@ -// 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/bessel_function.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. 240-245 - -#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC -#define _GLIBCXX_TR1_BESSEL_FUNCTION_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 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) / std::tr1::tgamma(_Tp(1) + __mu); - __gammi = _Tp(1) / std::tr1::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 = std::tr1::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$. - * - * References: - * (1) Handbook of Mathematical Functions, - * ed. Milton Abramowitz and Irene A. Stegun, - * Dover Publications, - * Section 9 p. 364, Equations 9.2.5-9.2.10 - * - * @param __nu The order of the Bessel functions. - * @param __x The argument of the Bessel functions. - * @param __Jnu The output Bessel function of the first kind. - * @param __Nnu The output Neumann function (Bessel function of the second kind). - */ - template <typename _Tp> - void - __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) - { - const _Tp __mu = _Tp(4) * __nu * __nu; - const _Tp __mum1 = __mu - _Tp(1); - const _Tp __mum9 = __mu - _Tp(9); - const _Tp __mum25 = __mu - _Tp(25); - const _Tp __mum49 = __mu - _Tp(49); - const _Tp __xx = _Tp(64) * __x * __x; - const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) - * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); - const _Tp __Q = __mum1 / (_Tp(8) * __x) - * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); - - const _Tp __chi = __x - (__nu + _Tp(0.5L)) - * __numeric_constants<_Tp>::__pi_2(); - const _Tp __c = std::cos(__chi); - const _Tp __s = std::sin(__chi); - - const _Tp __coef = std::sqrt(_Tp(2) - / (__numeric_constants<_Tp>::__pi() * __x)); - __Jnu = __coef * (__c * __P - __s * __Q); - __Nnu = __coef * (__s * __P + __c * __Q); - - return; - } - - - /** - * @brief This routine returns the cylindrical Bessel functions - * of order \f$ \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 -= std::tr1::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; - } - } - - _GLIBCXX_END_NAMESPACE_VERSION - } // namespace std::tr1::__detail -} -} - -#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |