| // Special functions -*- C++ -*- |
| |
| // Copyright (C) 2006, 2007, 2008, 2009 |
| // 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. |
| * You should not attempt to use it directly. |
| */ |
| |
| // |
| // 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 |
| { |
| namespace tr1 |
| { |
| |
| // [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(const _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(const _Tp __nu, const _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$. |
| * |
| * References: |
| * (1) Handbook of Mathematical Functions, |
| * ed. Milton Abramowitz and Irene A. Stegun, |
| * Dover Publications, |
| * Section 9 p. 364, Equations 9.2.5-9.2.10 |
| * |
| * @param __nu The order of the Bessel functions. |
| * @param __x The argument of the Bessel functions. |
| * @param __Jnu The output Bessel function of the first kind. |
| * @param __Nnu The output Neumann function (Bessel function of the second kind). |
| */ |
| template <typename _Tp> |
| void |
| __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x, |
| _Tp & __Jnu, _Tp & __Nnu) |
| { |
| const _Tp __coef = std::sqrt(_Tp(2) |
| / (__numeric_constants<_Tp>::__pi() * __x)); |
| const _Tp __mu = _Tp(4) * __nu * __nu; |
| const _Tp __mum1 = __mu - _Tp(1); |
| const _Tp __mum9 = __mu - _Tp(9); |
| const _Tp __mum25 = __mu - _Tp(25); |
| const _Tp __mum49 = __mu - _Tp(49); |
| const _Tp __xx = _Tp(64) * __x * __x; |
| const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) |
| * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); |
| const _Tp __Q = __mum1 / (_Tp(8) * __x) |
| * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); |
| |
| const _Tp __chi = __x - (__nu + _Tp(0.5L)) |
| * __numeric_constants<_Tp>::__pi_2(); |
| const _Tp __c = std::cos(__chi); |
| const _Tp __s = std::sin(__chi); |
| |
| __Jnu = __coef * (__c * __P - __s * __Q); |
| __Nnu = __coef * (__s * __P + __c * __Q); |
| |
| return; |
| } |
| |
| |
| /** |
| * @brief This routine returns the cylindrical Bessel functions |
| * of order \f$ \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(const _Tp __nu, const _Tp __x, const _Tp __sgn, |
| const unsigned int __max_iter) |
| { |
| |
| 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$. |
| * |
| * The cylindrical Bessel function is: |
| * @f[ |
| * J_{\nu}(x) = \sum_{k=0}^{\infty} |
| * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
| * @f] |
| * |
| * @param __nu The order of the Bessel function. |
| * @param __x The argument of the Bessel function. |
| * @return The output Bessel function. |
| */ |
| template<typename _Tp> |
| _Tp |
| __cyl_bessel_j(const _Tp __nu, const _Tp __x) |
| { |
| if (__nu < _Tp(0) || __x < _Tp(0)) |
| std::__throw_domain_error(__N("Bad argument " |
| "in __cyl_bessel_j.")); |
| else if (__isnan(__nu) || __isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) |
| return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); |
| else if (__x > _Tp(1000)) |
| { |
| _Tp __J_nu, __N_nu; |
| __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
| return __J_nu; |
| } |
| else |
| { |
| _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
| __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| return __J_nu; |
| } |
| } |
| |
| |
| /** |
| * @brief Return the Neumann function of order \f$ \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$. |
| * |
| * @param __nu The order of the Neumann function. |
| * @param __x The argument of the Neumann function. |
| * @return The output Neumann function. |
| */ |
| template<typename _Tp> |
| _Tp |
| __cyl_neumann_n(const _Tp __nu, const _Tp __x) |
| { |
| if (__nu < _Tp(0) || __x < _Tp(0)) |
| std::__throw_domain_error(__N("Bad argument " |
| "in __cyl_neumann_n.")); |
| else if (__isnan(__nu) || __isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else if (__x > _Tp(1000)) |
| { |
| _Tp __J_nu, __N_nu; |
| __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
| return __N_nu; |
| } |
| else |
| { |
| _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
| __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| return __N_nu; |
| } |
| } |
| |
| |
| /** |
| * @brief Compute the spherical Bessel @f$ 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(const unsigned int __n, const _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(const unsigned int __n, const _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(const unsigned int __n, const _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 std::tr1::__detail |
| } |
| } |
| |
| #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
