// Special functions -*- C++ -*-// Copyright (C) 2006, 2007, 2008// 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 2, 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.//// You should have received a copy of the GNU General Public License along// with this library; see the file COPYING. If not, write to the Free// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,// USA.//// As a special exception, you may use this file as part of a free software// library without restriction. Specifically, if other files instantiate// templates or use macros or inline functions from this file, or you compile// this file and link it with other files to produce an executable, this// file does not by itself cause the resulting executable to be covered by// the GNU General Public License. This exception does not however// invalidate any other reasons why the executable file might be covered by// the GNU General Public License./** @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$*/template <typename _Tp>void__gamma_temme(const _Tp __mu,_Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi){#if _GLIBCXX_USE_C99_MATH_TR1__gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu);__gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu);#else__gampl = _Tp(1) / __gamma(_Tp(1) + __mu);__gammi = _Tp(1) / __gamma(_Tp(1) - __mu);#endifif (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())__gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());else__gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);__gam2 = (__gammi + __gampl) / (_Tp(2));return;}/*** @brief Compute the Bessel @f$ 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);#elseif (__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);elsereturn _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