// Special functions -*- C++ -*- // Copyright (C) 2006-2020 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$. * * References: * (1) Handbook of Mathematical Functions, * ed. Milton Abramowitz and Irene A. Stegun, * Dover Publications, * Section 9 p. 364, Equations 9.2.5-9.2.10 * * @param __nu The order of the Bessel functions. * @param __x The argument of the Bessel functions. * @param __Jnu The output Bessel function of the first kind. * @param __Nnu The output Neumann function (Bessel function of the second kind). */ template <typename _Tp> void __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) { const _Tp __mu = _Tp(4) * __nu * __nu; const _Tp __8x = _Tp(8) * __x; _Tp __P = _Tp(0); _Tp __Q = _Tp(0); _Tp __k = _Tp(0); _Tp __term = _Tp(1); int __epsP = 0; int __epsQ = 0; _Tp __eps = std::numeric_limits<_Tp>::epsilon(); do { __term *= (__k == 0 ? _Tp(1) : -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x)); __epsP = std::abs(__term) < __eps * std::abs(__P); __P += __term; __k++; __term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x); __epsQ = std::abs(__term) < __eps * std::abs(__Q); __Q += __term; if (__epsP && __epsQ && __k > (__nu / 2.)) break; __k++; } while (__k < 1000); const _Tp __chi = __x - (__nu + _Tp(0.5L)) * __numeric_constants<_Tp>::__pi_2(); const _Tp __c = std::cos(__chi); const _Tp __s = std::sin(__chi); const _Tp __coef = std::sqrt(_Tp(2) / (__numeric_constants<_Tp>::__pi() * __x)); __Jnu = __coef * (__c * __P - __s * __Q); __Nnu = __coef * (__s * __P + __c * __Q); return; } /** * @brief This routine returns the cylindrical Bessel functions * of order \f$ \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