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diff --git a/uclibc-crosstools-gcc-4.4.2-1/usr/mips-linux-uclibc/include/c++/4.4.2/tr1/bessel_function.tcc b/uclibc-crosstools-gcc-4.4.2-1/usr/mips-linux-uclibc/include/c++/4.4.2/tr1/bessel_function.tcc new file mode 100644 index 0000000..2510215 --- /dev/null +++ b/uclibc-crosstools-gcc-4.4.2-1/usr/mips-linux-uclibc/include/c++/4.4.2/tr1/bessel_function.tcc @@ -0,0 +1,627 @@ +// 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 |