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// Special functions -*- C++ -*- |
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|
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// Copyright (C) 2006-2021 Free Software Foundation, Inc. |
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// |
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// This file is part of the GNU ISO C++ Library. This library is free |
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// software; you can redistribute it and/or modify it under the |
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// terms of the GNU General Public License as published by the |
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// Free Software Foundation; either version 3, or (at your option) |
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// any later version. |
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// |
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// This library is distributed in the hope that it will be useful, |
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// but WITHOUT ANY WARRANTY; without even the implied warranty of |
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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// GNU General Public License for more details. |
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// |
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// Under Section 7 of GPL version 3, you are granted additional |
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// permissions described in the GCC Runtime Library Exception, version |
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// 3.1, as published by the Free Software Foundation. |
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|
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// You should have received a copy of the GNU General Public License and |
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// a copy of the GCC Runtime Library Exception along with this program; |
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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// <http://www.gnu.org/licenses/>. |
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|
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/** @file tr1/bessel_function.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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*/ |
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|
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/* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c |
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* Copyright (C) 1996-2003 Gerard Jungman |
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*/ |
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|
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// |
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// ISO C++ 14882 TR1: 5.2 Special functions |
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// |
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|
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// Written by Edward Smith-Rowland. |
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// |
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// References: |
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// (1) Handbook of Mathematical Functions, |
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// ed. Milton Abramowitz and Irene A. Stegun, |
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// Dover Publications, |
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// Section 9, pp. 355-434, Section 10 pp. 435-478 |
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, |
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), |
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// 2nd ed, pp. 240-245 |
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|
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#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
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#define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 |
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|
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#include <tr1/special_function_util.h> |
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|
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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|
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#if _GLIBCXX_USE_STD_SPEC_FUNCS |
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# define _GLIBCXX_MATH_NS ::std |
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#elif defined(_GLIBCXX_TR1_CMATH) |
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namespace tr1 |
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{ |
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# define _GLIBCXX_MATH_NS ::std::tr1 |
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#else |
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# error do not include this header directly, use <cmath> or <tr1/cmath> |
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#endif |
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// [5.2] Special functions |
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|
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// Implementation-space details. |
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namespace __detail |
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{ |
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/** |
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* @brief Compute the gamma functions required by the Temme series |
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* expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. |
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* @f[ |
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* \Gamma_1 = \frac{1}{2\mu} |
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* [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] |
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* @f] |
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* and |
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* @f[ |
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* \Gamma_2 = \frac{1}{2} |
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* [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] |
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* @f] |
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* where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. |
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* is the nearest integer to @f$ \nu @f$. |
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* The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ |
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* are returned as well. |
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* |
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* The accuracy requirements on this are exquisite. |
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* |
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* @param __mu The input parameter of the gamma functions. |
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* @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ |
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* @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ |
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* @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ |
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* @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ |
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*/ |
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template <typename _Tp> |
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void |
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__gamma_temme(_Tp __mu, |
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_Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) |
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{ |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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__gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu); |
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__gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu); |
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#else |
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__gampl = _Tp(1) / __gamma(_Tp(1) + __mu); |
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__gammi = _Tp(1) / __gamma(_Tp(1) - __mu); |
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#endif |
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|
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if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) |
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__gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); |
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else |
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__gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); |
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|
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__gam2 = (__gammi + __gampl) / (_Tp(2)); |
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|
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return; |
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} |
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|
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|
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/** |
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* @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann |
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* @f$ N_\nu(x) @f$ functions and their first derivatives |
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* @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. |
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* These four functions are computed together for numerical |
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* stability. |
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* |
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* @param __nu The order of the Bessel functions. |
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* @param __x The argument of the Bessel functions. |
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* @param __Jnu The output Bessel function of the first kind. |
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* @param __Nnu The output Neumann function (Bessel function of the second kind). |
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* @param __Jpnu The output derivative of the Bessel function of the first kind. |
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* @param __Npnu The output derivative of the Neumann function. |
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*/ |
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template <typename _Tp> |
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void |
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__bessel_jn(_Tp __nu, _Tp __x, |
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_Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) |
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{ |
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if (__x == _Tp(0)) |
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{ |
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if (__nu == _Tp(0)) |
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{ |
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__Jnu = _Tp(1); |
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__Jpnu = _Tp(0); |
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} |
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else if (__nu == _Tp(1)) |
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{ |
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__Jnu = _Tp(0); |
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__Jpnu = _Tp(0.5L); |
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} |
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else |
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{ |
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__Jnu = _Tp(0); |
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__Jpnu = _Tp(0); |
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} |
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__Nnu = -std::numeric_limits<_Tp>::infinity(); |
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__Npnu = std::numeric_limits<_Tp>::infinity(); |
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return; |
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} |
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|
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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// When the multiplier is N i.e. |
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// fp_min = N * min() |
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// Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! |
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//const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); |
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const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); |
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const int __max_iter = 15000; |
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const _Tp __x_min = _Tp(2); |
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|
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const int __nl = (__x < __x_min |
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? static_cast<int>(__nu + _Tp(0.5L)) |
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: std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); |
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|
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const _Tp __mu = __nu - __nl; |
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const _Tp __mu2 = __mu * __mu; |
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const _Tp __xi = _Tp(1) / __x; |
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const _Tp __xi2 = _Tp(2) * __xi; |
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_Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); |
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int __isign = 1; |
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_Tp __h = __nu * __xi; |
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if (__h < __fp_min) |
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__h = __fp_min; |
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_Tp __b = __xi2 * __nu; |
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_Tp __d = _Tp(0); |
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_Tp __c = __h; |
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int __i; |
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for (__i = 1; __i <= __max_iter; ++__i) |
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{ |
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__b += __xi2; |
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__d = __b - __d; |
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if (std::abs(__d) < __fp_min) |
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__d = __fp_min; |
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__c = __b - _Tp(1) / __c; |
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if (std::abs(__c) < __fp_min) |
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__c = __fp_min; |
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__d = _Tp(1) / __d; |
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const _Tp __del = __c * __d; |
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__h *= __del; |
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if (__d < _Tp(0)) |
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__isign = -__isign; |
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if (std::abs(__del - _Tp(1)) < __eps) |
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break; |
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} |
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if (__i > __max_iter) |
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std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " |
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"try asymptotic expansion.")); |
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_Tp __Jnul = __isign * __fp_min; |
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_Tp __Jpnul = __h * __Jnul; |
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_Tp __Jnul1 = __Jnul; |
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_Tp __Jpnu1 = __Jpnul; |
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_Tp __fact = __nu * __xi; |
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for ( int __l = __nl; __l >= 1; --__l ) |
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{ |
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const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; |
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__fact -= __xi; |
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__Jpnul = __fact * __Jnutemp - __Jnul; |
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__Jnul = __Jnutemp; |
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} |
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if (__Jnul == _Tp(0)) |
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__Jnul = __eps; |
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_Tp __f= __Jpnul / __Jnul; |
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_Tp __Nmu, __Nnu1, __Npmu, __Jmu; |
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if (__x < __x_min) |
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{ |
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const _Tp __x2 = __x / _Tp(2); |
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const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; |
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_Tp __fact = (std::abs(__pimu) < __eps |
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? _Tp(1) : __pimu / std::sin(__pimu)); |
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_Tp __d = -std::log(__x2); |
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_Tp __e = __mu * __d; |
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_Tp __fact2 = (std::abs(__e) < __eps |
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? _Tp(1) : std::sinh(__e) / __e); |
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_Tp __gam1, __gam2, __gampl, __gammi; |
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__gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); |
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_Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) |
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* __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); |
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__e = std::exp(__e); |
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_Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); |
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_Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); |
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const _Tp __pimu2 = __pimu / _Tp(2); |
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_Tp __fact3 = (std::abs(__pimu2) < __eps |
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? _Tp(1) : std::sin(__pimu2) / __pimu2 ); |
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_Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; |
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_Tp __c = _Tp(1); |
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__d = -__x2 * __x2; |
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_Tp __sum = __ff + __r * __q; |
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_Tp __sum1 = __p; |
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for (__i = 1; __i <= __max_iter; ++__i) |
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{ |
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__ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); |
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__c *= __d / _Tp(__i); |
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__p /= _Tp(__i) - __mu; |
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__q /= _Tp(__i) + __mu; |
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const _Tp __del = __c * (__ff + __r * __q); |
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__sum += __del; |
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const _Tp __del1 = __c * __p - __i * __del; |
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__sum1 += __del1; |
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if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) |
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break; |
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} |
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if ( __i > __max_iter ) |
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std::__throw_runtime_error(__N("Bessel y series failed to converge " |
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"in __bessel_jn.")); |
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__Nmu = -__sum; |
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__Nnu1 = -__sum1 * __xi2; |
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__Npmu = __mu * __xi * __Nmu - __Nnu1; |
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__Jmu = __w / (__Npmu - __f * __Nmu); |
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} |
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else |
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{ |
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_Tp __a = _Tp(0.25L) - __mu2; |
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_Tp __q = _Tp(1); |
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_Tp __p = -__xi / _Tp(2); |
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_Tp __br = _Tp(2) * __x; |
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_Tp __bi = _Tp(2); |
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_Tp __fact = __a * __xi / (__p * __p + __q * __q); |
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_Tp __cr = __br + __q * __fact; |
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_Tp __ci = __bi + __p * __fact; |
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_Tp __den = __br * __br + __bi * __bi; |
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_Tp __dr = __br / __den; |
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_Tp __di = -__bi / __den; |
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_Tp __dlr = __cr * __dr - __ci * __di; |
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_Tp __dli = __cr * __di + __ci * __dr; |
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_Tp __temp = __p * __dlr - __q * __dli; |
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__q = __p * __dli + __q * __dlr; |
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__p = __temp; |
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int __i; |
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for (__i = 2; __i <= __max_iter; ++__i) |
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{ |
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__a += _Tp(2 * (__i - 1)); |
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__bi += _Tp(2); |
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__dr = __a * __dr + __br; |
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__di = __a * __di + __bi; |
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if (std::abs(__dr) + std::abs(__di) < __fp_min) |
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__dr = __fp_min; |
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__fact = __a / (__cr * __cr + __ci * __ci); |
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__cr = __br + __cr * __fact; |
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__ci = __bi - __ci * __fact; |
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if (std::abs(__cr) + std::abs(__ci) < __fp_min) |
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__cr = __fp_min; |
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__den = __dr * __dr + __di * __di; |
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__dr /= __den; |
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__di /= -__den; |
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__dlr = __cr * __dr - __ci * __di; |
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__dli = __cr * __di + __ci * __dr; |
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__temp = __p * __dlr - __q * __dli; |
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__q = __p * __dli + __q * __dlr; |
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__p = __temp; |
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if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) |
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break; |
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} |
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if (__i > __max_iter) |
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std::__throw_runtime_error(__N("Lentz's method failed " |
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"in __bessel_jn.")); |
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const _Tp __gam = (__p - __f) / __q; |
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__Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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__Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul); |
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#else |
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if (__Jmu * __Jnul < _Tp(0)) |
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__Jmu = -__Jmu; |
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#endif |
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__Nmu = __gam * __Jmu; |
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__Npmu = (__p + __q / __gam) * __Nmu; |
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__Nnu1 = __mu * __xi * __Nmu - __Npmu; |
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} |
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__fact = __Jmu / __Jnul; |
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__Jnu = __fact * __Jnul1; |
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__Jpnu = __fact * __Jpnu1; |
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for (__i = 1; __i <= __nl; ++__i) |
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{ |
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const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; |
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__Nmu = __Nnu1; |
| 336 |
__Nnu1 = __Nnutemp; |
| 337 |
} |
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__Nnu = __Nmu; |
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__Npnu = __nu * __xi * __Nmu - __Nnu1; |
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|
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return; |
| 342 |
} |
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|
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|
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/** |
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* @brief This routine computes the asymptotic cylindrical Bessel |
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* and Neumann functions of order nu: \f$ J_{\nu} \f$, |
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* \f$ N_{\nu} \f$. |
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* |
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* References: |
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* (1) Handbook of Mathematical Functions, |
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* ed. Milton Abramowitz and Irene A. Stegun, |
| 353 |
* Dover Publications, |
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* Section 9 p. 364, Equations 9.2.5-9.2.10 |
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* |
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* @param __nu The order of the Bessel functions. |
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* @param __x The argument of the Bessel functions. |
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* @param __Jnu The output Bessel function of the first kind. |
| 359 |
* @param __Nnu The output Neumann function (Bessel function of the second kind). |
| 360 |
*/ |
| 361 |
template <typename _Tp> |
| 362 |
void |
| 363 |
__cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) |
| 364 |
{ |
| 365 |
const _Tp __mu = _Tp(4) * __nu * __nu; |
| 366 |
const _Tp __8x = _Tp(8) * __x; |
| 367 |
|
| 368 |
_Tp __P = _Tp(0); |
| 369 |
_Tp __Q = _Tp(0); |
| 370 |
|
| 371 |
_Tp __k = _Tp(0); |
| 372 |
_Tp __term = _Tp(1); |
| 373 |
|
| 374 |
int __epsP = 0; |
| 375 |
int __epsQ = 0; |
| 376 |
|
| 377 |
_Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
| 378 |
|
| 379 |
do |
| 380 |
{ |
| 381 |
__term *= (__k == 0 |
| 382 |
? _Tp(1) |
| 383 |
: -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x)); |
| 384 |
|
| 385 |
__epsP = std::abs(__term) < __eps * std::abs(__P); |
| 386 |
__P += __term; |
| 387 |
|
| 388 |
__k++; |
| 389 |
|
| 390 |
__term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x); |
| 391 |
__epsQ = std::abs(__term) < __eps * std::abs(__Q); |
| 392 |
__Q += __term; |
| 393 |
|
| 394 |
if (__epsP && __epsQ && __k > (__nu / 2.)) |
| 395 |
break; |
| 396 |
|
| 397 |
__k++; |
| 398 |
} |
| 399 |
while (__k < 1000); |
| 400 |
|
| 401 |
const _Tp __chi = __x - (__nu + _Tp(0.5L)) |
| 402 |
* __numeric_constants<_Tp>::__pi_2(); |
| 403 |
|
| 404 |
const _Tp __c = std::cos(__chi); |
| 405 |
const _Tp __s = std::sin(__chi); |
| 406 |
|
| 407 |
const _Tp __coef = std::sqrt(_Tp(2) |
| 408 |
/ (__numeric_constants<_Tp>::__pi() * __x)); |
| 409 |
|
| 410 |
__Jnu = __coef * (__c * __P - __s * __Q); |
| 411 |
__Nnu = __coef * (__s * __P + __c * __Q); |
| 412 |
|
| 413 |
return; |
| 414 |
} |
| 415 |
|
| 416 |
|
| 417 |
/** |
| 418 |
* @brief This routine returns the cylindrical Bessel functions |
| 419 |
* of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ |
| 420 |
* by series expansion. |
| 421 |
* |
| 422 |
* The modified cylindrical Bessel function is: |
| 423 |
* @f[ |
| 424 |
* Z_{\nu}(x) = \sum_{k=0}^{\infty} |
| 425 |
* \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
| 426 |
* @f] |
| 427 |
* where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for |
| 428 |
* \f$ Z = I \f$ or \f$ J \f$ respectively. |
| 429 |
* |
| 430 |
* See Abramowitz & Stegun, 9.1.10 |
| 431 |
* Abramowitz & Stegun, 9.6.7 |
| 432 |
* (1) Handbook of Mathematical Functions, |
| 433 |
* ed. Milton Abramowitz and Irene A. Stegun, |
| 434 |
* Dover Publications, |
| 435 |
* Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 |
| 436 |
* |
| 437 |
* @param __nu The order of the Bessel function. |
| 438 |
* @param __x The argument of the Bessel function. |
| 439 |
* @param __sgn The sign of the alternate terms |
| 440 |
* -1 for the Bessel function of the first kind. |
| 441 |
* +1 for the modified Bessel function of the first kind. |
| 442 |
* @return The output Bessel function. |
| 443 |
*/ |
| 444 |
template <typename _Tp> |
| 445 |
_Tp |
| 446 |
__cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, |
| 447 |
unsigned int __max_iter) |
| 448 |
{ |
| 449 |
if (__x == _Tp(0)) |
| 450 |
return __nu == _Tp(0) ? _Tp(1) : _Tp(0); |
| 451 |
|
| 452 |
const _Tp __x2 = __x / _Tp(2); |
| 453 |
_Tp __fact = __nu * std::log(__x2); |
| 454 |
#if _GLIBCXX_USE_C99_MATH_TR1 |
| 455 |
__fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1)); |
| 456 |
#else |
| 457 |
__fact -= __log_gamma(__nu + _Tp(1)); |
| 458 |
#endif |
| 459 |
__fact = std::exp(__fact); |
| 460 |
const _Tp __xx4 = __sgn * __x2 * __x2; |
| 461 |
_Tp __Jn = _Tp(1); |
| 462 |
_Tp __term = _Tp(1); |
| 463 |
|
| 464 |
for (unsigned int __i = 1; __i < __max_iter; ++__i) |
| 465 |
{ |
| 466 |
__term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); |
| 467 |
__Jn += __term; |
| 468 |
if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) |
| 469 |
break; |
| 470 |
} |
| 471 |
|
| 472 |
return __fact * __Jn; |
| 473 |
} |
| 474 |
|
| 475 |
|
| 476 |
/** |
| 477 |
* @brief Return the Bessel function of order \f$ \nu \f$: |
| 478 |
* \f$ J_{\nu}(x) \f$. |
| 479 |
* |
| 480 |
* The cylindrical Bessel function is: |
| 481 |
* @f[ |
| 482 |
* J_{\nu}(x) = \sum_{k=0}^{\infty} |
| 483 |
* \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
| 484 |
* @f] |
| 485 |
* |
| 486 |
* @param __nu The order of the Bessel function. |
| 487 |
* @param __x The argument of the Bessel function. |
| 488 |
* @return The output Bessel function. |
| 489 |
*/ |
| 490 |
template<typename _Tp> |
| 491 |
_Tp |
| 492 |
__cyl_bessel_j(_Tp __nu, _Tp __x) |
| 493 |
{ |
| 494 |
if (__nu < _Tp(0) || __x < _Tp(0)) |
| 495 |
std::__throw_domain_error(__N("Bad argument " |
| 496 |
"in __cyl_bessel_j.")); |
| 497 |
else if (__isnan(__nu) || __isnan(__x)) |
| 498 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 499 |
else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) |
| 500 |
return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); |
| 501 |
else if (__x > _Tp(1000)) |
| 502 |
{ |
| 503 |
_Tp __J_nu, __N_nu; |
| 504 |
__cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
| 505 |
return __J_nu; |
| 506 |
} |
| 507 |
else |
| 508 |
{ |
| 509 |
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
| 510 |
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| 511 |
return __J_nu; |
| 512 |
} |
| 513 |
} |
| 514 |
|
| 515 |
|
| 516 |
/** |
| 517 |
* @brief Return the Neumann function of order \f$ \nu \f$: |
| 518 |
* \f$ N_{\nu}(x) \f$. |
| 519 |
* |
| 520 |
* The Neumann function is defined by: |
| 521 |
* @f[ |
| 522 |
* N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} |
| 523 |
* {\sin \nu\pi} |
| 524 |
* @f] |
| 525 |
* where for integral \f$ \nu = n \f$ a limit is taken: |
| 526 |
* \f$ lim_{\nu \to n} \f$. |
| 527 |
* |
| 528 |
* @param __nu The order of the Neumann function. |
| 529 |
* @param __x The argument of the Neumann function. |
| 530 |
* @return The output Neumann function. |
| 531 |
*/ |
| 532 |
template<typename _Tp> |
| 533 |
_Tp |
| 534 |
__cyl_neumann_n(_Tp __nu, _Tp __x) |
| 535 |
{ |
| 536 |
if (__nu < _Tp(0) || __x < _Tp(0)) |
| 537 |
std::__throw_domain_error(__N("Bad argument " |
| 538 |
"in __cyl_neumann_n.")); |
| 539 |
else if (__isnan(__nu) || __isnan(__x)) |
| 540 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 541 |
else if (__x > _Tp(1000)) |
| 542 |
{ |
| 543 |
_Tp __J_nu, __N_nu; |
| 544 |
__cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
| 545 |
return __N_nu; |
| 546 |
} |
| 547 |
else |
| 548 |
{ |
| 549 |
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
| 550 |
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| 551 |
return __N_nu; |
| 552 |
} |
| 553 |
} |
| 554 |
|
| 555 |
|
| 556 |
/** |
| 557 |
* @brief Compute the spherical Bessel @f$ j_n(x) @f$ |
| 558 |
* and Neumann @f$ n_n(x) @f$ functions and their first |
| 559 |
* derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ |
| 560 |
* respectively. |
| 561 |
* |
| 562 |
* @param __n The order of the spherical Bessel function. |
| 563 |
* @param __x The argument of the spherical Bessel function. |
| 564 |
* @param __j_n The output spherical Bessel function. |
| 565 |
* @param __n_n The output spherical Neumann function. |
| 566 |
* @param __jp_n The output derivative of the spherical Bessel function. |
| 567 |
* @param __np_n The output derivative of the spherical Neumann function. |
| 568 |
*/ |
| 569 |
template <typename _Tp> |
| 570 |
void |
| 571 |
__sph_bessel_jn(unsigned int __n, _Tp __x, |
| 572 |
_Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) |
| 573 |
{ |
| 574 |
const _Tp __nu = _Tp(__n) + _Tp(0.5L); |
| 575 |
|
| 576 |
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
| 577 |
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| 578 |
|
| 579 |
const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() |
| 580 |
/ std::sqrt(__x); |
| 581 |
|
| 582 |
__j_n = __factor * __J_nu; |
| 583 |
__n_n = __factor * __N_nu; |
| 584 |
__jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); |
| 585 |
__np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); |
| 586 |
|
| 587 |
return; |
| 588 |
} |
| 589 |
|
| 590 |
|
| 591 |
/** |
| 592 |
* @brief Return the spherical Bessel function |
| 593 |
* @f$ j_n(x) @f$ of order n. |
| 594 |
* |
| 595 |
* The spherical Bessel function is defined by: |
| 596 |
* @f[ |
| 597 |
* j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) |
| 598 |
* @f] |
| 599 |
* |
| 600 |
* @param __n The order of the spherical Bessel function. |
| 601 |
* @param __x The argument of the spherical Bessel function. |
| 602 |
* @return The output spherical Bessel function. |
| 603 |
*/ |
| 604 |
template <typename _Tp> |
| 605 |
_Tp |
| 606 |
__sph_bessel(unsigned int __n, _Tp __x) |
| 607 |
{ |
| 608 |
if (__x < _Tp(0)) |
| 609 |
std::__throw_domain_error(__N("Bad argument " |
| 610 |
"in __sph_bessel.")); |
| 611 |
else if (__isnan(__x)) |
| 612 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 613 |
else if (__x == _Tp(0)) |
| 614 |
{ |
| 615 |
if (__n == 0) |
| 616 |
return _Tp(1); |
| 617 |
else |
| 618 |
return _Tp(0); |
| 619 |
} |
| 620 |
else |
| 621 |
{ |
| 622 |
_Tp __j_n, __n_n, __jp_n, __np_n; |
| 623 |
__sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); |
| 624 |
return __j_n; |
| 625 |
} |
| 626 |
} |
| 627 |
|
| 628 |
|
| 629 |
/** |
| 630 |
* @brief Return the spherical Neumann function |
| 631 |
* @f$ n_n(x) @f$. |
| 632 |
* |
| 633 |
* The spherical Neumann function is defined by: |
| 634 |
* @f[ |
| 635 |
* n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) |
| 636 |
* @f] |
| 637 |
* |
| 638 |
* @param __n The order of the spherical Neumann function. |
| 639 |
* @param __x The argument of the spherical Neumann function. |
| 640 |
* @return The output spherical Neumann function. |
| 641 |
*/ |
| 642 |
template <typename _Tp> |
| 643 |
_Tp |
| 644 |
__sph_neumann(unsigned int __n, _Tp __x) |
| 645 |
{ |
| 646 |
if (__x < _Tp(0)) |
| 647 |
std::__throw_domain_error(__N("Bad argument " |
| 648 |
"in __sph_neumann.")); |
| 649 |
else if (__isnan(__x)) |
| 650 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 651 |
else if (__x == _Tp(0)) |
| 652 |
return -std::numeric_limits<_Tp>::infinity(); |
| 653 |
else |
| 654 |
{ |
| 655 |
_Tp __j_n, __n_n, __jp_n, __np_n; |
| 656 |
__sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); |
| 657 |
return __n_n; |
| 658 |
} |
| 659 |
} |
| 660 |
} // namespace __detail |
| 661 |
#undef _GLIBCXX_MATH_NS |
| 662 |
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
| 663 |
} // namespace tr1 |
| 664 |
#endif |
| 665 |
|
| 666 |
_GLIBCXX_END_NAMESPACE_VERSION |
| 667 |
} |
| 668 |
|
| 669 |
#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
