X-Git-Url: https://oss.titaniummirror.com/gitweb?a=blobdiff_plain;f=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fpoly_laguerre.tcc;fp=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fpoly_laguerre.tcc;h=9c999dbdfed96ea0cd711d83b6bbc79403b17f40;hb=6fed43773c9b0ce596dca5686f37ac3fc0fa11c0;hp=0000000000000000000000000000000000000000;hpb=27b11d56b743098deb193d510b337ba22dc52e5c;p=msp430-gcc.git diff --git a/libstdc++-v3/include/tr1/poly_laguerre.tcc b/libstdc++-v3/include/tr1/poly_laguerre.tcc new file mode 100644 index 00000000..9c999dbd --- /dev/null +++ b/libstdc++-v3/include/tr1/poly_laguerre.tcc @@ -0,0 +1,329 @@ +// 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 +// . + +/** @file tr1/poly_laguerre.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 based on: +// (1) Handbook of Mathematical Functions, +// Ed. Milton Abramowitz and Irene A. Stegun, +// Dover Publications, +// Section 13, pp. 509-510, Section 22 pp. 773-802 +// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl + +#ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC +#define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 + +namespace std +{ +namespace tr1 +{ + + // [5.2] Special functions + + // Implementation-space details. + namespace __detail + { + + + /** + * @brief This routine returns the associated Laguerre polynomial + * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. + * Abramowitz & Stegun, 13.5.21 + * + * @param __n The order of the Laguerre function. + * @param __alpha The degree of the Laguerre function. + * @param __x The argument of the Laguerre function. + * @return The value of the Laguerre function of order n, + * degree @f$ \alpha @f$, and argument x. + * + * This is from the GNU Scientific Library. + */ + template + _Tp + __poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1, + const _Tp __x) + { + const _Tp __a = -_Tp(__n); + const _Tp __b = _Tp(__alpha1) + _Tp(1); + const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; + const _Tp __cos2th = __x / __eta; + const _Tp __sin2th = _Tp(1) - __cos2th; + const _Tp __th = std::acos(std::sqrt(__cos2th)); + const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() + * __numeric_constants<_Tp>::__pi_2() + * __eta * __eta * __cos2th * __sin2th; + +#if _GLIBCXX_USE_C99_MATH_TR1 + const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b); + const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1)); +#else + const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); + const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); +#endif + + _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) + * std::log(_Tp(0.25L) * __x * __eta); + _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); + _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x + + __pre_term1 - __pre_term2; + _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); + _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta + * (_Tp(2) * __th + - std::sin(_Tp(2) * __th)) + + __numeric_constants<_Tp>::__pi_4()); + _Tp __ser = __ser_term1 + __ser_term2; + + return std::exp(__lnpre) * __ser; + } + + + /** + * @brief Evaluate the polynomial based on the confluent hypergeometric + * function in a safe way, with no restriction on the arguments. + * + * The associated Laguerre function is defined by + * @f[ + * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} + * _1F_1(-n; \alpha + 1; x) + * @f] + * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and + * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. + * + * This function assumes x != 0. + * + * This is from the GNU Scientific Library. + */ + template + _Tp + __poly_laguerre_hyperg(const unsigned int __n, const _Tpa __alpha1, + const _Tp __x) + { + const _Tp __b = _Tp(__alpha1) + _Tp(1); + const _Tp __mx = -__x; + const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) + : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); + // Get |x|^n/n! + _Tp __tc = _Tp(1); + const _Tp __ax = std::abs(__x); + for (unsigned int __k = 1; __k <= __n; ++__k) + __tc *= (__ax / __k); + + _Tp __term = __tc * __tc_sgn; + _Tp __sum = __term; + for (int __k = int(__n) - 1; __k >= 0; --__k) + { + __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) + * _Tp(__k + 1) / __mx; + __sum += __term; + } + + return __sum; + } + + + /** + * @brief This routine returns the associated Laguerre polynomial + * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ + * by recursion. + * + * The associated Laguerre function is defined by + * @f[ + * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} + * _1F_1(-n; \alpha + 1; x) + * @f] + * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and + * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. + * + * The associated Laguerre polynomial is defined for integral + * @f$ \alpha = m @f$ by: + * @f[ + * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) + * @f] + * where the Laguerre polynomial is defined by: + * @f[ + * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) + * @f] + * + * @param __n The order of the Laguerre function. + * @param __alpha The degree of the Laguerre function. + * @param __x The argument of the Laguerre function. + * @return The value of the Laguerre function of order n, + * degree @f$ \alpha @f$, and argument x. + */ + template + _Tp + __poly_laguerre_recursion(const unsigned int __n, + const _Tpa __alpha1, const _Tp __x) + { + // Compute l_0. + _Tp __l_0 = _Tp(1); + if (__n == 0) + return __l_0; + + // Compute l_1^alpha. + _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); + if (__n == 1) + return __l_1; + + // Compute l_n^alpha by recursion on n. + _Tp __l_n2 = __l_0; + _Tp __l_n1 = __l_1; + _Tp __l_n = _Tp(0); + for (unsigned int __nn = 2; __nn <= __n; ++__nn) + { + __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) + * __l_n1 / _Tp(__nn) + - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); + __l_n2 = __l_n1; + __l_n1 = __l_n; + } + + return __l_n; + } + + + /** + * @brief This routine returns the associated Laguerre polynomial + * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. + * + * The associated Laguerre function is defined by + * @f[ + * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} + * _1F_1(-n; \alpha + 1; x) + * @f] + * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and + * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. + * + * The associated Laguerre polynomial is defined for integral + * @f$ \alpha = m @f$ by: + * @f[ + * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) + * @f] + * where the Laguerre polynomial is defined by: + * @f[ + * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) + * @f] + * + * @param __n The order of the Laguerre function. + * @param __alpha The degree of the Laguerre function. + * @param __x The argument of the Laguerre function. + * @return The value of the Laguerre function of order n, + * degree @f$ \alpha @f$, and argument x. + */ + template + inline _Tp + __poly_laguerre(const unsigned int __n, const _Tpa __alpha1, + const _Tp __x) + { + if (__x < _Tp(0)) + std::__throw_domain_error(__N("Negative argument " + "in __poly_laguerre.")); + // Return NaN on NaN input. + else if (__isnan(__x)) + return std::numeric_limits<_Tp>::quiet_NaN(); + else if (__n == 0) + return _Tp(1); + else if (__n == 1) + return _Tp(1) + _Tp(__alpha1) - __x; + else if (__x == _Tp(0)) + { + _Tp __prod = _Tp(__alpha1) + _Tp(1); + for (unsigned int __k = 2; __k <= __n; ++__k) + __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); + return __prod; + } + else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) + && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) + return __poly_laguerre_large_n(__n, __alpha1, __x); + else if (_Tp(__alpha1) >= _Tp(0) + || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) + return __poly_laguerre_recursion(__n, __alpha1, __x); + else + return __poly_laguerre_hyperg(__n, __alpha1, __x); + } + + + /** + * @brief This routine returns the associated Laguerre polynomial + * of order n, degree m: @f$ L_n^m(x) @f$. + * + * The associated Laguerre polynomial is defined for integral + * @f$ \alpha = m @f$ by: + * @f[ + * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) + * @f] + * where the Laguerre polynomial is defined by: + * @f[ + * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) + * @f] + * + * @param __n The order of the Laguerre polynomial. + * @param __m The degree of the Laguerre polynomial. + * @param __x The argument of the Laguerre polynomial. + * @return The value of the associated Laguerre polynomial of order n, + * degree m, and argument x. + */ + template + inline _Tp + __assoc_laguerre(const unsigned int __n, const unsigned int __m, + const _Tp __x) + { + return __poly_laguerre(__n, __m, __x); + } + + + /** + * @brief This routine returns the Laguerre polynomial + * of order n: @f$ L_n(x) @f$. + * + * The Laguerre polynomial is defined by: + * @f[ + * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) + * @f] + * + * @param __n The order of the Laguerre polynomial. + * @param __x The argument of the Laguerre polynomial. + * @return The value of the Laguerre polynomial of order n + * and argument x. + */ + template + inline _Tp + __laguerre(const unsigned int __n, const _Tp __x) + { + return __poly_laguerre(__n, 0, __x); + } + + } // namespace std::tr1::__detail +} +} + +#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC