1*38fd1498Szrj // Special functions -*- C++ -*- 2*38fd1498Szrj 3*38fd1498Szrj // Copyright (C) 2006-2018 Free Software Foundation, Inc. 4*38fd1498Szrj // 5*38fd1498Szrj // This file is part of the GNU ISO C++ Library. This library is free 6*38fd1498Szrj // software; you can redistribute it and/or modify it under the 7*38fd1498Szrj // terms of the GNU General Public License as published by the 8*38fd1498Szrj // Free Software Foundation; either version 3, or (at your option) 9*38fd1498Szrj // any later version. 10*38fd1498Szrj // 11*38fd1498Szrj // This library is distributed in the hope that it will be useful, 12*38fd1498Szrj // but WITHOUT ANY WARRANTY; without even the implied warranty of 13*38fd1498Szrj // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 14*38fd1498Szrj // GNU General Public License for more details. 15*38fd1498Szrj // 16*38fd1498Szrj // Under Section 7 of GPL version 3, you are granted additional 17*38fd1498Szrj // permissions described in the GCC Runtime Library Exception, version 18*38fd1498Szrj // 3.1, as published by the Free Software Foundation. 19*38fd1498Szrj 20*38fd1498Szrj // You should have received a copy of the GNU General Public License and 21*38fd1498Szrj // a copy of the GCC Runtime Library Exception along with this program; 22*38fd1498Szrj // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 23*38fd1498Szrj // <http://www.gnu.org/licenses/>. 24*38fd1498Szrj 25*38fd1498Szrj /** @file tr1/poly_laguerre.tcc 26*38fd1498Szrj * This is an internal header file, included by other library headers. 27*38fd1498Szrj * Do not attempt to use it directly. @headername{tr1/cmath} 28*38fd1498Szrj */ 29*38fd1498Szrj 30*38fd1498Szrj // 31*38fd1498Szrj // ISO C++ 14882 TR1: 5.2 Special functions 32*38fd1498Szrj // 33*38fd1498Szrj 34*38fd1498Szrj // Written by Edward Smith-Rowland based on: 35*38fd1498Szrj // (1) Handbook of Mathematical Functions, 36*38fd1498Szrj // Ed. Milton Abramowitz and Irene A. Stegun, 37*38fd1498Szrj // Dover Publications, 38*38fd1498Szrj // Section 13, pp. 509-510, Section 22 pp. 773-802 39*38fd1498Szrj // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 40*38fd1498Szrj 41*38fd1498Szrj #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC 42*38fd1498Szrj #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 43*38fd1498Szrj 44*38fd1498Szrj namespace std _GLIBCXX_VISIBILITY(default) 45*38fd1498Szrj { 46*38fd1498Szrj _GLIBCXX_BEGIN_NAMESPACE_VERSION 47*38fd1498Szrj 48*38fd1498Szrj #if _GLIBCXX_USE_STD_SPEC_FUNCS 49*38fd1498Szrj # define _GLIBCXX_MATH_NS ::std 50*38fd1498Szrj #elif defined(_GLIBCXX_TR1_CMATH) 51*38fd1498Szrj namespace tr1 52*38fd1498Szrj { 53*38fd1498Szrj # define _GLIBCXX_MATH_NS ::std::tr1 54*38fd1498Szrj #else 55*38fd1498Szrj # error do not include this header directly, use <cmath> or <tr1/cmath> 56*38fd1498Szrj #endif 57*38fd1498Szrj // [5.2] Special functions 58*38fd1498Szrj 59*38fd1498Szrj // Implementation-space details. 60*38fd1498Szrj namespace __detail 61*38fd1498Szrj { 62*38fd1498Szrj /** 63*38fd1498Szrj * @brief This routine returns the associated Laguerre polynomial 64*38fd1498Szrj * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. 65*38fd1498Szrj * Abramowitz & Stegun, 13.5.21 66*38fd1498Szrj * 67*38fd1498Szrj * @param __n The order of the Laguerre function. 68*38fd1498Szrj * @param __alpha The degree of the Laguerre function. 69*38fd1498Szrj * @param __x The argument of the Laguerre function. 70*38fd1498Szrj * @return The value of the Laguerre function of order n, 71*38fd1498Szrj * degree @f$ \alpha @f$, and argument x. 72*38fd1498Szrj * 73*38fd1498Szrj * This is from the GNU Scientific Library. 74*38fd1498Szrj */ 75*38fd1498Szrj template<typename _Tpa, typename _Tp> 76*38fd1498Szrj _Tp __poly_laguerre_large_n(unsigned __n,_Tpa __alpha1,_Tp __x)77*38fd1498Szrj __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) 78*38fd1498Szrj { 79*38fd1498Szrj const _Tp __a = -_Tp(__n); 80*38fd1498Szrj const _Tp __b = _Tp(__alpha1) + _Tp(1); 81*38fd1498Szrj const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; 82*38fd1498Szrj const _Tp __cos2th = __x / __eta; 83*38fd1498Szrj const _Tp __sin2th = _Tp(1) - __cos2th; 84*38fd1498Szrj const _Tp __th = std::acos(std::sqrt(__cos2th)); 85*38fd1498Szrj const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() 86*38fd1498Szrj * __numeric_constants<_Tp>::__pi_2() 87*38fd1498Szrj * __eta * __eta * __cos2th * __sin2th; 88*38fd1498Szrj 89*38fd1498Szrj #if _GLIBCXX_USE_C99_MATH_TR1 90*38fd1498Szrj const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b); 91*38fd1498Szrj const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); 92*38fd1498Szrj #else 93*38fd1498Szrj const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); 94*38fd1498Szrj const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); 95*38fd1498Szrj #endif 96*38fd1498Szrj 97*38fd1498Szrj _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) 98*38fd1498Szrj * std::log(_Tp(0.25L) * __x * __eta); 99*38fd1498Szrj _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); 100*38fd1498Szrj _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x 101*38fd1498Szrj + __pre_term1 - __pre_term2; 102*38fd1498Szrj _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); 103*38fd1498Szrj _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta 104*38fd1498Szrj * (_Tp(2) * __th 105*38fd1498Szrj - std::sin(_Tp(2) * __th)) 106*38fd1498Szrj + __numeric_constants<_Tp>::__pi_4()); 107*38fd1498Szrj _Tp __ser = __ser_term1 + __ser_term2; 108*38fd1498Szrj 109*38fd1498Szrj return std::exp(__lnpre) * __ser; 110*38fd1498Szrj } 111*38fd1498Szrj 112*38fd1498Szrj 113*38fd1498Szrj /** 114*38fd1498Szrj * @brief Evaluate the polynomial based on the confluent hypergeometric 115*38fd1498Szrj * function in a safe way, with no restriction on the arguments. 116*38fd1498Szrj * 117*38fd1498Szrj * The associated Laguerre function is defined by 118*38fd1498Szrj * @f[ 119*38fd1498Szrj * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} 120*38fd1498Szrj * _1F_1(-n; \alpha + 1; x) 121*38fd1498Szrj * @f] 122*38fd1498Szrj * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and 123*38fd1498Szrj * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. 124*38fd1498Szrj * 125*38fd1498Szrj * This function assumes x != 0. 126*38fd1498Szrj * 127*38fd1498Szrj * This is from the GNU Scientific Library. 128*38fd1498Szrj */ 129*38fd1498Szrj template<typename _Tpa, typename _Tp> 130*38fd1498Szrj _Tp __poly_laguerre_hyperg(unsigned int __n,_Tpa __alpha1,_Tp __x)131*38fd1498Szrj __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) 132*38fd1498Szrj { 133*38fd1498Szrj const _Tp __b = _Tp(__alpha1) + _Tp(1); 134*38fd1498Szrj const _Tp __mx = -__x; 135*38fd1498Szrj const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) 136*38fd1498Szrj : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); 137*38fd1498Szrj // Get |x|^n/n! 138*38fd1498Szrj _Tp __tc = _Tp(1); 139*38fd1498Szrj const _Tp __ax = std::abs(__x); 140*38fd1498Szrj for (unsigned int __k = 1; __k <= __n; ++__k) 141*38fd1498Szrj __tc *= (__ax / __k); 142*38fd1498Szrj 143*38fd1498Szrj _Tp __term = __tc * __tc_sgn; 144*38fd1498Szrj _Tp __sum = __term; 145*38fd1498Szrj for (int __k = int(__n) - 1; __k >= 0; --__k) 146*38fd1498Szrj { 147*38fd1498Szrj __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) 148*38fd1498Szrj * _Tp(__k + 1) / __mx; 149*38fd1498Szrj __sum += __term; 150*38fd1498Szrj } 151*38fd1498Szrj 152*38fd1498Szrj return __sum; 153*38fd1498Szrj } 154*38fd1498Szrj 155*38fd1498Szrj 156*38fd1498Szrj /** 157*38fd1498Szrj * @brief This routine returns the associated Laguerre polynomial 158*38fd1498Szrj * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ 159*38fd1498Szrj * by recursion. 160*38fd1498Szrj * 161*38fd1498Szrj * The associated Laguerre function is defined by 162*38fd1498Szrj * @f[ 163*38fd1498Szrj * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} 164*38fd1498Szrj * _1F_1(-n; \alpha + 1; x) 165*38fd1498Szrj * @f] 166*38fd1498Szrj * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and 167*38fd1498Szrj * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. 168*38fd1498Szrj * 169*38fd1498Szrj * The associated Laguerre polynomial is defined for integral 170*38fd1498Szrj * @f$ \alpha = m @f$ by: 171*38fd1498Szrj * @f[ 172*38fd1498Szrj * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) 173*38fd1498Szrj * @f] 174*38fd1498Szrj * where the Laguerre polynomial is defined by: 175*38fd1498Szrj * @f[ 176*38fd1498Szrj * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) 177*38fd1498Szrj * @f] 178*38fd1498Szrj * 179*38fd1498Szrj * @param __n The order of the Laguerre function. 180*38fd1498Szrj * @param __alpha The degree of the Laguerre function. 181*38fd1498Szrj * @param __x The argument of the Laguerre function. 182*38fd1498Szrj * @return The value of the Laguerre function of order n, 183*38fd1498Szrj * degree @f$ \alpha @f$, and argument x. 184*38fd1498Szrj */ 185*38fd1498Szrj template<typename _Tpa, typename _Tp> 186*38fd1498Szrj _Tp __poly_laguerre_recursion(unsigned int __n,_Tpa __alpha1,_Tp __x)187*38fd1498Szrj __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) 188*38fd1498Szrj { 189*38fd1498Szrj // Compute l_0. 190*38fd1498Szrj _Tp __l_0 = _Tp(1); 191*38fd1498Szrj if (__n == 0) 192*38fd1498Szrj return __l_0; 193*38fd1498Szrj 194*38fd1498Szrj // Compute l_1^alpha. 195*38fd1498Szrj _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); 196*38fd1498Szrj if (__n == 1) 197*38fd1498Szrj return __l_1; 198*38fd1498Szrj 199*38fd1498Szrj // Compute l_n^alpha by recursion on n. 200*38fd1498Szrj _Tp __l_n2 = __l_0; 201*38fd1498Szrj _Tp __l_n1 = __l_1; 202*38fd1498Szrj _Tp __l_n = _Tp(0); 203*38fd1498Szrj for (unsigned int __nn = 2; __nn <= __n; ++__nn) 204*38fd1498Szrj { 205*38fd1498Szrj __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) 206*38fd1498Szrj * __l_n1 / _Tp(__nn) 207*38fd1498Szrj - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); 208*38fd1498Szrj __l_n2 = __l_n1; 209*38fd1498Szrj __l_n1 = __l_n; 210*38fd1498Szrj } 211*38fd1498Szrj 212*38fd1498Szrj return __l_n; 213*38fd1498Szrj } 214*38fd1498Szrj 215*38fd1498Szrj 216*38fd1498Szrj /** 217*38fd1498Szrj * @brief This routine returns the associated Laguerre polynomial 218*38fd1498Szrj * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. 219*38fd1498Szrj * 220*38fd1498Szrj * The associated Laguerre function is defined by 221*38fd1498Szrj * @f[ 222*38fd1498Szrj * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} 223*38fd1498Szrj * _1F_1(-n; \alpha + 1; x) 224*38fd1498Szrj * @f] 225*38fd1498Szrj * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and 226*38fd1498Szrj * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. 227*38fd1498Szrj * 228*38fd1498Szrj * The associated Laguerre polynomial is defined for integral 229*38fd1498Szrj * @f$ \alpha = m @f$ by: 230*38fd1498Szrj * @f[ 231*38fd1498Szrj * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) 232*38fd1498Szrj * @f] 233*38fd1498Szrj * where the Laguerre polynomial is defined by: 234*38fd1498Szrj * @f[ 235*38fd1498Szrj * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) 236*38fd1498Szrj * @f] 237*38fd1498Szrj * 238*38fd1498Szrj * @param __n The order of the Laguerre function. 239*38fd1498Szrj * @param __alpha The degree of the Laguerre function. 240*38fd1498Szrj * @param __x The argument of the Laguerre function. 241*38fd1498Szrj * @return The value of the Laguerre function of order n, 242*38fd1498Szrj * degree @f$ \alpha @f$, and argument x. 243*38fd1498Szrj */ 244*38fd1498Szrj template<typename _Tpa, typename _Tp> 245*38fd1498Szrj _Tp __poly_laguerre(unsigned int __n,_Tpa __alpha1,_Tp __x)246*38fd1498Szrj __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) 247*38fd1498Szrj { 248*38fd1498Szrj if (__x < _Tp(0)) 249*38fd1498Szrj std::__throw_domain_error(__N("Negative argument " 250*38fd1498Szrj "in __poly_laguerre.")); 251*38fd1498Szrj // Return NaN on NaN input. 252*38fd1498Szrj else if (__isnan(__x)) 253*38fd1498Szrj return std::numeric_limits<_Tp>::quiet_NaN(); 254*38fd1498Szrj else if (__n == 0) 255*38fd1498Szrj return _Tp(1); 256*38fd1498Szrj else if (__n == 1) 257*38fd1498Szrj return _Tp(1) + _Tp(__alpha1) - __x; 258*38fd1498Szrj else if (__x == _Tp(0)) 259*38fd1498Szrj { 260*38fd1498Szrj _Tp __prod = _Tp(__alpha1) + _Tp(1); 261*38fd1498Szrj for (unsigned int __k = 2; __k <= __n; ++__k) 262*38fd1498Szrj __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); 263*38fd1498Szrj return __prod; 264*38fd1498Szrj } 265*38fd1498Szrj else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) 266*38fd1498Szrj && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) 267*38fd1498Szrj return __poly_laguerre_large_n(__n, __alpha1, __x); 268*38fd1498Szrj else if (_Tp(__alpha1) >= _Tp(0) 269*38fd1498Szrj || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) 270*38fd1498Szrj return __poly_laguerre_recursion(__n, __alpha1, __x); 271*38fd1498Szrj else 272*38fd1498Szrj return __poly_laguerre_hyperg(__n, __alpha1, __x); 273*38fd1498Szrj } 274*38fd1498Szrj 275*38fd1498Szrj 276*38fd1498Szrj /** 277*38fd1498Szrj * @brief This routine returns the associated Laguerre polynomial 278*38fd1498Szrj * of order n, degree m: @f$ L_n^m(x) @f$. 279*38fd1498Szrj * 280*38fd1498Szrj * The associated Laguerre polynomial is defined for integral 281*38fd1498Szrj * @f$ \alpha = m @f$ by: 282*38fd1498Szrj * @f[ 283*38fd1498Szrj * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) 284*38fd1498Szrj * @f] 285*38fd1498Szrj * where the Laguerre polynomial is defined by: 286*38fd1498Szrj * @f[ 287*38fd1498Szrj * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) 288*38fd1498Szrj * @f] 289*38fd1498Szrj * 290*38fd1498Szrj * @param __n The order of the Laguerre polynomial. 291*38fd1498Szrj * @param __m The degree of the Laguerre polynomial. 292*38fd1498Szrj * @param __x The argument of the Laguerre polynomial. 293*38fd1498Szrj * @return The value of the associated Laguerre polynomial of order n, 294*38fd1498Szrj * degree m, and argument x. 295*38fd1498Szrj */ 296*38fd1498Szrj template<typename _Tp> 297*38fd1498Szrj inline _Tp __assoc_laguerre(unsigned int __n,unsigned int __m,_Tp __x)298*38fd1498Szrj __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) 299*38fd1498Szrj { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } 300*38fd1498Szrj 301*38fd1498Szrj 302*38fd1498Szrj /** 303*38fd1498Szrj * @brief This routine returns the Laguerre polynomial 304*38fd1498Szrj * of order n: @f$ L_n(x) @f$. 305*38fd1498Szrj * 306*38fd1498Szrj * The Laguerre polynomial is defined by: 307*38fd1498Szrj * @f[ 308*38fd1498Szrj * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) 309*38fd1498Szrj * @f] 310*38fd1498Szrj * 311*38fd1498Szrj * @param __n The order of the Laguerre polynomial. 312*38fd1498Szrj * @param __x The argument of the Laguerre polynomial. 313*38fd1498Szrj * @return The value of the Laguerre polynomial of order n 314*38fd1498Szrj * and argument x. 315*38fd1498Szrj */ 316*38fd1498Szrj template<typename _Tp> 317*38fd1498Szrj inline _Tp __laguerre(unsigned int __n,_Tp __x)318*38fd1498Szrj __laguerre(unsigned int __n, _Tp __x) 319*38fd1498Szrj { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } 320*38fd1498Szrj } // namespace __detail 321*38fd1498Szrj #undef _GLIBCXX_MATH_NS 322*38fd1498Szrj #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 323*38fd1498Szrj } // namespace tr1 324*38fd1498Szrj #endif 325*38fd1498Szrj 326*38fd1498Szrj _GLIBCXX_END_NAMESPACE_VERSION 327*38fd1498Szrj } 328*38fd1498Szrj 329*38fd1498Szrj #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC 330