1 | // Special functions -*- C++ -*- |
---|---|

2 | |

3 | // Copyright (C) 2006-2018 Free Software Foundation, Inc. |

4 | // |

5 | // This file is part of the GNU ISO C++ Library. This library is free |

6 | // software; you can redistribute it and/or modify it under the |

7 | // terms of the GNU General Public License as published by the |

8 | // Free Software Foundation; either version 3, or (at your option) |

9 | // any later version. |

10 | // |

11 | // This library is distributed in the hope that it will be useful, |

12 | // but WITHOUT ANY WARRANTY; without even the implied warranty of |

13 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |

14 | // GNU General Public License for more details. |

15 | // |

16 | // Under Section 7 of GPL version 3, you are granted additional |

17 | // permissions described in the GCC Runtime Library Exception, version |

18 | // 3.1, as published by the Free Software Foundation. |

19 | |

20 | // You should have received a copy of the GNU General Public License and |

21 | // a copy of the GCC Runtime Library Exception along with this program; |

22 | // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |

23 | // <http://www.gnu.org/licenses/>. |

24 | |

25 | /** @file tr1/poly_laguerre.tcc |

26 | * This is an internal header file, included by other library headers. |

27 | * Do not attempt to use it directly. @headername{tr1/cmath} |

28 | */ |

29 | |

30 | // |

31 | // ISO C++ 14882 TR1: 5.2 Special functions |

32 | // |

33 | |

34 | // Written by Edward Smith-Rowland based on: |

35 | // (1) Handbook of Mathematical Functions, |

36 | // Ed. Milton Abramowitz and Irene A. Stegun, |

37 | // Dover Publications, |

38 | // Section 13, pp. 509-510, Section 22 pp. 773-802 |

39 | // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |

40 | |

41 | #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC |

42 | #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 |

43 | |

44 | namespace std _GLIBCXX_VISIBILITY(default) |

45 | { |

46 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |

47 | |

48 | #if _GLIBCXX_USE_STD_SPEC_FUNCS |

49 | # define _GLIBCXX_MATH_NS ::std |

50 | #elif defined(_GLIBCXX_TR1_CMATH) |

51 | namespace tr1 |

52 | { |

53 | # define _GLIBCXX_MATH_NS ::std::tr1 |

54 | #else |

55 | # error do not include this header directly, use <cmath> or <tr1/cmath> |

56 | #endif |

57 | // [5.2] Special functions |

58 | |

59 | // Implementation-space details. |

60 | namespace __detail |

61 | { |

62 | /** |

63 | * @brief This routine returns the associated Laguerre polynomial |

64 | * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. |

65 | * Abramowitz & Stegun, 13.5.21 |

66 | * |

67 | * @param __n The order of the Laguerre function. |

68 | * @param __alpha The degree of the Laguerre function. |

69 | * @param __x The argument of the Laguerre function. |

70 | * @return The value of the Laguerre function of order n, |

71 | * degree @f$ \alpha @f$, and argument x. |

72 | * |

73 | * This is from the GNU Scientific Library. |

74 | */ |

75 | template<typename _Tpa, typename _Tp> |

76 | _Tp |

77 | __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) |

78 | { |

79 | const _Tp __a = -_Tp(__n); |

80 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |

81 | const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; |

82 | const _Tp __cos2th = __x / __eta; |

83 | const _Tp __sin2th = _Tp(1) - __cos2th; |

84 | const _Tp __th = std::acos(std::sqrt(__cos2th)); |

85 | const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() |

86 | * __numeric_constants<_Tp>::__pi_2() |

87 | * __eta * __eta * __cos2th * __sin2th; |

88 | |

89 | #if _GLIBCXX_USE_C99_MATH_TR1 |

90 | const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b); |

91 | const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); |

92 | #else |

93 | const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); |

94 | const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); |

95 | #endif |

96 | |

97 | _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) |

98 | * std::log(_Tp(0.25L) * __x * __eta); |

99 | _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); |

100 | _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x |

101 | + __pre_term1 - __pre_term2; |

102 | _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); |

103 | _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta |

104 | * (_Tp(2) * __th |

105 | - std::sin(_Tp(2) * __th)) |

106 | + __numeric_constants<_Tp>::__pi_4()); |

107 | _Tp __ser = __ser_term1 + __ser_term2; |

108 | |

109 | return std::exp(__lnpre) * __ser; |

110 | } |

111 | |

112 | |

113 | /** |

114 | * @brief Evaluate the polynomial based on the confluent hypergeometric |

115 | * function in a safe way, with no restriction on the arguments. |

116 | * |

117 | * The associated Laguerre function is defined by |

118 | * @f[ |

119 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |

120 | * _1F_1(-n; \alpha + 1; x) |

121 | * @f] |

122 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |

123 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |

124 | * |

125 | * This function assumes x != 0. |

126 | * |

127 | * This is from the GNU Scientific Library. |

128 | */ |

129 | template<typename _Tpa, typename _Tp> |

130 | _Tp |

131 | __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) |

132 | { |

133 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |

134 | const _Tp __mx = -__x; |

135 | const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) |

136 | : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); |

137 | // Get |x|^n/n! |

138 | _Tp __tc = _Tp(1); |

139 | const _Tp __ax = std::abs(__x); |

140 | for (unsigned int __k = 1; __k <= __n; ++__k) |

141 | __tc *= (__ax / __k); |

142 | |

143 | _Tp __term = __tc * __tc_sgn; |

144 | _Tp __sum = __term; |

145 | for (int __k = int(__n) - 1; __k >= 0; --__k) |

146 | { |

147 | __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) |

148 | * _Tp(__k + 1) / __mx; |

149 | __sum += __term; |

150 | } |

151 | |

152 | return __sum; |

153 | } |

154 | |

155 | |

156 | /** |

157 | * @brief This routine returns the associated Laguerre polynomial |

158 | * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ |

159 | * by recursion. |

160 | * |

161 | * The associated Laguerre function is defined by |

162 | * @f[ |

163 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |

164 | * _1F_1(-n; \alpha + 1; x) |

165 | * @f] |

166 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |

167 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |

168 | * |

169 | * The associated Laguerre polynomial is defined for integral |

170 | * @f$ \alpha = m @f$ by: |

171 | * @f[ |

172 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |

173 | * @f] |

174 | * where the Laguerre polynomial is defined by: |

175 | * @f[ |

176 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |

177 | * @f] |

178 | * |

179 | * @param __n The order of the Laguerre function. |

180 | * @param __alpha The degree of the Laguerre function. |

181 | * @param __x The argument of the Laguerre function. |

182 | * @return The value of the Laguerre function of order n, |

183 | * degree @f$ \alpha @f$, and argument x. |

184 | */ |

185 | template<typename _Tpa, typename _Tp> |

186 | _Tp |

187 | __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) |

188 | { |

189 | // Compute l_0. |

190 | _Tp __l_0 = _Tp(1); |

191 | if (__n == 0) |

192 | return __l_0; |

193 | |

194 | // Compute l_1^alpha. |

195 | _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); |

196 | if (__n == 1) |

197 | return __l_1; |

198 | |

199 | // Compute l_n^alpha by recursion on n. |

200 | _Tp __l_n2 = __l_0; |

201 | _Tp __l_n1 = __l_1; |

202 | _Tp __l_n = _Tp(0); |

203 | for (unsigned int __nn = 2; __nn <= __n; ++__nn) |

204 | { |

205 | __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) |

206 | * __l_n1 / _Tp(__nn) |

207 | - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); |

208 | __l_n2 = __l_n1; |

209 | __l_n1 = __l_n; |

210 | } |

211 | |

212 | return __l_n; |

213 | } |

214 | |

215 | |

216 | /** |

217 | * @brief This routine returns the associated Laguerre polynomial |

218 | * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. |

219 | * |

220 | * The associated Laguerre function is defined by |

221 | * @f[ |

222 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |

223 | * _1F_1(-n; \alpha + 1; x) |

224 | * @f] |

225 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |

226 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |

227 | * |

228 | * The associated Laguerre polynomial is defined for integral |

229 | * @f$ \alpha = m @f$ by: |

230 | * @f[ |

231 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |

232 | * @f] |

233 | * where the Laguerre polynomial is defined by: |

234 | * @f[ |

235 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |

236 | * @f] |

237 | * |

238 | * @param __n The order of the Laguerre function. |

239 | * @param __alpha The degree of the Laguerre function. |

240 | * @param __x The argument of the Laguerre function. |

241 | * @return The value of the Laguerre function of order n, |

242 | * degree @f$ \alpha @f$, and argument x. |

243 | */ |

244 | template<typename _Tpa, typename _Tp> |

245 | _Tp |

246 | __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) |

247 | { |

248 | if (__x < _Tp(0)) |

249 | std::__throw_domain_error(__N("Negative argument " |

250 | "in __poly_laguerre.")); |

251 | // Return NaN on NaN input. |

252 | else if (__isnan(__x)) |

253 | return std::numeric_limits<_Tp>::quiet_NaN(); |

254 | else if (__n == 0) |

255 | return _Tp(1); |

256 | else if (__n == 1) |

257 | return _Tp(1) + _Tp(__alpha1) - __x; |

258 | else if (__x == _Tp(0)) |

259 | { |

260 | _Tp __prod = _Tp(__alpha1) + _Tp(1); |

261 | for (unsigned int __k = 2; __k <= __n; ++__k) |

262 | __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); |

263 | return __prod; |

264 | } |

265 | else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) |

266 | && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) |

267 | return __poly_laguerre_large_n(__n, __alpha1, __x); |

268 | else if (_Tp(__alpha1) >= _Tp(0) |

269 | || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) |

270 | return __poly_laguerre_recursion(__n, __alpha1, __x); |

271 | else |

272 | return __poly_laguerre_hyperg(__n, __alpha1, __x); |

273 | } |

274 | |

275 | |

276 | /** |

277 | * @brief This routine returns the associated Laguerre polynomial |

278 | * of order n, degree m: @f$ L_n^m(x) @f$. |

279 | * |

280 | * The associated Laguerre polynomial is defined for integral |

281 | * @f$ \alpha = m @f$ by: |

282 | * @f[ |

283 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |

284 | * @f] |

285 | * where the Laguerre polynomial is defined by: |

286 | * @f[ |

287 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |

288 | * @f] |

289 | * |

290 | * @param __n The order of the Laguerre polynomial. |

291 | * @param __m The degree of the Laguerre polynomial. |

292 | * @param __x The argument of the Laguerre polynomial. |

293 | * @return The value of the associated Laguerre polynomial of order n, |

294 | * degree m, and argument x. |

295 | */ |

296 | template<typename _Tp> |

297 | inline _Tp |

298 | __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |

299 | { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } |

300 | |

301 | |

302 | /** |

303 | * @brief This routine returns the Laguerre polynomial |

304 | * of order n: @f$ L_n(x) @f$. |

305 | * |

306 | * The Laguerre polynomial is defined by: |

307 | * @f[ |

308 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |

309 | * @f] |

310 | * |

311 | * @param __n The order of the Laguerre polynomial. |

312 | * @param __x The argument of the Laguerre polynomial. |

313 | * @return The value of the Laguerre polynomial of order n |

314 | * and argument x. |

315 | */ |

316 | template<typename _Tp> |

317 | inline _Tp |

318 | __laguerre(unsigned int __n, _Tp __x) |

319 | { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } |

320 | } // namespace __detail |

321 | #undef _GLIBCXX_MATH_NS |

322 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |

323 | } // namespace tr1 |

324 | #endif |

325 | |

326 | _GLIBCXX_END_NAMESPACE_VERSION |

327 | } |

328 | |

329 | #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC |

330 |