| 1 |
// Special functions -*- C++ -*- |
| 2 |
|
| 3 |
// Copyright (C) 2006-2021 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/gamma.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 6, pp. 253-266 |
| 39 |
// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
| 40 |
// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, |
| 41 |
// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), |
| 42 |
// 2nd ed, pp. 213-216 |
| 43 |
// (4) Gamma, Exploring Euler's Constant, Julian Havil, |
| 44 |
// Princeton, 2003. |
| 45 |
|
| 46 |
#ifndef _GLIBCXX_TR1_GAMMA_TCC |
| 47 |
#define _GLIBCXX_TR1_GAMMA_TCC 1 |
| 48 |
|
| 49 |
#include <tr1/special_function_util.h> |
| 50 |
|
| 51 |
namespace std _GLIBCXX_VISIBILITY(default) |
| 52 |
{ |
| 53 |
_GLIBCXX_BEGIN_NAMESPACE_VERSION |
| 54 |
|
| 55 |
#if _GLIBCXX_USE_STD_SPEC_FUNCS |
| 56 |
# define _GLIBCXX_MATH_NS ::std |
| 57 |
#elif defined(_GLIBCXX_TR1_CMATH) |
| 58 |
namespace tr1 |
| 59 |
{ |
| 60 |
# define _GLIBCXX_MATH_NS ::std::tr1 |
| 61 |
#else |
| 62 |
# error do not include this header directly, use <cmath> or <tr1/cmath> |
| 63 |
#endif |
| 64 |
// Implementation-space details. |
| 65 |
namespace __detail |
| 66 |
{ |
| 67 |
/** |
| 68 |
* @brief This returns Bernoulli numbers from a table or by summation |
| 69 |
* for larger values. |
| 70 |
* |
| 71 |
* Recursion is unstable. |
| 72 |
* |
| 73 |
* @param __n the order n of the Bernoulli number. |
| 74 |
* @return The Bernoulli number of order n. |
| 75 |
*/ |
| 76 |
template <typename _Tp> |
| 77 |
_Tp |
| 78 |
__bernoulli_series(unsigned int __n) |
| 79 |
{ |
| 80 |
|
| 81 |
static const _Tp __num[28] = { |
| 82 |
_Tp(1UL), -_Tp(1UL) / _Tp(2UL), |
| 83 |
_Tp(1UL) / _Tp(6UL), _Tp(0UL), |
| 84 |
-_Tp(1UL) / _Tp(30UL), _Tp(0UL), |
| 85 |
_Tp(1UL) / _Tp(42UL), _Tp(0UL), |
| 86 |
-_Tp(1UL) / _Tp(30UL), _Tp(0UL), |
| 87 |
_Tp(5UL) / _Tp(66UL), _Tp(0UL), |
| 88 |
-_Tp(691UL) / _Tp(2730UL), _Tp(0UL), |
| 89 |
_Tp(7UL) / _Tp(6UL), _Tp(0UL), |
| 90 |
-_Tp(3617UL) / _Tp(510UL), _Tp(0UL), |
| 91 |
_Tp(43867UL) / _Tp(798UL), _Tp(0UL), |
| 92 |
-_Tp(174611) / _Tp(330UL), _Tp(0UL), |
| 93 |
_Tp(854513UL) / _Tp(138UL), _Tp(0UL), |
| 94 |
-_Tp(236364091UL) / _Tp(2730UL), _Tp(0UL), |
| 95 |
_Tp(8553103UL) / _Tp(6UL), _Tp(0UL) |
| 96 |
}; |
| 97 |
|
| 98 |
if (__n == 0) |
| 99 |
return _Tp(1); |
| 100 |
|
| 101 |
if (__n == 1) |
| 102 |
return -_Tp(1) / _Tp(2); |
| 103 |
|
| 104 |
// Take care of the rest of the odd ones. |
| 105 |
if (__n % 2 == 1) |
| 106 |
return _Tp(0); |
| 107 |
|
| 108 |
// Take care of some small evens that are painful for the series. |
| 109 |
if (__n < 28) |
| 110 |
return __num[__n]; |
| 111 |
|
| 112 |
|
| 113 |
_Tp __fact = _Tp(1); |
| 114 |
if ((__n / 2) % 2 == 0) |
| 115 |
__fact *= _Tp(-1); |
| 116 |
for (unsigned int __k = 1; __k <= __n; ++__k) |
| 117 |
__fact *= __k / (_Tp(2) * __numeric_constants<_Tp>::__pi()); |
| 118 |
__fact *= _Tp(2); |
| 119 |
|
| 120 |
_Tp __sum = _Tp(0); |
| 121 |
for (unsigned int __i = 1; __i < 1000; ++__i) |
| 122 |
{ |
| 123 |
_Tp __term = std::pow(_Tp(__i), -_Tp(__n)); |
| 124 |
if (__term < std::numeric_limits<_Tp>::epsilon()) |
| 125 |
break; |
| 126 |
__sum += __term; |
| 127 |
} |
| 128 |
|
| 129 |
return __fact * __sum; |
| 130 |
} |
| 131 |
|
| 132 |
|
| 133 |
/** |
| 134 |
* @brief This returns Bernoulli number \f$B_n\f$. |
| 135 |
* |
| 136 |
* @param __n the order n of the Bernoulli number. |
| 137 |
* @return The Bernoulli number of order n. |
| 138 |
*/ |
| 139 |
template<typename _Tp> |
| 140 |
inline _Tp |
| 141 |
__bernoulli(int __n) |
| 142 |
{ return __bernoulli_series<_Tp>(__n); } |
| 143 |
|
| 144 |
|
| 145 |
/** |
| 146 |
* @brief Return \f$log(\Gamma(x))\f$ by asymptotic expansion |
| 147 |
* with Bernoulli number coefficients. This is like |
| 148 |
* Sterling's approximation. |
| 149 |
* |
| 150 |
* @param __x The argument of the log of the gamma function. |
| 151 |
* @return The logarithm of the gamma function. |
| 152 |
*/ |
| 153 |
template<typename _Tp> |
| 154 |
_Tp |
| 155 |
__log_gamma_bernoulli(_Tp __x) |
| 156 |
{ |
| 157 |
_Tp __lg = (__x - _Tp(0.5L)) * std::log(__x) - __x |
| 158 |
+ _Tp(0.5L) * std::log(_Tp(2) |
| 159 |
* __numeric_constants<_Tp>::__pi()); |
| 160 |
|
| 161 |
const _Tp __xx = __x * __x; |
| 162 |
_Tp __help = _Tp(1) / __x; |
| 163 |
for ( unsigned int __i = 1; __i < 20; ++__i ) |
| 164 |
{ |
| 165 |
const _Tp __2i = _Tp(2 * __i); |
| 166 |
__help /= __2i * (__2i - _Tp(1)) * __xx; |
| 167 |
__lg += __bernoulli<_Tp>(2 * __i) * __help; |
| 168 |
} |
| 169 |
|
| 170 |
return __lg; |
| 171 |
} |
| 172 |
|
| 173 |
|
| 174 |
/** |
| 175 |
* @brief Return \f$log(\Gamma(x))\f$ by the Lanczos method. |
| 176 |
* This method dominates all others on the positive axis I think. |
| 177 |
* |
| 178 |
* @param __x The argument of the log of the gamma function. |
| 179 |
* @return The logarithm of the gamma function. |
| 180 |
*/ |
| 181 |
template<typename _Tp> |
| 182 |
_Tp |
| 183 |
__log_gamma_lanczos(_Tp __x) |
| 184 |
{ |
| 185 |
const _Tp __xm1 = __x - _Tp(1); |
| 186 |
|
| 187 |
static const _Tp __lanczos_cheb_7[9] = { |
| 188 |
_Tp( 0.99999999999980993227684700473478L), |
| 189 |
_Tp( 676.520368121885098567009190444019L), |
| 190 |
_Tp(-1259.13921672240287047156078755283L), |
| 191 |
_Tp( 771.3234287776530788486528258894L), |
| 192 |
_Tp(-176.61502916214059906584551354L), |
| 193 |
_Tp( 12.507343278686904814458936853L), |
| 194 |
_Tp(-0.13857109526572011689554707L), |
| 195 |
_Tp( 9.984369578019570859563e-6L), |
| 196 |
_Tp( 1.50563273514931155834e-7L) |
| 197 |
}; |
| 198 |
|
| 199 |
static const _Tp __LOGROOT2PI |
| 200 |
= _Tp(0.9189385332046727417803297364056176L); |
| 201 |
|
| 202 |
_Tp __sum = __lanczos_cheb_7[0]; |
| 203 |
for(unsigned int __k = 1; __k < 9; ++__k) |
| 204 |
__sum += __lanczos_cheb_7[__k] / (__xm1 + __k); |
| 205 |
|
| 206 |
const _Tp __term1 = (__xm1 + _Tp(0.5L)) |
| 207 |
* std::log((__xm1 + _Tp(7.5L)) |
| 208 |
/ __numeric_constants<_Tp>::__euler()); |
| 209 |
const _Tp __term2 = __LOGROOT2PI + std::log(__sum); |
| 210 |
const _Tp __result = __term1 + (__term2 - _Tp(7)); |
| 211 |
|
| 212 |
return __result; |
| 213 |
} |
| 214 |
|
| 215 |
|
| 216 |
/** |
| 217 |
* @brief Return \f$ log(|\Gamma(x)|) \f$. |
| 218 |
* This will return values even for \f$ x < 0 \f$. |
| 219 |
* To recover the sign of \f$ \Gamma(x) \f$ for |
| 220 |
* any argument use @a __log_gamma_sign. |
| 221 |
* |
| 222 |
* @param __x The argument of the log of the gamma function. |
| 223 |
* @return The logarithm of the gamma function. |
| 224 |
*/ |
| 225 |
template<typename _Tp> |
| 226 |
_Tp |
| 227 |
__log_gamma(_Tp __x) |
| 228 |
{ |
| 229 |
if (__x > _Tp(0.5L)) |
| 230 |
return __log_gamma_lanczos(__x); |
| 231 |
else |
| 232 |
{ |
| 233 |
const _Tp __sin_fact |
| 234 |
= std::abs(std::sin(__numeric_constants<_Tp>::__pi() * __x)); |
| 235 |
if (__sin_fact == _Tp(0)) |
| 236 |
std::__throw_domain_error(__N("Argument is nonpositive integer " |
| 237 |
"in __log_gamma")); |
| 238 |
return __numeric_constants<_Tp>::__lnpi() |
| 239 |
- std::log(__sin_fact) |
| 240 |
- __log_gamma_lanczos(_Tp(1) - __x); |
| 241 |
} |
| 242 |
} |
| 243 |
|
| 244 |
|
| 245 |
/** |
| 246 |
* @brief Return the sign of \f$ \Gamma(x) \f$. |
| 247 |
* At nonpositive integers zero is returned. |
| 248 |
* |
| 249 |
* @param __x The argument of the gamma function. |
| 250 |
* @return The sign of the gamma function. |
| 251 |
*/ |
| 252 |
template<typename _Tp> |
| 253 |
_Tp |
| 254 |
__log_gamma_sign(_Tp __x) |
| 255 |
{ |
| 256 |
if (__x > _Tp(0)) |
| 257 |
return _Tp(1); |
| 258 |
else |
| 259 |
{ |
| 260 |
const _Tp __sin_fact |
| 261 |
= std::sin(__numeric_constants<_Tp>::__pi() * __x); |
| 262 |
if (__sin_fact > _Tp(0)) |
| 263 |
return (1); |
| 264 |
else if (__sin_fact < _Tp(0)) |
| 265 |
return -_Tp(1); |
| 266 |
else |
| 267 |
return _Tp(0); |
| 268 |
} |
| 269 |
} |
| 270 |
|
| 271 |
|
| 272 |
/** |
| 273 |
* @brief Return the logarithm of the binomial coefficient. |
| 274 |
* The binomial coefficient is given by: |
| 275 |
* @f[ |
| 276 |
* \left( \right) = \frac{n!}{(n-k)! k!} |
| 277 |
* @f] |
| 278 |
* |
| 279 |
* @param __n The first argument of the binomial coefficient. |
| 280 |
* @param __k The second argument of the binomial coefficient. |
| 281 |
* @return The binomial coefficient. |
| 282 |
*/ |
| 283 |
template<typename _Tp> |
| 284 |
_Tp |
| 285 |
__log_bincoef(unsigned int __n, unsigned int __k) |
| 286 |
{ |
| 287 |
// Max e exponent before overflow. |
| 288 |
static const _Tp __max_bincoeff |
| 289 |
= std::numeric_limits<_Tp>::max_exponent10 |
| 290 |
* std::log(_Tp(10)) - _Tp(1); |
| 291 |
#if _GLIBCXX_USE_C99_MATH_TR1 |
| 292 |
_Tp __coeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __n)) |
| 293 |
- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __k)) |
| 294 |
- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __n - __k)); |
| 295 |
#else |
| 296 |
_Tp __coeff = __log_gamma(_Tp(1 + __n)) |
| 297 |
- __log_gamma(_Tp(1 + __k)) |
| 298 |
- __log_gamma(_Tp(1 + __n - __k)); |
| 299 |
#endif |
| 300 |
} |
| 301 |
|
| 302 |
|
| 303 |
/** |
| 304 |
* @brief Return the binomial coefficient. |
| 305 |
* The binomial coefficient is given by: |
| 306 |
* @f[ |
| 307 |
* \left( \right) = \frac{n!}{(n-k)! k!} |
| 308 |
* @f] |
| 309 |
* |
| 310 |
* @param __n The first argument of the binomial coefficient. |
| 311 |
* @param __k The second argument of the binomial coefficient. |
| 312 |
* @return The binomial coefficient. |
| 313 |
*/ |
| 314 |
template<typename _Tp> |
| 315 |
_Tp |
| 316 |
__bincoef(unsigned int __n, unsigned int __k) |
| 317 |
{ |
| 318 |
// Max e exponent before overflow. |
| 319 |
static const _Tp __max_bincoeff |
| 320 |
= std::numeric_limits<_Tp>::max_exponent10 |
| 321 |
* std::log(_Tp(10)) - _Tp(1); |
| 322 |
|
| 323 |
const _Tp __log_coeff = __log_bincoef<_Tp>(__n, __k); |
| 324 |
if (__log_coeff > __max_bincoeff) |
| 325 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 326 |
else |
| 327 |
return std::exp(__log_coeff); |
| 328 |
} |
| 329 |
|
| 330 |
|
| 331 |
/** |
| 332 |
* @brief Return \f$ \Gamma(x) \f$. |
| 333 |
* |
| 334 |
* @param __x The argument of the gamma function. |
| 335 |
* @return The gamma function. |
| 336 |
*/ |
| 337 |
template<typename _Tp> |
| 338 |
inline _Tp |
| 339 |
__gamma(_Tp __x) |
| 340 |
{ return std::exp(__log_gamma(__x)); } |
| 341 |
|
| 342 |
|
| 343 |
/** |
| 344 |
* @brief Return the digamma function by series expansion. |
| 345 |
* The digamma or @f$ \psi(x) @f$ function is defined by |
| 346 |
* @f[ |
| 347 |
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} |
| 348 |
* @f] |
| 349 |
* |
| 350 |
* The series is given by: |
| 351 |
* @f[ |
| 352 |
* \psi(x) = -\gamma_E - \frac{1}{x} |
| 353 |
* \sum_{k=1}^{\infty} \frac{x}{k(x + k)} |
| 354 |
* @f] |
| 355 |
*/ |
| 356 |
template<typename _Tp> |
| 357 |
_Tp |
| 358 |
__psi_series(_Tp __x) |
| 359 |
{ |
| 360 |
_Tp __sum = -__numeric_constants<_Tp>::__gamma_e() - _Tp(1) / __x; |
| 361 |
const unsigned int __max_iter = 100000; |
| 362 |
for (unsigned int __k = 1; __k < __max_iter; ++__k) |
| 363 |
{ |
| 364 |
const _Tp __term = __x / (__k * (__k + __x)); |
| 365 |
__sum += __term; |
| 366 |
if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon()) |
| 367 |
break; |
| 368 |
} |
| 369 |
return __sum; |
| 370 |
} |
| 371 |
|
| 372 |
|
| 373 |
/** |
| 374 |
* @brief Return the digamma function for large argument. |
| 375 |
* The digamma or @f$ \psi(x) @f$ function is defined by |
| 376 |
* @f[ |
| 377 |
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} |
| 378 |
* @f] |
| 379 |
* |
| 380 |
* The asymptotic series is given by: |
| 381 |
* @f[ |
| 382 |
* \psi(x) = \ln(x) - \frac{1}{2x} |
| 383 |
* - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}} |
| 384 |
* @f] |
| 385 |
*/ |
| 386 |
template<typename _Tp> |
| 387 |
_Tp |
| 388 |
__psi_asymp(_Tp __x) |
| 389 |
{ |
| 390 |
_Tp __sum = std::log(__x) - _Tp(0.5L) / __x; |
| 391 |
const _Tp __xx = __x * __x; |
| 392 |
_Tp __xp = __xx; |
| 393 |
const unsigned int __max_iter = 100; |
| 394 |
for (unsigned int __k = 1; __k < __max_iter; ++__k) |
| 395 |
{ |
| 396 |
const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp); |
| 397 |
__sum -= __term; |
| 398 |
if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon()) |
| 399 |
break; |
| 400 |
__xp *= __xx; |
| 401 |
} |
| 402 |
return __sum; |
| 403 |
} |
| 404 |
|
| 405 |
|
| 406 |
/** |
| 407 |
* @brief Return the digamma function. |
| 408 |
* The digamma or @f$ \psi(x) @f$ function is defined by |
| 409 |
* @f[ |
| 410 |
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} |
| 411 |
* @f] |
| 412 |
* For negative argument the reflection formula is used: |
| 413 |
* @f[ |
| 414 |
* \psi(x) = \psi(1-x) - \pi \cot(\pi x) |
| 415 |
* @f] |
| 416 |
*/ |
| 417 |
template<typename _Tp> |
| 418 |
_Tp |
| 419 |
__psi(_Tp __x) |
| 420 |
{ |
| 421 |
const int __n = static_cast<int>(__x + 0.5L); |
| 422 |
const _Tp __eps = _Tp(4) * std::numeric_limits<_Tp>::epsilon(); |
| 423 |
if (__n <= 0 && std::abs(__x - _Tp(__n)) < __eps) |
| 424 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 425 |
else if (__x < _Tp(0)) |
| 426 |
{ |
| 427 |
const _Tp __pi = __numeric_constants<_Tp>::__pi(); |
| 428 |
return __psi(_Tp(1) - __x) |
| 429 |
- __pi * std::cos(__pi * __x) / std::sin(__pi * __x); |
| 430 |
} |
| 431 |
else if (__x > _Tp(100)) |
| 432 |
return __psi_asymp(__x); |
| 433 |
else |
| 434 |
return __psi_series(__x); |
| 435 |
} |
| 436 |
|
| 437 |
|
| 438 |
/** |
| 439 |
* @brief Return the polygamma function @f$ \psi^{(n)}(x) @f$. |
| 440 |
* |
| 441 |
* The polygamma function is related to the Hurwitz zeta function: |
| 442 |
* @f[ |
| 443 |
* \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x) |
| 444 |
* @f] |
| 445 |
*/ |
| 446 |
template<typename _Tp> |
| 447 |
_Tp |
| 448 |
__psi(unsigned int __n, _Tp __x) |
| 449 |
{ |
| 450 |
if (__x <= _Tp(0)) |
| 451 |
std::__throw_domain_error(__N("Argument out of range " |
| 452 |
"in __psi")); |
| 453 |
else if (__n == 0) |
| 454 |
return __psi(__x); |
| 455 |
else |
| 456 |
{ |
| 457 |
const _Tp __hzeta = __hurwitz_zeta(_Tp(__n + 1), __x); |
| 458 |
#if _GLIBCXX_USE_C99_MATH_TR1 |
| 459 |
const _Tp __ln_nfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); |
| 460 |
#else |
| 461 |
const _Tp __ln_nfact = __log_gamma(_Tp(__n + 1)); |
| 462 |
#endif |
| 463 |
_Tp __result = std::exp(__ln_nfact) * __hzeta; |
| 464 |
if (__n % 2 == 1) |
| 465 |
__result = -__result; |
| 466 |
return __result; |
| 467 |
} |
| 468 |
} |
| 469 |
} // namespace __detail |
| 470 |
#undef _GLIBCXX_MATH_NS |
| 471 |
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
| 472 |
} // namespace tr1 |
| 473 |
#endif |
| 474 |
|
| 475 |
_GLIBCXX_END_NAMESPACE_VERSION |
| 476 |
} // namespace std |
| 477 |
|
| 478 |
#endif // _GLIBCXX_TR1_GAMMA_TCC |
| 479 |
|
