libstdc++
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00001 // Special functions -*- C++ -*- 00002 00003 // Copyright (C) 2006, 2007, 2008, 2009 00004 // Free Software Foundation, Inc. 00005 // 00006 // This file is part of the GNU ISO C++ Library. This library is free 00007 // software; you can redistribute it and/or modify it under the 00008 // terms of the GNU General Public License as published by the 00009 // Free Software Foundation; either version 3, or (at your option) 00010 // any later version. 00011 // 00012 // This library is distributed in the hope that it will be useful, 00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of 00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00015 // GNU General Public License for more details. 00016 // 00017 // Under Section 7 of GPL version 3, you are granted additional 00018 // permissions described in the GCC Runtime Library Exception, version 00019 // 3.1, as published by the Free Software Foundation. 00020 00021 // You should have received a copy of the GNU General Public License and 00022 // a copy of the GCC Runtime Library Exception along with this program; 00023 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 00024 // <http://www.gnu.org/licenses/>. 00025 00026 /** @file tr1/hypergeometric.tcc 00027 * This is an internal header file, included by other library headers. 00028 * You should not attempt to use it directly. 00029 */ 00030 00031 // 00032 // ISO C++ 14882 TR1: 5.2 Special functions 00033 // 00034 00035 // Written by Edward Smith-Rowland based: 00036 // (1) Handbook of Mathematical Functions, 00037 // ed. Milton Abramowitz and Irene A. Stegun, 00038 // Dover Publications, 00039 // Section 6, pp. 555-566 00040 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 00041 00042 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 00043 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1 00044 00045 namespace std 00046 { 00047 namespace tr1 00048 { 00049 00050 // [5.2] Special functions 00051 00052 // Implementation-space details. 00053 namespace __detail 00054 { 00055 00056 /** 00057 * @brief This routine returns the confluent hypergeometric function 00058 * by series expansion. 00059 * 00060 * @f[ 00061 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} 00062 * \sum_{n=0}^{\infty} 00063 * \frac{\Gamma(a+n)}{\Gamma(c+n)} 00064 * \frac{x^n}{n!} 00065 * @f] 00066 * 00067 * If a and b are integers and a < 0 and either b > 0 or b < a then the 00068 * series is a polynomial with a finite number of terms. If b is an integer 00069 * and b <= 0 the confluent hypergeometric function is undefined. 00070 * 00071 * @param __a The "numerator" parameter. 00072 * @param __c The "denominator" parameter. 00073 * @param __x The argument of the confluent hypergeometric function. 00074 * @return The confluent hypergeometric function. 00075 */ 00076 template<typename _Tp> 00077 _Tp 00078 __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x) 00079 { 00080 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00081 00082 _Tp __term = _Tp(1); 00083 _Tp __Fac = _Tp(1); 00084 const unsigned int __max_iter = 100000; 00085 unsigned int __i; 00086 for (__i = 0; __i < __max_iter; ++__i) 00087 { 00088 __term *= (__a + _Tp(__i)) * __x 00089 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 00090 if (std::abs(__term) < __eps) 00091 { 00092 break; 00093 } 00094 __Fac += __term; 00095 } 00096 if (__i == __max_iter) 00097 std::__throw_runtime_error(__N("Series failed to converge " 00098 "in __conf_hyperg_series.")); 00099 00100 return __Fac; 00101 } 00102 00103 00104 /** 00105 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 00106 * by an iterative procedure described in 00107 * Luke, Algorithms for the Computation of Mathematical Functions. 00108 * 00109 * Like the case of the 2F1 rational approximations, these are 00110 * probably guaranteed to converge for x < 0, barring gross 00111 * numerical instability in the pre-asymptotic regime. 00112 */ 00113 template<typename _Tp> 00114 _Tp 00115 __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin) 00116 { 00117 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 00118 const int __nmax = 20000; 00119 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00120 const _Tp __x = -__xin; 00121 const _Tp __x3 = __x * __x * __x; 00122 const _Tp __t0 = __a / __c; 00123 const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); 00124 const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); 00125 _Tp __F = _Tp(1); 00126 _Tp __prec; 00127 00128 _Tp __Bnm3 = _Tp(1); 00129 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 00130 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 00131 00132 _Tp __Anm3 = _Tp(1); 00133 _Tp __Anm2 = __Bnm2 - __t0 * __x; 00134 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 00135 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 00136 00137 int __n = 3; 00138 while(1) 00139 { 00140 _Tp __npam1 = _Tp(__n - 1) + __a; 00141 _Tp __npcm1 = _Tp(__n - 1) + __c; 00142 _Tp __npam2 = _Tp(__n - 2) + __a; 00143 _Tp __npcm2 = _Tp(__n - 2) + __c; 00144 _Tp __tnm1 = _Tp(2 * __n - 1); 00145 _Tp __tnm3 = _Tp(2 * __n - 3); 00146 _Tp __tnm5 = _Tp(2 * __n - 5); 00147 _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); 00148 _Tp __F2 = (_Tp(__n) + __a) * __npam1 00149 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 00150 _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) 00151 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 00152 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 00153 _Tp __E = -__npam1 * (_Tp(__n - 1) - __c) 00154 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 00155 00156 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 00157 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 00158 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 00159 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 00160 _Tp __r = __An / __Bn; 00161 00162 __prec = std::abs((__F - __r) / __F); 00163 __F = __r; 00164 00165 if (__prec < __eps || __n > __nmax) 00166 break; 00167 00168 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 00169 { 00170 __An /= __big; 00171 __Bn /= __big; 00172 __Anm1 /= __big; 00173 __Bnm1 /= __big; 00174 __Anm2 /= __big; 00175 __Bnm2 /= __big; 00176 __Anm3 /= __big; 00177 __Bnm3 /= __big; 00178 } 00179 else if (std::abs(__An) < _Tp(1) / __big 00180 || std::abs(__Bn) < _Tp(1) / __big) 00181 { 00182 __An *= __big; 00183 __Bn *= __big; 00184 __Anm1 *= __big; 00185 __Bnm1 *= __big; 00186 __Anm2 *= __big; 00187 __Bnm2 *= __big; 00188 __Anm3 *= __big; 00189 __Bnm3 *= __big; 00190 } 00191 00192 ++__n; 00193 __Bnm3 = __Bnm2; 00194 __Bnm2 = __Bnm1; 00195 __Bnm1 = __Bn; 00196 __Anm3 = __Anm2; 00197 __Anm2 = __Anm1; 00198 __Anm1 = __An; 00199 } 00200 00201 if (__n >= __nmax) 00202 std::__throw_runtime_error(__N("Iteration failed to converge " 00203 "in __conf_hyperg_luke.")); 00204 00205 return __F; 00206 } 00207 00208 00209 /** 00210 * @brief Return the confluent hypogeometric function 00211 * @f$ _1F_1(a;c;x) @f$. 00212 * 00213 * @todo Handle b == nonpositive integer blowup - return NaN. 00214 * 00215 * @param __a The "numerator" parameter. 00216 * @param __c The "denominator" parameter. 00217 * @param __x The argument of the confluent hypergeometric function. 00218 * @return The confluent hypergeometric function. 00219 */ 00220 template<typename _Tp> 00221 inline _Tp 00222 __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x) 00223 { 00224 #if _GLIBCXX_USE_C99_MATH_TR1 00225 const _Tp __c_nint = std::tr1::nearbyint(__c); 00226 #else 00227 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 00228 #endif 00229 if (__isnan(__a) || __isnan(__c) || __isnan(__x)) 00230 return std::numeric_limits<_Tp>::quiet_NaN(); 00231 else if (__c_nint == __c && __c_nint <= 0) 00232 return std::numeric_limits<_Tp>::infinity(); 00233 else if (__a == _Tp(0)) 00234 return _Tp(1); 00235 else if (__c == __a) 00236 return std::exp(__x); 00237 else if (__x < _Tp(0)) 00238 return __conf_hyperg_luke(__a, __c, __x); 00239 else 00240 return __conf_hyperg_series(__a, __c, __x); 00241 } 00242 00243 00244 /** 00245 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 00246 * by series expansion. 00247 * 00248 * The hypogeometric function is defined by 00249 * @f[ 00250 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 00251 * \sum_{n=0}^{\infty} 00252 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 00253 * \frac{x^n}{n!} 00254 * @f] 00255 * 00256 * This works and it's pretty fast. 00257 * 00258 * @param __a The first "numerator" parameter. 00259 * @param __a The second "numerator" parameter. 00260 * @param __c The "denominator" parameter. 00261 * @param __x The argument of the confluent hypergeometric function. 00262 * @return The confluent hypergeometric function. 00263 */ 00264 template<typename _Tp> 00265 _Tp 00266 __hyperg_series(const _Tp __a, const _Tp __b, 00267 const _Tp __c, const _Tp __x) 00268 { 00269 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00270 00271 _Tp __term = _Tp(1); 00272 _Tp __Fabc = _Tp(1); 00273 const unsigned int __max_iter = 100000; 00274 unsigned int __i; 00275 for (__i = 0; __i < __max_iter; ++__i) 00276 { 00277 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x 00278 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 00279 if (std::abs(__term) < __eps) 00280 { 00281 break; 00282 } 00283 __Fabc += __term; 00284 } 00285 if (__i == __max_iter) 00286 std::__throw_runtime_error(__N("Series failed to converge " 00287 "in __hyperg_series.")); 00288 00289 return __Fabc; 00290 } 00291 00292 00293 /** 00294 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 00295 * by an iterative procedure described in 00296 * Luke, Algorithms for the Computation of Mathematical Functions. 00297 */ 00298 template<typename _Tp> 00299 _Tp 00300 __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c, 00301 const _Tp __xin) 00302 { 00303 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 00304 const int __nmax = 20000; 00305 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00306 const _Tp __x = -__xin; 00307 const _Tp __x3 = __x * __x * __x; 00308 const _Tp __t0 = __a * __b / __c; 00309 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); 00310 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) 00311 / (_Tp(2) * (__c + _Tp(1))); 00312 00313 _Tp __F = _Tp(1); 00314 00315 _Tp __Bnm3 = _Tp(1); 00316 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 00317 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 00318 00319 _Tp __Anm3 = _Tp(1); 00320 _Tp __Anm2 = __Bnm2 - __t0 * __x; 00321 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 00322 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 00323 00324 int __n = 3; 00325 while (1) 00326 { 00327 const _Tp __npam1 = _Tp(__n - 1) + __a; 00328 const _Tp __npbm1 = _Tp(__n - 1) + __b; 00329 const _Tp __npcm1 = _Tp(__n - 1) + __c; 00330 const _Tp __npam2 = _Tp(__n - 2) + __a; 00331 const _Tp __npbm2 = _Tp(__n - 2) + __b; 00332 const _Tp __npcm2 = _Tp(__n - 2) + __c; 00333 const _Tp __tnm1 = _Tp(2 * __n - 1); 00334 const _Tp __tnm3 = _Tp(2 * __n - 3); 00335 const _Tp __tnm5 = _Tp(2 * __n - 5); 00336 const _Tp __n2 = __n * __n; 00337 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n 00338 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) 00339 / (_Tp(2) * __tnm3 * __npcm1); 00340 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n 00341 + _Tp(2) - __a * __b) * __npam1 * __npbm1 00342 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 00343 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 00344 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) 00345 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 00346 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 00347 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) 00348 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 00349 00350 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 00351 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 00352 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 00353 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 00354 const _Tp __r = __An / __Bn; 00355 00356 const _Tp __prec = std::abs((__F - __r) / __F); 00357 __F = __r; 00358 00359 if (__prec < __eps || __n > __nmax) 00360 break; 00361 00362 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 00363 { 00364 __An /= __big; 00365 __Bn /= __big; 00366 __Anm1 /= __big; 00367 __Bnm1 /= __big; 00368 __Anm2 /= __big; 00369 __Bnm2 /= __big; 00370 __Anm3 /= __big; 00371 __Bnm3 /= __big; 00372 } 00373 else if (std::abs(__An) < _Tp(1) / __big 00374 || std::abs(__Bn) < _Tp(1) / __big) 00375 { 00376 __An *= __big; 00377 __Bn *= __big; 00378 __Anm1 *= __big; 00379 __Bnm1 *= __big; 00380 __Anm2 *= __big; 00381 __Bnm2 *= __big; 00382 __Anm3 *= __big; 00383 __Bnm3 *= __big; 00384 } 00385 00386 ++__n; 00387 __Bnm3 = __Bnm2; 00388 __Bnm2 = __Bnm1; 00389 __Bnm1 = __Bn; 00390 __Anm3 = __Anm2; 00391 __Anm2 = __Anm1; 00392 __Anm1 = __An; 00393 } 00394 00395 if (__n >= __nmax) 00396 std::__throw_runtime_error(__N("Iteration failed to converge " 00397 "in __hyperg_luke.")); 00398 00399 return __F; 00400 } 00401 00402 00403 /** 00404 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection 00405 * formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral 00406 * and formula 15.3.11 for d = c - a - b integral. 00407 * This assumes a, b, c != negative integer. 00408 * 00409 * The hypogeometric function is defined by 00410 * @f[ 00411 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 00412 * \sum_{n=0}^{\infty} 00413 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 00414 * \frac{x^n}{n!} 00415 * @f] 00416 * 00417 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: 00418 * @f[ 00419 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} 00420 * _2F_1(a,b;1-d;1-x) 00421 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} 00422 * _2F_1(c-a,c-b;1+d;1-x) 00423 * @f] 00424 * 00425 * The reflection formula for integral @f$ m = c - a - b @f$ is: 00426 * @f[ 00427 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} 00428 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} 00429 * - 00430 * @f] 00431 */ 00432 template<typename _Tp> 00433 _Tp 00434 __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c, 00435 const _Tp __x) 00436 { 00437 const _Tp __d = __c - __a - __b; 00438 const int __intd = std::floor(__d + _Tp(0.5L)); 00439 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00440 const _Tp __toler = _Tp(1000) * __eps; 00441 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); 00442 const bool __d_integer = (std::abs(__d - __intd) < __toler); 00443 00444 if (__d_integer) 00445 { 00446 const _Tp __ln_omx = std::log(_Tp(1) - __x); 00447 const _Tp __ad = std::abs(__d); 00448 _Tp __F1, __F2; 00449 00450 _Tp __d1, __d2; 00451 if (__d >= _Tp(0)) 00452 { 00453 __d1 = __d; 00454 __d2 = _Tp(0); 00455 } 00456 else 00457 { 00458 __d1 = _Tp(0); 00459 __d2 = __d; 00460 } 00461 00462 const _Tp __lng_c = __log_gamma(__c); 00463 00464 // Evaluate F1. 00465 if (__ad < __eps) 00466 { 00467 // d = c - a - b = 0. 00468 __F1 = _Tp(0); 00469 } 00470 else 00471 { 00472 00473 bool __ok_d1 = true; 00474 _Tp __lng_ad, __lng_ad1, __lng_bd1; 00475 __try 00476 { 00477 __lng_ad = __log_gamma(__ad); 00478 __lng_ad1 = __log_gamma(__a + __d1); 00479 __lng_bd1 = __log_gamma(__b + __d1); 00480 } 00481 __catch(...) 00482 { 00483 __ok_d1 = false; 00484 } 00485 00486 if (__ok_d1) 00487 { 00488 /* Gamma functions in the denominator are ok. 00489 * Proceed with evaluation. 00490 */ 00491 _Tp __sum1 = _Tp(1); 00492 _Tp __term = _Tp(1); 00493 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx 00494 - __lng_ad1 - __lng_bd1; 00495 00496 /* Do F1 sum. 00497 */ 00498 for (int __i = 1; __i < __ad; ++__i) 00499 { 00500 const int __j = __i - 1; 00501 __term *= (__a + __d2 + __j) * (__b + __d2 + __j) 00502 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); 00503 __sum1 += __term; 00504 } 00505 00506 if (__ln_pre1 > __log_max) 00507 std::__throw_runtime_error(__N("Overflow of gamma functions " 00508 "in __hyperg_luke.")); 00509 else 00510 __F1 = std::exp(__ln_pre1) * __sum1; 00511 } 00512 else 00513 { 00514 // Gamma functions in the denominator were not ok. 00515 // So the F1 term is zero. 00516 __F1 = _Tp(0); 00517 } 00518 } // end F1 evaluation 00519 00520 // Evaluate F2. 00521 bool __ok_d2 = true; 00522 _Tp __lng_ad2, __lng_bd2; 00523 __try 00524 { 00525 __lng_ad2 = __log_gamma(__a + __d2); 00526 __lng_bd2 = __log_gamma(__b + __d2); 00527 } 00528 __catch(...) 00529 { 00530 __ok_d2 = false; 00531 } 00532 00533 if (__ok_d2) 00534 { 00535 // Gamma functions in the denominator are ok. 00536 // Proceed with evaluation. 00537 const int __maxiter = 2000; 00538 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); 00539 const _Tp __psi_1pd = __psi(_Tp(1) + __ad); 00540 const _Tp __psi_apd1 = __psi(__a + __d1); 00541 const _Tp __psi_bpd1 = __psi(__b + __d1); 00542 00543 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 00544 - __psi_bpd1 - __ln_omx; 00545 _Tp __fact = _Tp(1); 00546 _Tp __sum2 = __psi_term; 00547 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx 00548 - __lng_ad2 - __lng_bd2; 00549 00550 // Do F2 sum. 00551 int __j; 00552 for (__j = 1; __j < __maxiter; ++__j) 00553 { 00554 // Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5 00555 const _Tp __term1 = _Tp(1) / _Tp(__j) 00556 + _Tp(1) / (__ad + __j); 00557 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) 00558 + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); 00559 __psi_term += __term1 - __term2; 00560 __fact *= (__a + __d1 + _Tp(__j - 1)) 00561 * (__b + __d1 + _Tp(__j - 1)) 00562 / ((__ad + __j) * __j) * (_Tp(1) - __x); 00563 const _Tp __delta = __fact * __psi_term; 00564 __sum2 += __delta; 00565 if (std::abs(__delta) < __eps * std::abs(__sum2)) 00566 break; 00567 } 00568 if (__j == __maxiter) 00569 std::__throw_runtime_error(__N("Sum F2 failed to converge " 00570 "in __hyperg_reflect")); 00571 00572 if (__sum2 == _Tp(0)) 00573 __F2 = _Tp(0); 00574 else 00575 __F2 = std::exp(__ln_pre2) * __sum2; 00576 } 00577 else 00578 { 00579 // Gamma functions in the denominator not ok. 00580 // So the F2 term is zero. 00581 __F2 = _Tp(0); 00582 } // end F2 evaluation 00583 00584 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); 00585 const _Tp __F = __F1 + __sgn_2 * __F2; 00586 00587 return __F; 00588 } 00589 else 00590 { 00591 // d = c - a - b not an integer. 00592 00593 // These gamma functions appear in the denominator, so we 00594 // catch their harmless domain errors and set the terms to zero. 00595 bool __ok1 = true; 00596 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); 00597 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); 00598 __try 00599 { 00600 __sgn_g1ca = __log_gamma_sign(__c - __a); 00601 __ln_g1ca = __log_gamma(__c - __a); 00602 __sgn_g1cb = __log_gamma_sign(__c - __b); 00603 __ln_g1cb = __log_gamma(__c - __b); 00604 } 00605 __catch(...) 00606 { 00607 __ok1 = false; 00608 } 00609 00610 bool __ok2 = true; 00611 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); 00612 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); 00613 __try 00614 { 00615 __sgn_g2a = __log_gamma_sign(__a); 00616 __ln_g2a = __log_gamma(__a); 00617 __sgn_g2b = __log_gamma_sign(__b); 00618 __ln_g2b = __log_gamma(__b); 00619 } 00620 __catch(...) 00621 { 00622 __ok2 = false; 00623 } 00624 00625 const _Tp __sgn_gc = __log_gamma_sign(__c); 00626 const _Tp __ln_gc = __log_gamma(__c); 00627 const _Tp __sgn_gd = __log_gamma_sign(__d); 00628 const _Tp __ln_gd = __log_gamma(__d); 00629 const _Tp __sgn_gmd = __log_gamma_sign(-__d); 00630 const _Tp __ln_gmd = __log_gamma(-__d); 00631 00632 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; 00633 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; 00634 00635 _Tp __pre1, __pre2; 00636 if (__ok1 && __ok2) 00637 { 00638 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 00639 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 00640 + __d * std::log(_Tp(1) - __x); 00641 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) 00642 { 00643 __pre1 = std::exp(__ln_pre1); 00644 __pre2 = std::exp(__ln_pre2); 00645 __pre1 *= __sgn1; 00646 __pre2 *= __sgn2; 00647 } 00648 else 00649 { 00650 std::__throw_runtime_error(__N("Overflow of gamma functions " 00651 "in __hyperg_reflect")); 00652 } 00653 } 00654 else if (__ok1 && !__ok2) 00655 { 00656 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 00657 if (__ln_pre1 < __log_max) 00658 { 00659 __pre1 = std::exp(__ln_pre1); 00660 __pre1 *= __sgn1; 00661 __pre2 = _Tp(0); 00662 } 00663 else 00664 { 00665 std::__throw_runtime_error(__N("Overflow of gamma functions " 00666 "in __hyperg_reflect")); 00667 } 00668 } 00669 else if (!__ok1 && __ok2) 00670 { 00671 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 00672 + __d * std::log(_Tp(1) - __x); 00673 if (__ln_pre2 < __log_max) 00674 { 00675 __pre1 = _Tp(0); 00676 __pre2 = std::exp(__ln_pre2); 00677 __pre2 *= __sgn2; 00678 } 00679 else 00680 { 00681 std::__throw_runtime_error(__N("Overflow of gamma functions " 00682 "in __hyperg_reflect")); 00683 } 00684 } 00685 else 00686 { 00687 __pre1 = _Tp(0); 00688 __pre2 = _Tp(0); 00689 std::__throw_runtime_error(__N("Underflow of gamma functions " 00690 "in __hyperg_reflect")); 00691 } 00692 00693 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, 00694 _Tp(1) - __x); 00695 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, 00696 _Tp(1) - __x); 00697 00698 const _Tp __F = __pre1 * __F1 + __pre2 * __F2; 00699 00700 return __F; 00701 } 00702 } 00703 00704 00705 /** 00706 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. 00707 * 00708 * The hypogeometric function is defined by 00709 * @f[ 00710 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 00711 * \sum_{n=0}^{\infty} 00712 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 00713 * \frac{x^n}{n!} 00714 * @f] 00715 * 00716 * @param __a The first "numerator" parameter. 00717 * @param __a The second "numerator" parameter. 00718 * @param __c The "denominator" parameter. 00719 * @param __x The argument of the confluent hypergeometric function. 00720 * @return The confluent hypergeometric function. 00721 */ 00722 template<typename _Tp> 00723 inline _Tp 00724 __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) 00725 { 00726 #if _GLIBCXX_USE_C99_MATH_TR1 00727 const _Tp __a_nint = std::tr1::nearbyint(__a); 00728 const _Tp __b_nint = std::tr1::nearbyint(__b); 00729 const _Tp __c_nint = std::tr1::nearbyint(__c); 00730 #else 00731 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); 00732 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); 00733 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 00734 #endif 00735 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); 00736 if (std::abs(__x) >= _Tp(1)) 00737 std::__throw_domain_error(__N("Argument outside unit circle " 00738 "in __hyperg.")); 00739 else if (__isnan(__a) || __isnan(__b) 00740 || __isnan(__c) || __isnan(__x)) 00741 return std::numeric_limits<_Tp>::quiet_NaN(); 00742 else if (__c_nint == __c && __c_nint <= _Tp(0)) 00743 return std::numeric_limits<_Tp>::infinity(); 00744 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) 00745 return std::pow(_Tp(1) - __x, __c - __a - __b); 00746 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) 00747 && __x >= _Tp(0) && __x < _Tp(0.995L)) 00748 return __hyperg_series(__a, __b, __c, __x); 00749 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) 00750 { 00751 // For integer a and b the hypergeometric function is a finite polynomial. 00752 if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) 00753 return __hyperg_series(__a_nint, __b, __c, __x); 00754 else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) 00755 return __hyperg_series(__a, __b_nint, __c, __x); 00756 else if (__x < -_Tp(0.25L)) 00757 return __hyperg_luke(__a, __b, __c, __x); 00758 else if (__x < _Tp(0.5L)) 00759 return __hyperg_series(__a, __b, __c, __x); 00760 else 00761 if (std::abs(__c) > _Tp(10)) 00762 return __hyperg_series(__a, __b, __c, __x); 00763 else 00764 return __hyperg_reflect(__a, __b, __c, __x); 00765 } 00766 else 00767 return __hyperg_luke(__a, __b, __c, __x); 00768 } 00769 00770 } // namespace std::tr1::__detail 00771 } 00772 } 00773 00774 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC