| // (C) Copyright John Maddock 2006. | |
| // Use, modification and distribution are subject to the | |
| // Boost Software License, Version 1.0. (See accompanying file | |
| // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | |
| #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL | |
| #define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL | |
| #ifdef _MSC_VER | |
| #pragma once | |
| #endif | |
| namespace boost{ namespace math{ namespace detail{ | |
| // | |
| // lgamma for small arguments: | |
| // | |
| template <class T, class Policy, class L> | |
| T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<64>&, const Policy& /* l */, const L&) | |
| { | |
| // This version uses rational approximations for small | |
| // values of z accurate enough for 64-bit mantissas | |
| // (80-bit long doubles), works well for 53-bit doubles as well. | |
| // L is only used to select the Lanczos function. | |
| BOOST_MATH_STD_USING // for ADL of std names | |
| T result = 0; | |
| if(z < tools::epsilon<T>()) | |
| { | |
| result = -log(z); | |
| } | |
| else if((zm1 == 0) || (zm2 == 0)) | |
| { | |
| // nothing to do, result is zero.... | |
| } | |
| else if(z > 2) | |
| { | |
| // | |
| // Begin by performing argument reduction until | |
| // z is in [2,3): | |
| // | |
| if(z >= 3) | |
| { | |
| do | |
| { | |
| z -= 1; | |
| zm2 -= 1; | |
| result += log(z); | |
| }while(z >= 3); | |
| // Update zm2, we need it below: | |
| zm2 = z - 2; | |
| } | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (z-2)(z+1)(Y + R(z-2)) | |
| // | |
| // where R(z-2) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // R(z-2) has the following properties: | |
| // | |
| // At double: Max error found: 4.231e-18 | |
| // At long double: Max error found: 1.987e-21 | |
| // Maximum Deviation Found (approximation error): 5.900e-24 | |
| // | |
| static const T P[] = { | |
| static_cast<T>(-0.180355685678449379109e-1L), | |
| static_cast<T>(0.25126649619989678683e-1L), | |
| static_cast<T>(0.494103151567532234274e-1L), | |
| static_cast<T>(0.172491608709613993966e-1L), | |
| static_cast<T>(-0.259453563205438108893e-3L), | |
| static_cast<T>(-0.541009869215204396339e-3L), | |
| static_cast<T>(-0.324588649825948492091e-4L) | |
| }; | |
| static const T Q[] = { | |
| static_cast<T>(0.1e1), | |
| static_cast<T>(0.196202987197795200688e1L), | |
| static_cast<T>(0.148019669424231326694e1L), | |
| static_cast<T>(0.541391432071720958364e0L), | |
| static_cast<T>(0.988504251128010129477e-1L), | |
| static_cast<T>(0.82130967464889339326e-2L), | |
| static_cast<T>(0.224936291922115757597e-3L), | |
| static_cast<T>(-0.223352763208617092964e-6L) | |
| }; | |
| static const float Y = 0.158963680267333984375e0f; | |
| T r = zm2 * (z + 1); | |
| T R = tools::evaluate_polynomial(P, zm2); | |
| R /= tools::evaluate_polynomial(Q, zm2); | |
| result += r * Y + r * R; | |
| } | |
| else | |
| { | |
| // | |
| // If z is less than 1 use recurrance to shift to | |
| // z in the interval [1,2]: | |
| // | |
| if(z < 1) | |
| { | |
| result += -log(z); | |
| zm2 = zm1; | |
| zm1 = z; | |
| z += 1; | |
| } | |
| // | |
| // Two approximations, on for z in [1,1.5] and | |
| // one for z in [1.5,2]: | |
| // | |
| if(z <= 1.5) | |
| { | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (z-1)(z-2)(Y + R(z-1)) | |
| // | |
| // where R(z-1) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // R(z-1) has the following properties: | |
| // | |
| // At double precision: Max error found: 1.230011e-17 | |
| // At 80-bit long double precision: Max error found: 5.631355e-21 | |
| // Maximum Deviation Found: 3.139e-021 | |
| // Expected Error Term: 3.139e-021 | |
| // | |
| static const float Y = 0.52815341949462890625f; | |
| static const T P[] = { | |
| static_cast<T>(0.490622454069039543534e-1L), | |
| static_cast<T>(-0.969117530159521214579e-1L), | |
| static_cast<T>(-0.414983358359495381969e0L), | |
| static_cast<T>(-0.406567124211938417342e0L), | |
| static_cast<T>(-0.158413586390692192217e0L), | |
| static_cast<T>(-0.240149820648571559892e-1L), | |
| static_cast<T>(-0.100346687696279557415e-2L) | |
| }; | |
| static const T Q[] = { | |
| static_cast<T>(0.1e1L), | |
| static_cast<T>(0.302349829846463038743e1L), | |
| static_cast<T>(0.348739585360723852576e1L), | |
| static_cast<T>(0.191415588274426679201e1L), | |
| static_cast<T>(0.507137738614363510846e0L), | |
| static_cast<T>(0.577039722690451849648e-1L), | |
| static_cast<T>(0.195768102601107189171e-2L) | |
| }; | |
| T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1); | |
| T prefix = zm1 * zm2; | |
| result += prefix * Y + prefix * r; | |
| } | |
| else | |
| { | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (2-z)(1-z)(Y + R(2-z)) | |
| // | |
| // where R(2-z) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // R(2-z) has the following properties: | |
| // | |
| // At double precision, max error found: 1.797565e-17 | |
| // At 80-bit long double precision, max error found: 9.306419e-21 | |
| // Maximum Deviation Found: 2.151e-021 | |
| // Expected Error Term: 2.150e-021 | |
| // | |
| static const float Y = 0.452017307281494140625f; | |
| static const T P[] = { | |
| static_cast<T>(-0.292329721830270012337e-1L), | |
| static_cast<T>(0.144216267757192309184e0L), | |
| static_cast<T>(-0.142440390738631274135e0L), | |
| static_cast<T>(0.542809694055053558157e-1L), | |
| static_cast<T>(-0.850535976868336437746e-2L), | |
| static_cast<T>(0.431171342679297331241e-3L) | |
| }; | |
| static const T Q[] = { | |
| static_cast<T>(0.1e1), | |
| static_cast<T>(-0.150169356054485044494e1L), | |
| static_cast<T>(0.846973248876495016101e0L), | |
| static_cast<T>(-0.220095151814995745555e0L), | |
| static_cast<T>(0.25582797155975869989e-1L), | |
| static_cast<T>(-0.100666795539143372762e-2L), | |
| static_cast<T>(-0.827193521891290553639e-6L) | |
| }; | |
| T r = zm2 * zm1; | |
| T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2); | |
| result += r * Y + r * R; | |
| } | |
| } | |
| return result; | |
| } | |
| template <class T, class Policy, class L> | |
| T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<113>&, const Policy& /* l */, const L&) | |
| { | |
| // | |
| // This version uses rational approximations for small | |
| // values of z accurate enough for 113-bit mantissas | |
| // (128-bit long doubles). | |
| // | |
| BOOST_MATH_STD_USING // for ADL of std names | |
| T result = 0; | |
| if(z < tools::epsilon<T>()) | |
| { | |
| result = -log(z); | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| } | |
| else if((zm1 == 0) || (zm2 == 0)) | |
| { | |
| // nothing to do, result is zero.... | |
| } | |
| else if(z > 2) | |
| { | |
| // | |
| // Begin by performing argument reduction until | |
| // z is in [2,3): | |
| // | |
| if(z >= 3) | |
| { | |
| do | |
| { | |
| z -= 1; | |
| result += log(z); | |
| }while(z >= 3); | |
| zm2 = z - 2; | |
| } | |
| BOOST_MATH_INSTRUMENT_CODE(zm2); | |
| BOOST_MATH_INSTRUMENT_CODE(z); | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (z-2)(z+1)(Y + R(z-2)) | |
| // | |
| // where R(z-2) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // Maximum Deviation Found (approximation error) 3.73e-37 | |
| static const T P[] = { | |
| -0.018035568567844937910504030027467476655L, | |
| 0.013841458273109517271750705401202404195L, | |
| 0.062031842739486600078866923383017722399L, | |
| 0.052518418329052161202007865149435256093L, | |
| 0.01881718142472784129191838493267755758L, | |
| 0.0025104830367021839316463675028524702846L, | |
| -0.00021043176101831873281848891452678568311L, | |
| -0.00010249622350908722793327719494037981166L, | |
| -0.11381479670982006841716879074288176994e-4L, | |
| -0.49999811718089980992888533630523892389e-6L, | |
| -0.70529798686542184668416911331718963364e-8L | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.5877485070422317542808137697939233685L, | |
| 2.8797959228352591788629602533153837126L, | |
| 1.8030885955284082026405495275461180977L, | |
| 0.69774331297747390169238306148355428436L, | |
| 0.17261566063277623942044077039756583802L, | |
| 0.02729301254544230229429621192443000121L, | |
| 0.0026776425891195270663133581960016620433L, | |
| 0.00015244249160486584591370355730402168106L, | |
| 0.43997034032479866020546814475414346627e-5L, | |
| 0.46295080708455613044541885534408170934e-7L, | |
| -0.93326638207459533682980757982834180952e-11L, | |
| 0.42316456553164995177177407325292867513e-13L | |
| }; | |
| T R = tools::evaluate_polynomial(P, zm2); | |
| R /= tools::evaluate_polynomial(Q, zm2); | |
| static const float Y = 0.158963680267333984375F; | |
| T r = zm2 * (z + 1); | |
| result += r * Y + r * R; | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| } | |
| else | |
| { | |
| // | |
| // If z is less than 1 use recurrance to shift to | |
| // z in the interval [1,2]: | |
| // | |
| if(z < 1) | |
| { | |
| result += -log(z); | |
| zm2 = zm1; | |
| zm1 = z; | |
| z += 1; | |
| } | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| BOOST_MATH_INSTRUMENT_CODE(z); | |
| BOOST_MATH_INSTRUMENT_CODE(zm2); | |
| // | |
| // Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1] | |
| // | |
| if(z <= 1.35) | |
| { | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (z-1)(z-2)(Y + R(z-1)) | |
| // | |
| // where R(z-1) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // R(z-1) has the following properties: | |
| // | |
| // Maximum Deviation Found (approximation error) 1.659e-36 | |
| // Expected Error Term (theoretical error) 1.343e-36 | |
| // Max error found at 128-bit long double precision 1.007e-35 | |
| // | |
| static const float Y = 0.54076099395751953125f; | |
| static const T P[] = { | |
| 0.036454670944013329356512090082402429697L, | |
| -0.066235835556476033710068679907798799959L, | |
| -0.67492399795577182387312206593595565371L, | |
| -1.4345555263962411429855341651960000166L, | |
| -1.4894319559821365820516771951249649563L, | |
| -0.87210277668067964629483299712322411566L, | |
| -0.29602090537771744401524080430529369136L, | |
| -0.0561832587517836908929331992218879676L, | |
| -0.0053236785487328044334381502530383140443L, | |
| -0.00018629360291358130461736386077971890789L, | |
| -0.10164985672213178500790406939467614498e-6L, | |
| 0.13680157145361387405588201461036338274e-8L | |
| }; | |
| static const T Q[] = { | |
| 1, | |
| 4.9106336261005990534095838574132225599L, | |
| 10.258804800866438510889341082793078432L, | |
| 11.88588976846826108836629960537466889L, | |
| 8.3455000546999704314454891036700998428L, | |
| 3.6428823682421746343233362007194282703L, | |
| 0.97465989807254572142266753052776132252L, | |
| 0.15121052897097822172763084966793352524L, | |
| 0.012017363555383555123769849654484594893L, | |
| 0.0003583032812720649835431669893011257277L | |
| }; | |
| T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1); | |
| T prefix = zm1 * zm2; | |
| result += prefix * Y + prefix * r; | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| } | |
| else if(z <= 1.625) | |
| { | |
| // | |
| // Use the following form: | |
| // | |
| // lgamma(z) = (2-z)(1-z)(Y + R(2-z)) | |
| // | |
| // where R(2-z) is a rational approximation optimised for | |
| // low absolute error - as long as it's absolute error | |
| // is small compared to the constant Y - then any rounding | |
| // error in it's computation will get wiped out. | |
| // | |
| // R(2-z) has the following properties: | |
| // | |
| // Max error found at 128-bit long double precision 9.634e-36 | |
| // Maximum Deviation Found (approximation error) 1.538e-37 | |
| // Expected Error Term (theoretical error) 2.350e-38 | |
| // | |
| static const float Y = 0.483787059783935546875f; | |
| static const T P[] = { | |
| -0.017977422421608624353488126610933005432L, | |
| 0.18484528905298309555089509029244135703L, | |
| -0.40401251514859546989565001431430884082L, | |
| 0.40277179799147356461954182877921388182L, | |
| -0.21993421441282936476709677700477598816L, | |
| 0.069595742223850248095697771331107571011L, | |
| -0.012681481427699686635516772923547347328L, | |
| 0.0012489322866834830413292771335113136034L, | |
| -0.57058739515423112045108068834668269608e-4L, | |
| 0.8207548771933585614380644961342925976e-6L | |
| }; | |
| static const T Q[] = { | |
| 1, | |
| -2.9629552288944259229543137757200262073L, | |
| 3.7118380799042118987185957298964772755L, | |
| -2.5569815272165399297600586376727357187L, | |
| 1.0546764918220835097855665680632153367L, | |
| -0.26574021300894401276478730940980810831L, | |
| 0.03996289731752081380552901986471233462L, | |
| -0.0033398680924544836817826046380586480873L, | |
| 0.00013288854760548251757651556792598235735L, | |
| -0.17194794958274081373243161848194745111e-5L | |
| }; | |
| T r = zm2 * zm1; | |
| T R = tools::evaluate_polynomial(P, 0.625 - zm1) / tools::evaluate_polynomial(Q, 0.625 - zm1); | |
| result += r * Y + r * R; | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| } | |
| else | |
| { | |
| // | |
| // Same form as above. | |
| // | |
| // Max error found (at 128-bit long double precision) 1.831e-35 | |
| // Maximum Deviation Found (approximation error) 8.588e-36 | |
| // Expected Error Term (theoretical error) 1.458e-36 | |
| // | |
| static const float Y = 0.443811893463134765625f; | |
| static const T P[] = { | |
| -0.021027558364667626231512090082402429494L, | |
| 0.15128811104498736604523586803722368377L, | |
| -0.26249631480066246699388544451126410278L, | |
| 0.21148748610533489823742352180628489742L, | |
| -0.093964130697489071999873506148104370633L, | |
| 0.024292059227009051652542804957550866827L, | |
| -0.0036284453226534839926304745756906117066L, | |
| 0.0002939230129315195346843036254392485984L, | |
| -0.11088589183158123733132268042570710338e-4L, | |
| 0.13240510580220763969511741896361984162e-6L | |
| }; | |
| static const T Q[] = { | |
| 1, | |
| -2.4240003754444040525462170802796471996L, | |
| 2.4868383476933178722203278602342786002L, | |
| -1.4047068395206343375520721509193698547L, | |
| 0.47583809087867443858344765659065773369L, | |
| -0.09865724264554556400463655444270700132L, | |
| 0.012238223514176587501074150988445109735L, | |
| -0.00084625068418239194670614419707491797097L, | |
| 0.2796574430456237061420839429225710602e-4L, | |
| -0.30202973883316730694433702165188835331e-6L | |
| }; | |
| // (2 - x) * (1 - x) * (c + R(2 - x)) | |
| T r = zm2 * zm1; | |
| T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2); | |
| result += r * Y + r * R; | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| } | |
| } | |
| BOOST_MATH_INSTRUMENT_CODE(result); | |
| return result; | |
| } | |
| template <class T, class Policy, class L> | |
| T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<0>&, const Policy& pol, const L&) | |
| { | |
| // | |
| // No rational approximations are available because either | |
| // T has no numeric_limits support (so we can't tell how | |
| // many digits it has), or T has more digits than we know | |
| // what to do with.... we do have a Lanczos approximation | |
| // though, and that can be used to keep errors under control. | |
| // | |
| BOOST_MATH_STD_USING // for ADL of std names | |
| T result = 0; | |
| if(z < tools::epsilon<T>()) | |
| { | |
| result = -log(z); | |
| } | |
| else if(z < 0.5) | |
| { | |
| // taking the log of tgamma reduces the error, no danger of overflow here: | |
| result = log(gamma_imp(z, pol, L())); | |
| } | |
| else if(z >= 3) | |
| { | |
| // taking the log of tgamma reduces the error, no danger of overflow here: | |
| result = log(gamma_imp(z, pol, L())); | |
| } | |
| else if(z >= 1.5) | |
| { | |
| // special case near 2: | |
| T dz = zm2; | |
| result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e<T>()); | |
| result += boost::math::log1p(dz / (L::g() + T(1.5)), pol) * T(1.5); | |
| result += boost::math::log1p(L::lanczos_sum_near_2(dz), pol); | |
| } | |
| else | |
| { | |
| // special case near 1: | |
| T dz = zm1; | |
| result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e<T>()); | |
| result += boost::math::log1p(dz / (L::g() + T(0.5)), pol) / 2; | |
| result += boost::math::log1p(L::lanczos_sum_near_1(dz), pol); | |
| } | |
| return result; | |
| } | |
| }}} // namespaces | |
| #endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL | |