// (C) Copyright John Maddock 2006-8. | |
// 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_NTL_DIGAMMA | |
#define BOOST_MATH_NTL_DIGAMMA | |
#include <boost/math/tools/rational.hpp> | |
#include <boost/math/policies/error_handling.hpp> | |
#include <boost/math/constants/constants.hpp> | |
namespace boost{ namespace math{ namespace detail{ | |
template <class T> | |
T big_digamma_helper(T x) | |
{ | |
static const T P[61] = { | |
boost::lexical_cast<T>("0.6660133691143982067148122682345055274952e81"), | |
boost::lexical_cast<T>("0.6365271516829242456324234577164675383137e81"), | |
boost::lexical_cast<T>("0.2991038873096202943405966144203628966976e81"), | |
boost::lexical_cast<T>("0.9211116495503170498076013367421231351115e80"), | |
boost::lexical_cast<T>("0.2090792764676090716286400360584443891749e80"), | |
boost::lexical_cast<T>("0.3730037777359591428226035156377978092809e79"), | |
boost::lexical_cast<T>("0.5446396536956682043376492370432031543834e78"), | |
boost::lexical_cast<T>("0.6692523966335177847425047827449069256345e77"), | |
boost::lexical_cast<T>("0.7062543624100864681625612653756619116848e76"), | |
boost::lexical_cast<T>("0.6499914905966283735005256964443226879158e75"), | |
boost::lexical_cast<T>("0.5280364564853225211197557708655426736091e74"), | |
boost::lexical_cast<T>("0.3823205608981176913075543599005095206953e73"), | |
boost::lexical_cast<T>("0.2486733714214237704739129972671154532415e72"), | |
boost::lexical_cast<T>("0.1462562139602039577983434547171318011675e71"), | |
boost::lexical_cast<T>("0.7821169065036815012381267259559910324285e69"), | |
boost::lexical_cast<T>("0.3820552182348155468636157988764435365078e68"), | |
boost::lexical_cast<T>("0.1711618296983598244658239925535632505062e67"), | |
boost::lexical_cast<T>("0.7056661618357643731419080738521475204245e65"), | |
boost::lexical_cast<T>("0.2685246896473614017356264531791459936036e64"), | |
boost::lexical_cast<T>("0.9455168125599643085283071944864977592391e62"), | |
boost::lexical_cast<T>("0.3087541626972538362237309145177486236219e61"), | |
boost::lexical_cast<T>("0.9367928873352980208052601301625005737407e59"), | |
boost::lexical_cast<T>("0.2645306130689794942883818547314327466007e58"), | |
boost::lexical_cast<T>("0.6961815141171454309161007351079576190079e56"), | |
boost::lexical_cast<T>("0.1709637824471794552313802669803885946843e55"), | |
boost::lexical_cast<T>("0.3921553258481531526663112728778759311158e53"), | |
boost::lexical_cast<T>("0.8409006354449988687714450897575728228696e51"), | |
boost::lexical_cast<T>("0.1686755204461325935742097669030363344927e50"), | |
boost::lexical_cast<T>("0.3166653542877070999007425197729038754254e48"), | |
boost::lexical_cast<T>("0.5566029092358215049069560272835654229637e46"), | |
boost::lexical_cast<T>("0.9161766287916328133080586672953875116242e44"), | |
boost::lexical_cast<T>("1412317772330871298317974693514430627922000"), | |
boost::lexical_cast<T>("20387991986727877473732570146112459874790"), | |
boost::lexical_cast<T>("275557928713904105182512535678580359839.3"), | |
boost::lexical_cast<T>("3485719851040516559072031256589598330.723"), | |
boost::lexical_cast<T>("41247046743564028399938106707656877.40859"), | |
boost::lexical_cast<T>("456274078125709314602601667471879.0147312"), | |
boost::lexical_cast<T>("4714450683242899367025707077155.310613012"), | |
boost::lexical_cast<T>("45453933537925041680009544258.75073849996"), | |
boost::lexical_cast<T>("408437900487067278846361972.302331241052"), | |
boost::lexical_cast<T>("3415719344386166273085838.705771571751035"), | |
boost::lexical_cast<T>("26541502879185876562320.93134691487351145"), | |
boost::lexical_cast<T>("191261415065918713661.1571433274648417668"), | |
boost::lexical_cast<T>("1275349770108718421.645275944284937551702"), | |
boost::lexical_cast<T>("7849171120971773.318910987434906905704272"), | |
boost::lexical_cast<T>("44455946386549.80866460312682983576538056"), | |
boost::lexical_cast<T>("230920362395.3198137186361608905136598046"), | |
boost::lexical_cast<T>("1095700096.240863858624279930600654130254"), | |
boost::lexical_cast<T>("4727085.467506050153744334085516289728134"), | |
boost::lexical_cast<T>("18440.75118859447173303252421991479005424"), | |
boost::lexical_cast<T>("64.62515887799460295677071749181651317052"), | |
boost::lexical_cast<T>("0.201851568864688406206528472883512147547"), | |
boost::lexical_cast<T>("0.0005565091674187978029138500039504078098143"), | |
boost::lexical_cast<T>("0.1338097668312907986354698683493366559613e-5"), | |
boost::lexical_cast<T>("0.276308225077464312820179030238305271638e-8"), | |
boost::lexical_cast<T>("0.4801582970473168520375942100071070575043e-11"), | |
boost::lexical_cast<T>("0.6829184144212920949740376186058541800175e-14"), | |
boost::lexical_cast<T>("0.7634080076358511276617829524639455399182e-17"), | |
boost::lexical_cast<T>("0.6290035083727140966418512608156646142409e-20"), | |
boost::lexical_cast<T>("0.339652245667538733044036638506893821352e-23"), | |
boost::lexical_cast<T>("0.9017518064256388530773585529891677854909e-27") | |
}; | |
static const T Q[61] = { | |
boost::lexical_cast<T>("0"), | |
boost::lexical_cast<T>("0.1386831185456898357379390197203894063459e81"), | |
boost::lexical_cast<T>("0.6467076379487574703291056110838151259438e81"), | |
boost::lexical_cast<T>("0.1394967823848615838336194279565285465161e82"), | |
boost::lexical_cast<T>("0.1872927317344192945218570366455046340458e82"), | |
boost::lexical_cast<T>("0.1772461045338946243584650759986310355937e82"), | |
boost::lexical_cast<T>("0.1267294892200258648315971144069595555118e82"), | |
boost::lexical_cast<T>("0.7157764838362416821508872117623058626589e81"), | |
boost::lexical_cast<T>("0.329447266909948668265277828268378274513e81"), | |
boost::lexical_cast<T>("0.1264376077317689779509250183194342571207e81"), | |
boost::lexical_cast<T>("0.4118230304191980787640446056583623228873e80"), | |
boost::lexical_cast<T>("0.1154393529762694616405952270558316515261e80"), | |
boost::lexical_cast<T>("0.281655612889423906125295485693696744275e79"), | |
boost::lexical_cast<T>("0.6037483524928743102724159846414025482077e78"), | |
boost::lexical_cast<T>("0.1145927995397835468123576831800276999614e78"), | |
boost::lexical_cast<T>("0.1938624296151985600348534009382865995154e77"), | |
boost::lexical_cast<T>("0.293980925856227626211879961219188406675e76"), | |
boost::lexical_cast<T>("0.4015574518216966910319562902099567437832e75"), | |
boost::lexical_cast<T>("0.4961475457509727343545565970423431880907e74"), | |
boost::lexical_cast<T>("0.5565482348278933960215521991000378896338e73"), | |
boost::lexical_cast<T>("0.5686112924615820754631098622770303094938e72"), | |
boost::lexical_cast<T>("0.5305988545844796293285410303747469932856e71"), | |
boost::lexical_cast<T>("0.4533363413802585060568537458067343491358e70"), | |
boost::lexical_cast<T>("0.3553932059473516064068322757331575565718e69"), | |
boost::lexical_cast<T>("0.2561198565218704414618802902533972354203e68"), | |
boost::lexical_cast<T>("0.1699519313292900324098102065697454295572e67"), | |
boost::lexical_cast<T>("0.1039830160862334505389615281373574959236e66"), | |
boost::lexical_cast<T>("0.5873082967977428281000961954715372504986e64"), | |
boost::lexical_cast<T>("0.3065255179030575882202133042549783442446e63"), | |
boost::lexical_cast<T>("0.1479494813481364701208655943688307245459e62"), | |
boost::lexical_cast<T>("0.6608150467921598615495180659808895663164e60"), | |
boost::lexical_cast<T>("0.2732535313770902021791888953487787496976e59"), | |
boost::lexical_cast<T>("0.1046402297662493314531194338414508049069e58"), | |
boost::lexical_cast<T>("0.3711375077192882936085049147920021549622e56"), | |
boost::lexical_cast<T>("0.1219154482883895482637944309702972234576e55"), | |
boost::lexical_cast<T>("0.3708359374149458741391374452286837880162e53"), | |
boost::lexical_cast<T>("0.1044095509971707189716913168889769471468e52"), | |
boost::lexical_cast<T>("0.271951506225063286130946773813524945052e50"), | |
boost::lexical_cast<T>("0.6548016291215163843464133978454065823866e48"), | |
boost::lexical_cast<T>("0.1456062447610542135403751730809295219344e47"), | |
boost::lexical_cast<T>("0.2986690175077969760978388356833006028929e45"), | |
boost::lexical_cast<T>("5643149706574013350061247429006443326844000"), | |
boost::lexical_cast<T>("98047545414467090421964387960743688053480"), | |
boost::lexical_cast<T>("1563378767746846395507385099301468978550"), | |
boost::lexical_cast<T>("22823360528584500077862274918382796495"), | |
boost::lexical_cast<T>("304215527004115213046601295970388750"), | |
boost::lexical_cast<T>("3690289075895685793844344966820325"), | |
boost::lexical_cast<T>("40584512015702371433911456606050"), | |
boost::lexical_cast<T>("402834190897282802772754873905"), | |
boost::lexical_cast<T>("3589522158493606918146495750"), | |
boost::lexical_cast<T>("28530557707503483723634725"), | |
boost::lexical_cast<T>("200714561335055753000730"), | |
boost::lexical_cast<T>("1237953783437761888641"), | |
boost::lexical_cast<T>("6614698701445762950"), | |
boost::lexical_cast<T>("30155495647727505"), | |
boost::lexical_cast<T>("114953256021450"), | |
boost::lexical_cast<T>("356398020013"), | |
boost::lexical_cast<T>("863113950"), | |
boost::lexical_cast<T>("1531345"), | |
boost::lexical_cast<T>("1770"), | |
boost::lexical_cast<T>("1") | |
}; | |
static const T PD[60] = { | |
boost::lexical_cast<T>("0.6365271516829242456324234577164675383137e81"), | |
2*boost::lexical_cast<T>("0.2991038873096202943405966144203628966976e81"), | |
3*boost::lexical_cast<T>("0.9211116495503170498076013367421231351115e80"), | |
4*boost::lexical_cast<T>("0.2090792764676090716286400360584443891749e80"), | |
5*boost::lexical_cast<T>("0.3730037777359591428226035156377978092809e79"), | |
6*boost::lexical_cast<T>("0.5446396536956682043376492370432031543834e78"), | |
7*boost::lexical_cast<T>("0.6692523966335177847425047827449069256345e77"), | |
8*boost::lexical_cast<T>("0.7062543624100864681625612653756619116848e76"), | |
9*boost::lexical_cast<T>("0.6499914905966283735005256964443226879158e75"), | |
10*boost::lexical_cast<T>("0.5280364564853225211197557708655426736091e74"), | |
11*boost::lexical_cast<T>("0.3823205608981176913075543599005095206953e73"), | |
12*boost::lexical_cast<T>("0.2486733714214237704739129972671154532415e72"), | |
13*boost::lexical_cast<T>("0.1462562139602039577983434547171318011675e71"), | |
14*boost::lexical_cast<T>("0.7821169065036815012381267259559910324285e69"), | |
15*boost::lexical_cast<T>("0.3820552182348155468636157988764435365078e68"), | |
16*boost::lexical_cast<T>("0.1711618296983598244658239925535632505062e67"), | |
17*boost::lexical_cast<T>("0.7056661618357643731419080738521475204245e65"), | |
18*boost::lexical_cast<T>("0.2685246896473614017356264531791459936036e64"), | |
19*boost::lexical_cast<T>("0.9455168125599643085283071944864977592391e62"), | |
20*boost::lexical_cast<T>("0.3087541626972538362237309145177486236219e61"), | |
21*boost::lexical_cast<T>("0.9367928873352980208052601301625005737407e59"), | |
22*boost::lexical_cast<T>("0.2645306130689794942883818547314327466007e58"), | |
23*boost::lexical_cast<T>("0.6961815141171454309161007351079576190079e56"), | |
24*boost::lexical_cast<T>("0.1709637824471794552313802669803885946843e55"), | |
25*boost::lexical_cast<T>("0.3921553258481531526663112728778759311158e53"), | |
26*boost::lexical_cast<T>("0.8409006354449988687714450897575728228696e51"), | |
27*boost::lexical_cast<T>("0.1686755204461325935742097669030363344927e50"), | |
28*boost::lexical_cast<T>("0.3166653542877070999007425197729038754254e48"), | |
29*boost::lexical_cast<T>("0.5566029092358215049069560272835654229637e46"), | |
30*boost::lexical_cast<T>("0.9161766287916328133080586672953875116242e44"), | |
31*boost::lexical_cast<T>("1412317772330871298317974693514430627922000"), | |
32*boost::lexical_cast<T>("20387991986727877473732570146112459874790"), | |
33*boost::lexical_cast<T>("275557928713904105182512535678580359839.3"), | |
34*boost::lexical_cast<T>("3485719851040516559072031256589598330.723"), | |
35*boost::lexical_cast<T>("41247046743564028399938106707656877.40859"), | |
36*boost::lexical_cast<T>("456274078125709314602601667471879.0147312"), | |
37*boost::lexical_cast<T>("4714450683242899367025707077155.310613012"), | |
38*boost::lexical_cast<T>("45453933537925041680009544258.75073849996"), | |
39*boost::lexical_cast<T>("408437900487067278846361972.302331241052"), | |
40*boost::lexical_cast<T>("3415719344386166273085838.705771571751035"), | |
41*boost::lexical_cast<T>("26541502879185876562320.93134691487351145"), | |
42*boost::lexical_cast<T>("191261415065918713661.1571433274648417668"), | |
43*boost::lexical_cast<T>("1275349770108718421.645275944284937551702"), | |
44*boost::lexical_cast<T>("7849171120971773.318910987434906905704272"), | |
45*boost::lexical_cast<T>("44455946386549.80866460312682983576538056"), | |
46*boost::lexical_cast<T>("230920362395.3198137186361608905136598046"), | |
47*boost::lexical_cast<T>("1095700096.240863858624279930600654130254"), | |
48*boost::lexical_cast<T>("4727085.467506050153744334085516289728134"), | |
49*boost::lexical_cast<T>("18440.75118859447173303252421991479005424"), | |
50*boost::lexical_cast<T>("64.62515887799460295677071749181651317052"), | |
51*boost::lexical_cast<T>("0.201851568864688406206528472883512147547"), | |
52*boost::lexical_cast<T>("0.0005565091674187978029138500039504078098143"), | |
53*boost::lexical_cast<T>("0.1338097668312907986354698683493366559613e-5"), | |
54*boost::lexical_cast<T>("0.276308225077464312820179030238305271638e-8"), | |
55*boost::lexical_cast<T>("0.4801582970473168520375942100071070575043e-11"), | |
56*boost::lexical_cast<T>("0.6829184144212920949740376186058541800175e-14"), | |
57*boost::lexical_cast<T>("0.7634080076358511276617829524639455399182e-17"), | |
58*boost::lexical_cast<T>("0.6290035083727140966418512608156646142409e-20"), | |
59*boost::lexical_cast<T>("0.339652245667538733044036638506893821352e-23"), | |
60*boost::lexical_cast<T>("0.9017518064256388530773585529891677854909e-27") | |
}; | |
static const T QD[60] = { | |
boost::lexical_cast<T>("0.1386831185456898357379390197203894063459e81"), | |
2*boost::lexical_cast<T>("0.6467076379487574703291056110838151259438e81"), | |
3*boost::lexical_cast<T>("0.1394967823848615838336194279565285465161e82"), | |
4*boost::lexical_cast<T>("0.1872927317344192945218570366455046340458e82"), | |
5*boost::lexical_cast<T>("0.1772461045338946243584650759986310355937e82"), | |
6*boost::lexical_cast<T>("0.1267294892200258648315971144069595555118e82"), | |
7*boost::lexical_cast<T>("0.7157764838362416821508872117623058626589e81"), | |
8*boost::lexical_cast<T>("0.329447266909948668265277828268378274513e81"), | |
9*boost::lexical_cast<T>("0.1264376077317689779509250183194342571207e81"), | |
10*boost::lexical_cast<T>("0.4118230304191980787640446056583623228873e80"), | |
11*boost::lexical_cast<T>("0.1154393529762694616405952270558316515261e80"), | |
12*boost::lexical_cast<T>("0.281655612889423906125295485693696744275e79"), | |
13*boost::lexical_cast<T>("0.6037483524928743102724159846414025482077e78"), | |
14*boost::lexical_cast<T>("0.1145927995397835468123576831800276999614e78"), | |
15*boost::lexical_cast<T>("0.1938624296151985600348534009382865995154e77"), | |
16*boost::lexical_cast<T>("0.293980925856227626211879961219188406675e76"), | |
17*boost::lexical_cast<T>("0.4015574518216966910319562902099567437832e75"), | |
18*boost::lexical_cast<T>("0.4961475457509727343545565970423431880907e74"), | |
19*boost::lexical_cast<T>("0.5565482348278933960215521991000378896338e73"), | |
20*boost::lexical_cast<T>("0.5686112924615820754631098622770303094938e72"), | |
21*boost::lexical_cast<T>("0.5305988545844796293285410303747469932856e71"), | |
22*boost::lexical_cast<T>("0.4533363413802585060568537458067343491358e70"), | |
23*boost::lexical_cast<T>("0.3553932059473516064068322757331575565718e69"), | |
24*boost::lexical_cast<T>("0.2561198565218704414618802902533972354203e68"), | |
25*boost::lexical_cast<T>("0.1699519313292900324098102065697454295572e67"), | |
26*boost::lexical_cast<T>("0.1039830160862334505389615281373574959236e66"), | |
27*boost::lexical_cast<T>("0.5873082967977428281000961954715372504986e64"), | |
28*boost::lexical_cast<T>("0.3065255179030575882202133042549783442446e63"), | |
29*boost::lexical_cast<T>("0.1479494813481364701208655943688307245459e62"), | |
30*boost::lexical_cast<T>("0.6608150467921598615495180659808895663164e60"), | |
31*boost::lexical_cast<T>("0.2732535313770902021791888953487787496976e59"), | |
32*boost::lexical_cast<T>("0.1046402297662493314531194338414508049069e58"), | |
33*boost::lexical_cast<T>("0.3711375077192882936085049147920021549622e56"), | |
34*boost::lexical_cast<T>("0.1219154482883895482637944309702972234576e55"), | |
35*boost::lexical_cast<T>("0.3708359374149458741391374452286837880162e53"), | |
36*boost::lexical_cast<T>("0.1044095509971707189716913168889769471468e52"), | |
37*boost::lexical_cast<T>("0.271951506225063286130946773813524945052e50"), | |
38*boost::lexical_cast<T>("0.6548016291215163843464133978454065823866e48"), | |
39*boost::lexical_cast<T>("0.1456062447610542135403751730809295219344e47"), | |
40*boost::lexical_cast<T>("0.2986690175077969760978388356833006028929e45"), | |
41*boost::lexical_cast<T>("5643149706574013350061247429006443326844000"), | |
42*boost::lexical_cast<T>("98047545414467090421964387960743688053480"), | |
43*boost::lexical_cast<T>("1563378767746846395507385099301468978550"), | |
44*boost::lexical_cast<T>("22823360528584500077862274918382796495"), | |
45*boost::lexical_cast<T>("304215527004115213046601295970388750"), | |
46*boost::lexical_cast<T>("3690289075895685793844344966820325"), | |
47*boost::lexical_cast<T>("40584512015702371433911456606050"), | |
48*boost::lexical_cast<T>("402834190897282802772754873905"), | |
49*boost::lexical_cast<T>("3589522158493606918146495750"), | |
50*boost::lexical_cast<T>("28530557707503483723634725"), | |
51*boost::lexical_cast<T>("200714561335055753000730"), | |
52*boost::lexical_cast<T>("1237953783437761888641"), | |
53*boost::lexical_cast<T>("6614698701445762950"), | |
54*boost::lexical_cast<T>("30155495647727505"), | |
55*boost::lexical_cast<T>("114953256021450"), | |
56*boost::lexical_cast<T>("356398020013"), | |
57*boost::lexical_cast<T>("863113950"), | |
58*boost::lexical_cast<T>("1531345"), | |
59*boost::lexical_cast<T>("1770"), | |
60*boost::lexical_cast<T>("1") | |
}; | |
static const double g = 63.192152; | |
T zgh = x + g - 0.5; | |
T result = (x - 0.5) / zgh; | |
result += log(zgh); | |
result += tools::evaluate_polynomial(PD, x) / tools::evaluate_polynomial(P, x); | |
result -= tools::evaluate_polynomial(QD, x) / tools::evaluate_polynomial(Q, x); | |
result -= 1; | |
return result; | |
} | |
template <class T> | |
T big_digamma(T x) | |
{ | |
BOOST_MATH_STD_USING | |
if(x < 0) | |
{ | |
return big_digamma_helper(static_cast<T>(1-x)) + constants::pi<T>() / tan(constants::pi<T>() * (1-x)); | |
} | |
return big_digamma_helper(x); | |
} | |
}}} | |
#endif // include guard |