| // (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 |