From a663232f38e971dd573bb5161d7f6519366f0d28 Mon Sep 17 00:00:00 2001 From: jzmaddock Date: Sat, 22 Nov 2014 18:37:31 +0000 Subject: [PATCH] [polygamma] Add two special cases for x = 1 and x = 0.5. --- doc/equations/polygamma8.mml | 52 +++++++++++++ doc/equations/polygamma8.png | Bin 0 -> 2767 bytes doc/equations/polygamma8.svg | 2 + doc/equations/polygamma9.mml | 69 ++++++++++++++++++ doc/equations/polygamma9.png | Bin 0 -> 4071 bytes doc/equations/polygamma9.svg | 2 + doc/html/index.html | 2 +- doc/html/indexes/s01.html | 2 +- doc/html/indexes/s02.html | 2 +- doc/html/indexes/s03.html | 2 +- doc/html/indexes/s04.html | 2 +- doc/html/indexes/s05.html | 2 +- doc/html/math_toolkit/conventions.html | 2 +- doc/html/math_toolkit/navigation.html | 2 +- doc/html/math_toolkit/sf_gamma/polygamma.html | 13 +++- doc/sf/polygamma.qbk | 10 ++- .../special_functions/detail/polygamma.hpp | 16 +++- test/test_polygamma.cpp | 2 +- test/test_polygamma.hpp | 6 +- 19 files changed, 173 insertions(+), 15 deletions(-) create mode 100644 doc/equations/polygamma8.mml create mode 100644 doc/equations/polygamma8.png create mode 100644 doc/equations/polygamma8.svg create mode 100644 doc/equations/polygamma9.mml create mode 100644 doc/equations/polygamma9.png create mode 100644 doc/equations/polygamma9.svg diff --git a/doc/equations/polygamma8.mml b/doc/equations/polygamma8.mml new file mode 100644 index 000000000..a76cb5f1c --- /dev/null +++ b/doc/equations/polygamma8.mml @@ -0,0 +1,52 @@ + +]> + +polygamma8 + + + + + + + ψ + + + n + + + + + + 1 + + + = + + + + − + 1 + + + + n + + + 1 + + + n + ! 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Last revised: November 20, 2014 at 10:35:49 GMT

+

Last revised: November 22, 2014 at 18:29:53 GMT

diff --git a/doc/html/indexes/s01.html b/doc/html/indexes/s01.html index 7ed2007d0..2b6714bc2 100644 --- a/doc/html/indexes/s01.html +++ b/doc/html/indexes/s01.html @@ -24,7 +24,7 @@

-Function Index

+Function Index

A B C D E F G H I J K L M N O P Q R S T U V Z

diff --git a/doc/html/indexes/s02.html b/doc/html/indexes/s02.html index 08a77ab5a..706f10955 100644 --- a/doc/html/indexes/s02.html +++ b/doc/html/indexes/s02.html @@ -24,7 +24,7 @@

-Class Index

+Class Index

B C D E F G H I L M N O P Q R S T U W

diff --git a/doc/html/indexes/s03.html b/doc/html/indexes/s03.html index 7bdc0e8cd..4c84fa6f5 100644 --- a/doc/html/indexes/s03.html +++ b/doc/html/indexes/s03.html @@ -24,7 +24,7 @@

-Typedef Index

+Typedef Index

A B C D E F G H I L N O P R S T U V W

diff --git a/doc/html/indexes/s04.html b/doc/html/indexes/s04.html index bc332ea5b..fd80d790d 100644 --- a/doc/html/indexes/s04.html +++ b/doc/html/indexes/s04.html @@ -24,7 +24,7 @@

-Macro Index

+Macro Index

B F

diff --git a/doc/html/indexes/s05.html b/doc/html/indexes/s05.html index 005563e7f..47275e61e 100644 --- a/doc/html/indexes/s05.html +++ b/doc/html/indexes/s05.html @@ -23,7 +23,7 @@

-Index

+Index

A B C D E F G H I J K L M N O P Q R S T U V W Z

diff --git a/doc/html/math_toolkit/conventions.html b/doc/html/math_toolkit/conventions.html index 36b191315..783456eef 100644 --- a/doc/html/math_toolkit/conventions.html +++ b/doc/html/math_toolkit/conventions.html @@ -27,7 +27,7 @@ Document Conventions

- +

This documentation aims to use of the following naming and formatting conventions. diff --git a/doc/html/math_toolkit/navigation.html b/doc/html/math_toolkit/navigation.html index 4b912f43f..307a922db 100644 --- a/doc/html/math_toolkit/navigation.html +++ b/doc/html/math_toolkit/navigation.html @@ -27,7 +27,7 @@ Navigation

- +

Boost.Math documentation is provided in both HTML and PDF formats. diff --git a/doc/html/math_toolkit/sf_gamma/polygamma.html b/doc/html/math_toolkit/sf_gamma/polygamma.html index a5528f5ca..013db9e7b 100644 --- a/doc/html/math_toolkit/sf_gamma/polygamma.html +++ b/doc/html/math_toolkit/sf_gamma/polygamma.html @@ -183,7 +183,9 @@ As shown above, error rates are generally very acceptable for moderately sized arguments. Error rates should stay low for exact inputs, however, please note that the function becomes exceptionally sensitive to small changes in - input for large n and negative x. + input for large n and negative x, indeed for cases where n! + would overflow, the function changes directly from -∞ to +∞ somewhere between + each negative integer - these cases are not handled correctly.

For these reasons results should be treated with extreme @@ -257,6 +259,15 @@

to make x large enough for the asymptotic expansion to be used.

+

+ There are also two special cases: +

+

+ +

+

+ +

diff --git a/doc/sf/polygamma.qbk b/doc/sf/polygamma.qbk index 1f44e02d2..873403216 100644 --- a/doc/sf/polygamma.qbk +++ b/doc/sf/polygamma.qbk @@ -49,7 +49,9 @@ than the one shown will have __zero_error. As shown above, error rates are generally very acceptable for moderately sized arguments. Error rates should stay low for exact inputs, however, please note that the -function becomes exceptionally sensitive to small changes in input for large n and negative x. +function becomes exceptionally sensitive to small changes in input for large n and negative x, +indeed for cases where ['n!] would overflow, the function changes directly from -[infin] to ++[infin] somewhere between each negative integer - ['these cases are not handled correctly]. [*For these reasons results should be treated with extreme caution when /n/ is large and x negative]. @@ -99,6 +101,12 @@ For x in-between the two extremes we use the relation: to make x large enough for the asymptotic expansion to be used. +There are also two special cases: + +[equation polygamma8] + +[equation polygamma9] + [endsect][/section:polygamma The Polygamma Function] [/ diff --git a/include/boost/math/special_functions/detail/polygamma.hpp b/include/boost/math/special_functions/detail/polygamma.hpp index 27003fc3a..f9ce7509b 100644 --- a/include/boost/math/special_functions/detail/polygamma.hpp +++ b/include/boost/math/special_functions/detail/polygamma.hpp @@ -426,8 +426,8 @@ for(int column = 0; column <= max_columns; ++column) { int cos_order = 2 * column + offset; // order of the cosine term in entry "column" - BOOST_ASSERT(column < table[i].size()); - BOOST_ASSERT((cos_order + 1) / 2 < table[i + 1].size()); + BOOST_ASSERT(column < (int)table[i].size()); + BOOST_ASSERT((cos_order + 1) / 2 < (int)table[i + 1].size()); table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1); if(cos_order) table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1); @@ -520,6 +520,18 @@ { return polygamma_atinfinityplus(n, x, pol, function); } + else if(x == 1) + { + return (n & 1 ? 1 : -1) * boost::math::factorial(n, pol) * boost::math::zeta(T(n + 1), pol); + } + else if(x == 0.5f) + { + T result = (n & 1 ? 1 : -1) * boost::math::factorial(n, pol) * boost::math::zeta(T(n + 1), pol); + if(fabs(result) >= ldexp(tools::max_value(), -n - 1)) + return boost::math::sign(result) * policies::raise_overflow_error(function, 0, pol); + result *= ldexp(T(1), n + 1) - 1; + return result; + } else { return polygamma_attransitionplus(n, x, pol, function); diff --git a/test/test_polygamma.cpp b/test/test_polygamma.cpp index c6a6f3950..f87ae95c6 100644 --- a/test/test_polygamma.cpp +++ b/test/test_polygamma.cpp @@ -65,7 +65,7 @@ void expected_results() ".*", // platform largest_type, // test type(s) ".*", // test data group - ".*", 20, 10); // test function + ".*", 30, 10); // test function // // Finish off by printing out the compiler/stdlib/platform names, // we do this to make it easier to mark up expected error rates. diff --git a/test/test_polygamma.hpp b/test/test_polygamma.hpp index 74296093f..e7559bf18 100644 --- a/test/test_polygamma.hpp +++ b/test/test_polygamma.hpp @@ -62,7 +62,7 @@ void test_polygamma(T, const char* name) { typedef typename table_type::type value_type; - boost::array, 384> data = + boost::array, 484> data = { { { 1, SC_(0.1250000), SC_(65.388133444988034473142999334395961) }, { 1, SC_(2.250000), SC_(0.55732915450711073927131911933522402) }, { 1, SC_(4.375000), SC_(0.25666408805722660683906428275458774) }, { 1, SC_(6.500000), SC_(0.16628453574995823763989666631218566) }, { 1, SC_(8.625000), SC_(0.12292237374423990274075995315923687) }, { 1, SC_(10.75000), SC_(0.097483848201852104395946001854344927) }, { 1, SC_(12.87500), SC_(0.080764208092843621858393487209278638) }, { 1, SC_(15.00000), SC_(0.068938227847683806226155216756371670) }, { 1, SC_(17.12500), SC_(0.060132263162293455894576107399989891) }, { 1, SC_(19.25000), SC_(0.053320703915617277211139745531295189) }, { 1, SC_(21.37500), SC_(0.047895038036916716105500109226810942) }, { 1, SC_(23.50000), SC_(0.043471416266946770249685779030294199) }, { 1, SC_(25.62500), SC_(0.039795743807625238080963836217545550) }, { 1, SC_(27.75000), SC_(0.036693131333593477569076090983653779) }, { 1, SC_(29.87500), SC_(0.034039266877179098641898178094001935) }, { 1, SC_(32.00000), SC_(0.031743366520302090126581680438741427) }, { 1, SC_(34.12500), SC_(0.029737585673522726661363528635488348) }, { 1, SC_(36.25000), SC_(0.027970204614894933106878169214392067) }, { 1, SC_(38.37500), SC_(0.026401106865951764123858232364900665) }, { 1, SC_(40.50000), SC_(0.024998698201356741322280011143883922) }, { 1, SC_(42.62500), SC_(0.023737757818075642720991864115881164) }, { 1, SC_(44.75000), SC_(0.022597908441287364284087900916682571) }, { 1, SC_(46.87500), SC_(0.021562506914486557632388530071308920) }, { 1, SC_(49.00000), SC_(0.020617826354560516060031453062401102) }, { 1, SC_(51.12500), SC_(0.019752444228552790805040230288386135) }, { 1, SC_(53.25000), SC_(0.018956778300513446216011889734099110) }, { 1, SC_(55.37500), SC_(0.018222730375562878627773770314126276) }, { 1, SC_(57.50000), SC_(0.017543409716574620734228673575882677) }, { 1, SC_(59.62500), SC_(0.016912916093398919581541485278641593) }, { 1, SC_(61.75000), SC_(0.016326167985389235938281994221076159) }, { 1, SC_(63.87500), SC_(0.015778765341125640054231784371915358) }, { 1, SC_(66.00000), SC_(0.015266879048806385777045778219279459) }, { 1, SC_(68.12500), SC_(0.014787161242916062152535888615850260) }, { 1, SC_(70.25000), SC_(0.014336672004276912093324664070489886) }, { 1, SC_(72.37500), SC_(0.013912819061256593888508241399678441) }, { 1, SC_(74.50000), SC_(0.013513307879079644573772830117155790) }, { 1, SC_(76.62500), SC_(0.013136100107643257539226305043251166) }, { 1, SC_(78.75000), SC_(0.012779378799112389298746113446648982) }, { 1, SC_(80.87500), SC_(0.012441519142554280549428650598112729) }, { 1, SC_(83.00000), SC_(0.012121063720980953787190412456820037) }, { 1, SC_(85.12500), SC_(0.011816701495952671887865236708275449) }, { 1, SC_(87.25000), SC_(0.011527249880640584213306489206202069) }, { 1, SC_(89.37500), SC_(0.011251639384481213426240562514542477) }, { 1, SC_(91.50000), SC_(0.010988900409103388104670465071922664) }, { 1, SC_(93.62500), SC_(0.010738151851930343087942933225290955) }, { 1, SC_(95.75000), SC_(0.010498591235178571927709342934117247) }, { 1, SC_(97.87500), SC_(0.010269486127251686167359605922136260) }, { 1, SC_(100.0000), SC_(0.010050166663333571395245668465701423) }, { 2, SC_(0.1250000), -SC_(1025.7533381181356825956689300565174) }, { 2, SC_(2.250000), -SC_(0.30373993753692033333796717884398989) }, { 2, SC_(4.375000), -SC_(0.065528725397877855792664680804766330) }, { 2, SC_(6.500000), -SC_(0.027587910706876798794117450572831562) }, { 2, SC_(8.625000), -SC_(0.015091062676061564388822078971884395) }, { 2, SC_(10.75000), -SC_(0.0094956196449265900776488965631791775) }, { 2, SC_(12.87500), -SC_(0.0065193261909178260169885198194705291) }, { 2, SC_(15.00000), -SC_(0.0047506027165515547467791223768191188) }, { 2, SC_(17.12500), -SC_(0.0036148020016626802195565448384834283) }, { 2, SC_(19.25000), -SC_(0.0028424250740909855631736845850335535) }, { 2, SC_(21.37500), -SC_(0.0022934967923297751145806065882712900) }, { 2, SC_(23.50000), -SC_(0.0018894667868625909895476094900477678) }, { 2, SC_(25.62500), -SC_(0.0015834924252652953218803111387922996) }, { 2, SC_(27.75000), -SC_(0.0013462349527320363170378913859580860) }, { 2, SC_(29.87500), -SC_(0.0011585598948326545653381467670779893) }, { 2, SC_(32.00000), -SC_(0.0010075567602140907392185110593117265) }, { 2, SC_(34.12500), -SC_(0.00088425886906787461365402045268994574) }, { 2, SC_(36.25000), -SC_(0.00078228136778540401396894643668418462) }, { 2, SC_(38.37500), -SC_(0.00069697797537723210697105880606724159) }, { 2, SC_(40.50000), -SC_(0.00062490237932923289658683588812998732) }, { 2, SC_(42.62500), -SC_(0.00056345469641484389456121935536309197) }, { 2, SC_(44.75000), -SC_(0.00051064374134295093656385520125286421) }, { 2, SC_(46.87500), -SC_(0.00046492369550612086723038302919171885) }, { 2, SC_(49.00000), -SC_(0.00042507970884222510504308867471824948) }, { 2, SC_(51.12500), -SC_(0.00039014637079445100439645401142194535) }, { 2, SC_(53.25000), -SC_(0.00035934868438211062777790080207505881) }, { 2, SC_(55.37500), -SC_(0.00033205871518117505559928737076624192) }, { 2, SC_(57.50000), -SC_(0.00030776333242771756953580263456220628) }, { 2, SC_(59.62500), -SC_(0.00028603991345613245802207114123372414) }, { 2, SC_(61.75000), -SC_(0.00026653784162151616772904552992231739) }, { 2, SC_(63.87500), -SC_(0.00024896427102287300629668349085350311) }, { 2, SC_(66.00000), -SC_(0.00023307306946180321476975587412226332) }, { 2, SC_(68.12500), -SC_(0.00021865615382096049855997233359992101) }, { 2, SC_(70.25000), -SC_(0.00020553664405312492990454318541834320) }, { 2, SC_(72.37500), -SC_(0.00019356341228034103704385457246984425) }, { 2, SC_(74.50000), -SC_(0.00018260671130395075946075978013398780) }, { 2, SC_(76.62500), -SC_(0.00017255464497899193114313963889049197) }, { 2, SC_(78.75000), -SC_(0.00016331030013937037094502788633900241) }, { 2, SC_(80.87500), -SC_(0.00015478940207215993505449988937898640) }, { 2, SC_(83.00000), -SC_(0.00014691838710034332570083901051760808) }, { 2, SC_(85.12500), -SC_(0.00013963280957351165370765015817038278) }, { 2, SC_(87.25000), -SC_(0.00013287601856558888563037791437997733) }, { 2, SC_(89.37500), -SC_(0.00012659805332837531702241737466058293) }, { 2, SC_(91.50000), -SC_(0.00012075471712784846612240303590084686) }, { 2, SC_(93.62500), -SC_(0.00011530679728326736002584104318350829) }, { 2, SC_(95.75000), -SC_(0.00011021940561565095981861374083495299) }, { 2, SC_(97.87500), -SC_(0.00010546141852085692313553484671110170) }, { 2, SC_(100.0000), -SC_(0.00010100499983334999700083300446059382) }, @@ -119,7 +119,9 @@ void test_polygamma(T, const char* name) { 9, SC_(0.1250000), SC_(3.8963943320506514766700086867372762e14) }, { 9, SC_(2.250000), SC_(112.10537259293726188704511169205760) }, { 9, SC_(4.375000), SC_(0.16363919906361224458550935569570297) }, { 9, SC_(6.500000), SC_(0.0036236228486465262827554601212241334) }, { 9, SC_(8.625000), SC_(0.00024724827512609608284324030730925865) }, { 9, SC_(10.75000), SC_(0.000031173409615380334086095590039151370) }, { 9, SC_(12.87500), SC_(5.7824557132102074668570643058203530e-6) }, { 9, SC_(15.00000), SC_(1.3980855499116564037825647833211369e-6) }, { 9, SC_(17.12500), SC_(4.1002910484982160273499509487840966e-7) }, { 9, SC_(19.25000), SC_(1.3928434419796988121776758599785632e-7) }, { 9, SC_(21.37500), SC_(5.3108201638858702713365922453188085e-8) }, { 9, SC_(23.50000), SC_(2.2228120959414339842295401083512930e-8) }, { 9, SC_(25.62500), SC_(1.0046019155325212007574589173916857e-8) }, { 9, SC_(27.75000), SC_(4.8421087709589697640039499715706350e-9) }, { 9, SC_(29.87500), SC_(2.4651114661905897591489828702449678e-9) }, { 9, SC_(32.00000), SC_(1.3154897461853304542190391130140025e-9) }, { 9, SC_(34.12500), SC_(7.3134444425276046658844002904667737e-10) }, { 9, SC_(36.25000), SC_(4.2146721224691157996140805304942419e-10) }, { 9, SC_(38.37500), SC_(2.5073387292785072270991813660631786e-10) }, { 9, SC_(40.50000), SC_(1.5344896969374692327054949367251260e-10) }, { 9, SC_(42.62500), SC_(9.6326225603670754883495513330468241e-11) }, { 9, SC_(44.75000), SC_(6.1868579168628392331266656512721531e-11) }, { 9, SC_(46.87500), SC_(4.0570340164556015067828668493424902e-11) }, { 9, SC_(49.00000), SC_(2.7111471879948743580114339016340155e-11) }, { 9, SC_(51.12500), SC_(1.8433178567812423799249519881074934e-11) }, { 9, SC_(53.25000), SC_(1.2733080158190412770374826663936684e-11) }, { 9, SC_(55.37500), SC_(8.9250208887659481803261595053553675e-12) }, { 9, SC_(57.50000), SC_(6.3408264036510339424452353875188095e-12) }, { 9, SC_(59.62500), SC_(4.5615718874164287974321379689375635e-12) }, { 9, SC_(61.75000), SC_(3.3199519699896432328216923633601725e-12) }, { 9, SC_(63.87500), SC_(2.4426063512782352683767956278168985e-12) }, { 9, SC_(66.00000), SC_(1.8153877149006577317987652651306379e-12) }, { 9, SC_(68.12500), SC_(1.3620688165735114474674715908706633e-12) }, { 9, SC_(70.25000), SC_(1.0310696339614726306516051424066462e-12) }, { 9, SC_(72.37500), SC_(7.8705425492600705644491688042239872e-13) }, { 9, SC_(74.50000), SC_(6.0553290145315817804599098427153448e-13) }, { 9, SC_(76.62500), SC_(4.6934676954218461465393390444338856e-13) }, { 9, SC_(78.75000), SC_(3.6634942521175284527379986400237305e-13) }, { 9, SC_(80.87500), SC_(2.8785887689513567008615786864512179e-13) }, { 9, SC_(83.00000), SC_(2.2761232508374172373558791779207099e-13) }, { 9, SC_(85.12500), SC_(1.8105271275784390697733304142062576e-13) }, { 9, SC_(87.25000), SC_(1.4483676610519321733096991924616972e-13) }, { 9, SC_(89.37500), SC_(1.1649247346426341778218700567243322e-13) }, { 9, SC_(91.50000), SC_(9.4178413859615655080323473554582586e-14) }, { 9, SC_(93.62500), SC_(7.6513170761508530170189422972149840e-14) }, { 9, SC_(95.75000), SC_(6.2453415533716351149737976646666822e-14) }, { 9, SC_(97.87500), SC_(5.1206083919072129278059542340974138e-14) }, { 9, SC_(100.0000), SC_(4.2164633350081151607323910418414347e-14) }, { SC_(12.0), SC_(0.1250000), SC_(-2.6333391446175784623707514121843937e20) }, { SC_(12.0), SC_(2.250000), SC_(-12755.934552347367694976392995238872) }, { SC_(12.0), SC_(4.375000), SC_(-2.3995726885358590731215736760290659) }, { SC_(12.0), SC_(6.500000), SC_(-0.015498964669830504389631890538267195) }, { SC_(12.0), SC_(8.625000), SC_(-0.00043875188128348404126256495384460962) }, { SC_(12.0), SC_(10.75000), SC_(-0.000027944407762544917990491373502083128) }, { SC_(12.0), SC_(12.87500), SC_(-2.9683309167523389772660608703858297e-6) }, { SC_(12.0), SC_(15.00000), SC_(-4.4822030612790888239848346241379762e-7) }, { SC_(12.0), SC_(17.12500), SC_(-8.7479493941258027953346726543995677e-8) }, { SC_(12.0), SC_(19.25000), SC_(-2.0757514198487629655709801633624060e-8) }, { SC_(12.0), SC_(21.37500), SC_(-5.7438353740388980670397346200747331e-9) }, { SC_(12.0), SC_(23.50000), SC_(-1.7993432749405246733914335090978470e-9) }, { SC_(12.0), SC_(25.62500), SC_(-6.2435444143873592979499739047338207e-10) }, { SC_(12.0), SC_(27.75000), SC_(-2.3603187097640550778749196581740161e-10) }, { SC_(12.0), SC_(29.87500), SC_(-9.5975541975412793814120279791068637e-11) }, { SC_(12.0), SC_(32.00000), SC_(-4.1552035282000578511439617844316334e-11) }, { SC_(12.0), SC_(34.12500), SC_(-1.8998432787826251814581286958206123e-11) }, { SC_(12.0), SC_(36.25000), SC_(-9.1125382347537588010833133716666482e-12) }, { SC_(12.0), SC_(38.37500), SC_(-4.5599501480319696078145812150042737e-12) }, { SC_(12.0), SC_(40.50000), SC_(-2.3695979124099134723655912747303232e-12) }, { SC_(12.0), SC_(42.62500), SC_(-1.2737626539048915719376237689570656e-12) }, { SC_(12.0), SC_(44.75000), SC_(-7.0592220046106837677774762353832582e-13) }, { SC_(12.0), SC_(46.87500), SC_(-4.0219605264396476495155503895759108e-13) }, { SC_(12.0), SC_(49.00000), SC_(-2.3499370395545407582624144041709511e-13) }, { SC_(12.0), SC_(51.12500), SC_(-1.4049959061173089543038862427082956e-13) }, { SC_(12.0), SC_(53.25000), SC_(-8.5797116983505159074486972598899288e-14) }, { SC_(12.0), SC_(55.37500), SC_(-5.3422568073471069670219359482696794e-14) }, { SC_(12.0), SC_(57.50000), SC_(-3.3867985175935679891876577636637264e-14) }, { SC_(12.0), SC_(59.62500), SC_(-2.1832074877153446073077836903455745e-14) }, { SC_(12.0), SC_(61.75000), SC_(-1.4293240909063248751348970309438269e-14) }, { SC_(12.0), SC_(63.87500), SC_(-9.4937480883301538244379833116307891e-15) }, { SC_(12.0), SC_(66.00000), SC_(-6.3914967852421984438071853365851405e-15) }, { SC_(12.0), SC_(68.12500), SC_(-4.3576413857619763879519099830883388e-15) }, { SC_(12.0), SC_(70.25000), SC_(-3.0063897012087057369199810835484853e-15) }, { SC_(12.0), SC_(72.37500), SC_(-2.0973711510201259912443465435969711e-15) }, { SC_(12.0), SC_(74.50000), SC_(-1.4786311290892013378232248983192765e-15) }, { SC_(12.0), SC_(76.62500), SC_(-1.0527885762581410791135439292370286e-15) }, { SC_(12.0), SC_(78.75000), SC_(-7.5662859409010038912157660753875556e-16) }, { SC_(12.0), SC_(80.87500), SC_(-5.4861427907366815368388985862444385e-16) }, { SC_(12.0), SC_(83.00000), SC_(-4.0113795601837316346341436637916290e-16) }, { SC_(12.0), SC_(85.12500), SC_(-2.9564974628089487210535317533851159e-16) }, { SC_(12.0), SC_(87.25000), SC_(-2.1955682892705039626630737602169576e-16) }, { SC_(12.0), SC_(89.37500), SC_(-1.6422644673043790572542870203840550e-16) }, { SC_(12.0), SC_(91.50000), SC_(-1.2368534305097629421391380808602245e-16) }, { SC_(12.0), SC_(93.62500), SC_(-9.3763732289494543868014071205929613e-17) }, { SC_(12.0), SC_(95.75000), SC_(-7.1526151114554258090139227356676697e-17) }, { SC_(12.0), SC_(97.87500), SC_(-5.4889394969371742548386275256109609e-17) }, { SC_(12.0), SC_(100.0000), SC_(-4.2363681689608104413899863907775333e-17) }, { SC_(21.0), SC_(0.1250000), SC_(3.7698461389048740847200205590867822e39) }, { SC_(21.0), SC_(2.250000), SC_(9.1298158507949915159597719407312508e11) }, { SC_(21.0), SC_(4.375000), SC_(408886.47063811418290988174462904689) }, { SC_(21.0), SC_(6.500000), SC_(69.783498668647519157069128850360732) }, { SC_(21.0), SC_(8.625000), SC_(0.14574011463957290617412866498392900) }, { SC_(21.0), SC_(10.75000), SC_(0.0012181648740805635977027563578320036) }, { SC_(21.0), SC_(12.87500), 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SC_(2.5862324151116909945729536986576770e59) }, { SC_(48.0), SC_(1.0), SC_(-1.2413915592536094722406207462376651e61) }, { SC_(49.0), SC_(1.0), SC_(6.0828186403426810113507774083359074e62) }, { SC_(50.0), SC_(1.0), SC_(-3.0414093201713391550183240542952737e64) }, + { SC_(1.0), SC_(0.5), SC_(4.9348022005446793094172454999380756) }, { SC_(2.0), SC_(0.5), SC_(-16.828796644234319995596334261160300) }, { SC_(3.0), SC_(0.5), SC_(97.409091034002437236440332688705111) }, { SC_(4.0), SC_(0.5), SC_(-771.47424982666722519053592192403342) }, { SC_(5.0), SC_(0.5), SC_(7691.1135486024354962417555492193592) }, { SC_(6.0), SC_(0.5), SC_(-92203.457923803023286231087958265416) }, { SC_(7.0), SC_(0.5), SC_(1.2904402181855980649694862685313201e6) }, { SC_(8.0), SC_(0.5), SC_(-2.0644899961760041426402517887935387e7) }, { SC_(9.0), SC_(0.5), SC_(3.7159545238509742722370782665375996e8) }, { SC_(10.0), SC_(0.5), SC_(-7.4318245088587689754917967712772148e9) }, { SC_(11.0), SC_(0.5), SC_(1.6349952113475880004602936491973290e11) }, { SC_(12.0), SC_(0.5), SC_(-3.9239835716776094268285475780118302e12) }, { SC_(13.0), SC_(0.5), SC_(1.0202353013465195041277151739878551e14) }, { SC_(14.0), SC_(0.5), SC_(-2.8566584452212481470833807159097089e15) }, { SC_(15.0), SC_(0.5), SC_(8.5699749372666517252032697437036695e16) }, { SC_(16.0), SC_(0.5), SC_(-2.7423919374393211160431177426964643e18) }, { SC_(17.0), SC_(0.5), SC_(9.3241325391494482576994960512370215e19) }, { SC_(18.0), SC_(0.5), SC_(-3.3566877083169638444274914449348672e21) }, { SC_(19.0), SC_(0.5), SC_(1.2755413284287493036311595679555294e23) }, { SC_(20.0), SC_(0.5), SC_(-5.1021653127394298787426098736355399e24) }, { SC_(21.0), SC_(0.5), SC_(2.1429094312139835232862558079810850e26) }, { SC_(22.0), SC_(0.5), SC_(-9.4288014971412166432678344430683531e27) }, { SC_(23.0), SC_(0.5), SC_(4.3372486886542455183876952472394031e29) }, { SC_(24.0), SC_(0.5), SC_(-2.0818793705491236054644197662022342e31) }, { SC_(25.0), SC_(0.5), SC_(1.0409396852737427640356547073711139e33) }, { SC_(26.0), SC_(0.5), SC_(-5.4128863634220427078493834793146517e34) }, { SC_(27.0), SC_(0.5), SC_(2.9229586362476475227234677536391722e36) }, { SC_(28.0), SC_(0.5), SC_(-1.6368568362986349120390762957016873e38) }, { SC_(29.0), SC_(0.5), SC_(9.4937696505319902685274359255316741e39) }, { SC_(30.0), SC_(0.5), SC_(-5.6962617903191757168598825608931775e41) }, { SC_(31.0), SC_(0.5), SC_(3.5316823099978851326405053801048261e43) }, { SC_(32.0), SC_(0.5), SC_(-2.2602766783986456717032825683189511e45) }, { SC_(33.0), SC_(0.5), SC_(1.4917826077431059644231123371238779e47) }, { SC_(34.0), SC_(0.5), SC_(-1.0144121732653120152568117095915465e49) }, { SC_(35.0), SC_(0.5), SC_(7.1008852128571840121789056825569522e50) }, { SC_(36.0), SC_(0.5), SC_(-5.1126373532571724660603059705232204e52) }, { SC_(37.0), SC_(0.5), SC_(3.7833516414103076192831949360932899e54) }, { SC_(38.0), SC_(0.5), SC_(-2.8753472474718337892361988468491097e56) }, { SC_(39.0), SC_(0.5), SC_(2.2427708530280303552352874820086301e58) }, { SC_(40.0), SC_(0.5), SC_(-1.7942166824224242840898439541031383e60) }, { SC_(41.0), SC_(0.5), SC_(1.4712576795863879129267798604374659e62) }, { SC_(42.0), SC_(0.5), SC_(-1.2358564508525658468509652718307801e64) }, { SC_(43.0), SC_(0.5), SC_(1.0628365477332066282896715879731180e66) }, { SC_(44.0), SC_(0.5), SC_(-9.3529616200522183289427782398136808e67) }, { SC_(45.0), SC_(0.5), SC_(8.4176654580469964960466008955275434e69) }, { SC_(46.0), SC_(0.5), SC_(-7.7442522214032367763622903043252397e71) }, { SC_(47.0), SC_(0.5), SC_(7.2795970881190425697803703632702328e73) }, { SC_(48.0), SC_(0.5), SC_(-6.9884132045942808669890971414448669e75) }, { SC_(49.0), SC_(0.5), SC_(6.8486449405023952496492961188997479e77) }, { SC_(50.0), SC_(0.5), SC_(-6.8486449405023952496492897589943408e79) }, }}; do_test_polygamma(data, name, "Mathematica Data");