gonum.org/v1/gonum@v0.14.0/mathext/internal/cephes/igam.go

// Derived from SciPy's special/cephes/igam.c and special/cephes/igam.h
// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c
// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.h
// Made freely available by Stephen L. Moshier without support or guarantee.

// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Copyright ©1985, ©1987 by Stephen L. Moshier
// Portions Copyright ©2016 The Gonum Authors. All rights reserved.

package cephes

import "math"

const (
	igamDimK       = 25
	igamDimN       = 25
	igam           = 1
	igamC          = 0
	igamSmall      = 20
	igamLarge      = 200
	igamSmallRatio = 0.3
	igamLargeRatio = 4.5
)

var igamCoefs = [igamDimK][igamDimN]float64{
	{-3.3333333333333333e-1, 8.3333333333333333e-2, -1.4814814814814815e-2, 1.1574074074074074e-3, 3.527336860670194e-4, -1.7875514403292181e-4, 3.9192631785224378e-5, -2.1854485106799922e-6, -1.85406221071516e-6, 8.296711340953086e-7, -1.7665952736826079e-7, 6.7078535434014986e-9, 1.0261809784240308e-8, -4.3820360184533532e-9, 9.1476995822367902e-10, -2.551419399494625e-11, -5.8307721325504251e-11, 2.4361948020667416e-11, -5.0276692801141756e-12, 1.1004392031956135e-13, 3.3717632624009854e-13, -1.3923887224181621e-13, 2.8534893807047443e-14, -5.1391118342425726e-16, -1.9752288294349443e-15},
	{-1.8518518518518519e-3, -3.4722222222222222e-3, 2.6455026455026455e-3, -9.9022633744855967e-4, 2.0576131687242798e-4, -4.0187757201646091e-7, -1.8098550334489978e-5, 7.6491609160811101e-6, -1.6120900894563446e-6, 4.6471278028074343e-9, 1.378633446915721e-7, -5.752545603517705e-8, 1.1951628599778147e-8, -1.7543241719747648e-11, -1.0091543710600413e-9, 4.1627929918425826e-10, -8.5639070264929806e-11, 6.0672151016047586e-14, 7.1624989648114854e-12, -2.9331866437714371e-12, 5.9966963656836887e-13, -2.1671786527323314e-16, -4.9783399723692616e-14, 2.0291628823713425e-14, -4.13125571381061e-15},
	{4.1335978835978836e-3, -2.6813271604938272e-3, 7.7160493827160494e-4, 2.0093878600823045e-6, -1.0736653226365161e-4, 5.2923448829120125e-5, -1.2760635188618728e-5, 3.4235787340961381e-8, 1.3721957309062933e-6, -6.298992138380055e-7, 1.4280614206064242e-7, -2.0477098421990866e-10, -1.4092529910867521e-8, 6.228974084922022e-9, -1.3670488396617113e-9, 9.4283561590146782e-13, 1.2872252400089318e-10, -5.5645956134363321e-11, 1.1975935546366981e-11, -4.1689782251838635e-15, -1.0940640427884594e-12, 4.6622399463901357e-13, -9.905105763906906e-14, 1.8931876768373515e-17, 8.8592218725911273e-15},
	{6.4943415637860082e-4, 2.2947209362139918e-4, -4.6918949439525571e-4, 2.6772063206283885e-4, -7.5618016718839764e-5, -2.3965051138672967e-7, 1.1082654115347302e-5, -5.6749528269915966e-6, 1.4230900732435884e-6, -2.7861080291528142e-11, -1.6958404091930277e-7, 8.0994649053880824e-8, -1.9111168485973654e-8, 2.3928620439808118e-12, 2.0620131815488798e-9, -9.4604966618551322e-10, 2.1541049775774908e-10, -1.388823336813903e-14, -2.1894761681963939e-11, 9.7909989511716851e-12, -2.1782191880180962e-12, 6.2088195734079014e-17, 2.126978363279737e-13, -9.3446887915174333e-14, 2.0453671226782849e-14},
	{-8.618882909167117e-4, 7.8403922172006663e-4, -2.9907248030319018e-4, -1.4638452578843418e-6, 6.6414982154651222e-5, -3.9683650471794347e-5, 1.1375726970678419e-5, 2.5074972262375328e-10, -1.6954149536558306e-6, 8.9075075322053097e-7, -2.2929348340008049e-7, 2.956794137544049e-11, 2.8865829742708784e-8, -1.4189739437803219e-8, 3.4463580499464897e-9, -2.3024517174528067e-13, -3.9409233028046405e-10, 1.8602338968504502e-10, -4.356323005056618e-11, 1.2786001016296231e-15, 4.6792750266579195e-12, -2.1492464706134829e-12, 4.9088156148096522e-13, -6.3385914848915603e-18, -5.0453320690800944e-14},
	{-3.3679855336635815e-4, -6.9728137583658578e-5, 2.7727532449593921e-4, -1.9932570516188848e-4, 6.7977804779372078e-5, 1.419062920643967e-7, -1.3594048189768693e-5, 8.0184702563342015e-6, -2.2914811765080952e-6, -3.252473551298454e-10, 3.4652846491085265e-7, -1.8447187191171343e-7, 4.8240967037894181e-8, -1.7989466721743515e-14, -6.3061945000135234e-9, 3.1624176287745679e-9, -7.8409242536974293e-10, 5.1926791652540407e-15, 9.3589442423067836e-11, -4.5134262161632782e-11, 1.0799129993116827e-11, -3.661886712685252e-17, -1.210902069055155e-12, 5.6807435849905643e-13, -1.3249659916340829e-13},
	{5.3130793646399222e-4, -5.9216643735369388e-4, 2.7087820967180448e-4, 7.9023532326603279e-7, -8.1539693675619688e-5, 5.6116827531062497e-5, -1.8329116582843376e-5, -3.0796134506033048e-9, 3.4651553688036091e-6, -2.0291327396058604e-6, 5.7887928631490037e-7, 2.338630673826657e-13, -8.8286007463304835e-8, 4.7435958880408128e-8, -1.2545415020710382e-8, 8.6496488580102925e-14, 1.6846058979264063e-9, -8.5754928235775947e-10, 2.1598224929232125e-10, -7.6132305204761539e-16, -2.6639822008536144e-11, 1.3065700536611057e-11, -3.1799163902367977e-12, 4.7109761213674315e-18, 3.6902800842763467e-13},
	{3.4436760689237767e-4, 5.1717909082605922e-5, -3.3493161081142236e-4, 2.812695154763237e-4, -1.0976582244684731e-4, -1.2741009095484485e-7, 2.7744451511563644e-5, -1.8263488805711333e-5, 5.7876949497350524e-6, 4.9387589339362704e-10, -1.0595367014026043e-6, 6.1667143761104075e-7, -1.7562973359060462e-7, -1.2974473287015439e-12, 2.695423606288966e-8, -1.4578352908731271e-8, 3.887645959386175e-9, -3.8810022510194121e-17, -5.3279941738772867e-10, 2.7437977643314845e-10, -6.9957960920705679e-11, 2.5899863874868481e-17, 8.8566890996696381e-12, -4.403168815871311e-12, 1.0865561947091654e-12},
	{-6.5262391859530942e-4, 8.3949872067208728e-4, -4.3829709854172101e-4, -6.969091458420552e-7, 1.6644846642067548e-4, -1.2783517679769219e-4, 4.6299532636913043e-5, 4.5579098679227077e-9, -1.0595271125805195e-5, 6.7833429048651666e-6, -2.1075476666258804e-6, -1.7213731432817145e-11, 3.7735877416110979e-7, -2.1867506700122867e-7, 6.2202288040189269e-8, 6.5977038267330006e-16, -9.5903864974256858e-9, 5.2132144922808078e-9, -1.3991589583935709e-9, 5.382058999060575e-16, 1.9484714275467745e-10, -1.0127287556389682e-10, 2.6077347197254926e-11, -5.0904186999932993e-18, -3.3721464474854592e-12},
	{-5.9676129019274625e-4, -7.2048954160200106e-5, 6.7823088376673284e-4, -6.4014752602627585e-4, 2.7750107634328704e-4, 1.8197008380465151e-7, -8.4795071170685032e-5, 6.105192082501531e-5, -2.1073920183404862e-5, -8.8585890141255994e-10, 4.5284535953805377e-6, -2.8427815022504408e-6, 8.7082341778646412e-7, 3.6886101871706965e-12, -1.5344695190702061e-7, 8.862466778790695e-8, -2.5184812301826817e-8, -1.0225912098215092e-14, 3.8969470758154777e-9, -2.1267304792235635e-9, 5.7370135528051385e-10, -1.887749850169741e-19, -8.0931538694657866e-11, 4.2382723283449199e-11, -1.1002224534207726e-11},
	{1.3324454494800656e-3, -1.9144384985654775e-3, 1.1089369134596637e-3, 9.932404122642299e-7, -5.0874501293093199e-4, 4.2735056665392884e-4, -1.6858853767910799e-4, -8.1301893922784998e-9, 4.5284402370562147e-5, -3.127053674781734e-5, 1.044986828530338e-5, 4.8435226265680926e-11, -2.1482565873456258e-6, 1.329369701097492e-6, -4.0295693092101029e-7, -1.7567877666323291e-13, 7.0145043163668257e-8, -4.040787734999483e-8, 1.1474026743371963e-8, 3.9642746853563325e-18, -1.7804938269892714e-9, 9.7480262548731646e-10, -2.6405338676507616e-10, 5.794875163403742e-18, 3.7647749553543836e-11},
	{1.579727660730835e-3, 1.6251626278391582e-4, -2.0633421035543276e-3, 2.1389686185689098e-3, -1.0108559391263003e-3, -3.9912705529919201e-7, 3.6235025084764691e-4, -2.8143901463712154e-4, 1.0449513336495887e-4, 2.1211418491830297e-9, -2.5779417251947842e-5, 1.7281818956040463e-5, -5.6413773872904282e-6, -1.1024320105776174e-11, 1.1223224418895175e-6, -6.8693396379526735e-7, 2.0653236975414887e-7, 4.6714772409838506e-14, -3.5609886164949055e-8, 2.0470855345905963e-8, -5.8091738633283358e-9, -1.332821287582869e-16, 9.0354604391335133e-10, -4.9598782517330834e-10, 1.3481607129399749e-10},
	{-4.0725121195140166e-3, 6.4033628338080698e-3, -4.0410161081676618e-3, -2.183732802866233e-6, 2.1740441801254639e-3, -1.9700440518418892e-3, 8.3595469747962458e-4, 1.9445447567109655e-8, -2.5779387120421696e-4, 1.9009987368139304e-4, -6.7696499937438965e-5, -1.4440629666426572e-10, 1.5712512518742269e-5, -1.0304008744776893e-5, 3.304517767401387e-6, 7.9829760242325709e-13, -6.4097794149313004e-7, 3.8894624761300056e-7, -1.1618347644948869e-7, -2.816808630596451e-15, 1.9878012911297093e-8, -1.1407719956357511e-8, 3.2355857064185555e-9, 4.1759468293455945e-20, -5.0423112718105824e-10},
	{-5.9475779383993003e-3, -5.4016476789260452e-4, 8.7910413550767898e-3, -9.8576315587856125e-3, 5.0134695031021538e-3, 1.2807521786221875e-6, -2.0626019342754683e-3, 1.7109128573523058e-3, -6.7695312714133799e-4, -6.9011545676562133e-9, 1.8855128143995902e-4, -1.3395215663491969e-4, 4.6263183033528039e-5, 4.0034230613321351e-11, -1.0255652921494033e-5, 6.612086372797651e-6, -2.0913022027253008e-6, -2.0951775649603837e-13, 3.9756029041993247e-7, -2.3956211978815887e-7, 7.1182883382145864e-8, 8.925574873053455e-16, -1.2101547235064676e-8, 6.9350618248334386e-9, -1.9661464453856102e-9},
	{1.7402027787522711e-2, -2.9527880945699121e-2, 2.0045875571402799e-2, 7.0289515966903407e-6, -1.2375421071343148e-2, 1.1976293444235254e-2, -5.4156038466518525e-3, -6.3290893396418616e-8, 1.8855118129005065e-3, -1.473473274825001e-3, 5.5515810097708387e-4, 5.2406834412550662e-10, -1.4357913535784836e-4, 9.9181293224943297e-5, -3.3460834749478311e-5, -3.5755837291098993e-12, 7.1560851960630076e-6, -4.5516802628155526e-6, 1.4236576649271475e-6, 1.8803149082089664e-14, -2.6623403898929211e-7, 1.5950642189595716e-7, -4.7187514673841102e-8, -6.5107872958755177e-17, 7.9795091026746235e-9},
	{3.0249124160905891e-2, 2.4817436002649977e-3, -4.9939134373457022e-2, 5.9915643009307869e-2, -3.2483207601623391e-2, -5.7212968652103441e-6, 1.5085251778569354e-2, -1.3261324005088445e-2, 5.5515262632426148e-3, 3.0263182257030016e-8, -1.7229548406756723e-3, 1.2893570099929637e-3, -4.6845138348319876e-4, -1.830259937893045e-10, 1.1449739014822654e-4, -7.7378565221244477e-5, 2.5625836246985201e-5, 1.0766165333192814e-12, -5.3246809282422621e-6, 3.349634863064464e-6, -1.0381253128684018e-6, -5.608909920621128e-15, 1.9150821930676591e-7, -1.1418365800203486e-7, 3.3654425209171788e-8},
	{-9.9051020880159045e-2, 1.7954011706123486e-1, -1.2989606383463778e-1, -3.1478872752284357e-5, 9.0510635276848131e-2, -9.2828824411184397e-2, 4.4412112839877808e-2, 2.7779236316835888e-7, -1.7229543805449697e-2, 1.4182925050891573e-2, -5.6214161633747336e-3, -2.39598509186381e-9, 1.6029634366079908e-3, -1.1606784674435773e-3, 4.1001337768153873e-4, 1.8365800754090661e-11, -9.5844256563655903e-5, 6.3643062337764708e-5, -2.076250624489065e-5, -1.1806020912804483e-13, 4.2131808239120649e-6, -2.6262241337012467e-6, 8.0770620494930662e-7, 6.0125912123632725e-16, -1.4729737374018841e-7},
	{-1.9994542198219728e-1, -1.5056113040026424e-2, 3.6470239469348489e-1, -4.6435192311733545e-1, 2.6640934719197893e-1, 3.4038266027147191e-5, -1.3784338709329624e-1, 1.276467178337056e-1, -5.6213828755200985e-2, -1.753150885483011e-7, 1.9235592956768113e-2, -1.5088821281095315e-2, 5.7401854451350123e-3, 1.0622382710310225e-9, -1.5335082692563998e-3, 1.0819320643228214e-3, -3.7372510193945659e-4, -6.6170909729031985e-12, 8.4263617380909628e-5, -5.5150706827483479e-5, 1.7769536448348069e-5, 3.8827923210205533e-14, -3.53513697488768e-6, 2.1865832130045269e-6, -6.6812849447625594e-7},
	{7.2438608504029431e-1, -1.3918010932653375, 1.0654143352413968, 1.876173868950258e-4, -8.2705501176152696e-1, 8.9352433347828414e-1, -4.4971003995291339e-1, -1.6107401567546652e-6, 1.9235590165271091e-1, -1.6597702160042609e-1, 6.8882222681814333e-2, 1.3910091724608687e-8, -2.146911561508663e-2, 1.6228980898865892e-2, -5.9796016172584256e-3, -1.1287469112826745e-10, 1.5167451119784857e-3, -1.0478634293553899e-3, 3.5539072889126421e-4, 8.1704322111801517e-13, -7.7773013442452395e-5, 5.0291413897007722e-5, -1.6035083867000518e-5, 1.2469354315487605e-14, 3.1369106244517615e-6},
	{1.6668949727276811, 1.165462765994632e-1, -3.3288393225018906, 4.4692325482864037, -2.6977693045875807, -2.600667859891061e-4, 1.5389017615694539, -1.4937962361134612, 6.8881964633233148e-1, 1.3077482004552385e-6, -2.5762963325596288e-1, 2.1097676102125449e-1, -8.3714408359219882e-2, -7.7920428881354753e-9, 2.4267923064833599e-2, -1.7813678334552311e-2, 6.3970330388900056e-3, 4.9430807090480523e-11, -1.5554602758465635e-3, 1.0561196919903214e-3, -3.5277184460472902e-4, 9.3002334645022459e-14, 7.5285855026557172e-5, -4.8186515569156351e-5, 1.5227271505597605e-5},
	{-6.6188298861372935, 1.3397985455142589e+1, -1.0789350606845146e+1, -1.4352254537875018e-3, 9.2333694596189809, -1.0456552819547769e+1, 5.5105526029033471, 1.2024439690716742e-5, -2.5762961164755816, 2.3207442745387179, -1.0045728797216284, -1.0207833290021914e-7, 3.3975092171169466e-1, -2.6720517450757468e-1, 1.0235252851562706e-1, 8.4329730484871625e-10, -2.7998284958442595e-2, 2.0066274144976813e-2, -7.0554368915086242e-3, 1.9402238183698188e-12, 1.6562888105449611e-3, -1.1082898580743683e-3, 3.654545161310169e-4, -5.1290032026971794e-11, -7.6340103696869031e-5},
	{-1.7112706061976095e+1, -1.1208044642899116, 3.7131966511885444e+1, -5.2298271025348962e+1, 3.3058589696624618e+1, 2.4791298976200222e-3, -2.061089403411526e+1, 2.088672775145582e+1, -1.0045703956517752e+1, -1.2238783449063012e-5, 4.0770134274221141, -3.473667358470195, 1.4329352617312006, 7.1359914411879712e-8, -4.4797257159115612e-1, 3.4112666080644461e-1, -1.2699786326594923e-1, -2.8953677269081528e-10, 3.3125776278259863e-2, -2.3274087021036101e-2, 8.0399993503648882e-3, -1.177805216235265e-9, -1.8321624891071668e-3, 1.2108282933588665e-3, -3.9479941246822517e-4},
	{7.389033153567425e+1, -1.5680141270402273e+2, 1.322177542759164e+2, 1.3692876877324546e-2, -1.2366496885920151e+2, 1.4620689391062729e+2, -8.0365587724865346e+1, -1.1259851148881298e-4, 4.0770132196179938e+1, -3.8210340013273034e+1, 1.719522294277362e+1, 9.3519707955168356e-7, -6.2716159907747034, 5.1168999071852637, -2.0319658112299095, -4.9507215582761543e-9, 5.9626397294332597e-1, -4.4220765337238094e-1, 1.6079998700166273e-1, -2.4733786203223402e-8, -4.0307574759979762e-2, 2.7849050747097869e-2, -9.4751858992054221e-3, 6.419922235909132e-6, 2.1250180774699461e-3},
	{2.1216837098382522e+2, 1.3107863022633868e+1, -4.9698285932871748e+2, 7.3121595266969204e+2, -4.8213821720890847e+2, -2.8817248692894889e-2, 3.2616720302947102e+2, -3.4389340280087117e+2, 1.7195193870816232e+2, 1.4038077378096158e-4, -7.52594195897599e+1, 6.651969984520934e+1, -2.8447519748152462e+1, -7.613702615875391e-7, 9.5402237105304373, -7.5175301113311376, 2.8943997568871961, -4.6612194999538201e-7, -8.0615149598794088e-1, 5.8483006570631029e-1, -2.0845408972964956e-1, 1.4765818959305817e-4, 5.1000433863753019e-2, -3.3066252141883665e-2, 1.5109265210467774e-2},
	{-9.8959643098322368e+2, 2.1925555360905233e+3, -1.9283586782723356e+3, -1.5925738122215253e-1, 1.9569985945919857e+3, -2.4072514765081556e+3, 1.3756149959336496e+3, 1.2920735237496668e-3, -7.525941715948055e+2, 7.3171668742208716e+2, -3.4137023466220065e+2, -9.9857390260608043e-6, 1.3356313181291573e+2, -1.1276295161252794e+2, 4.6310396098204458e+1, -7.9237387133614756e-6, -1.4510726927018646e+1, 1.1111771248100563e+1, -4.1690817945270892, 3.1008219800117808e-3, 1.1220095449981468, -7.6052379926149916e-1, 3.6262236505085254e-1, 2.216867741940747e-1, 4.8683443692930507e-1},
}

// Igam computes the incomplete Gamma integral.
//
//	Igam(a,x) = (1/ Γ(a)) \int_0^x e^{-t} t^{a-1} dt
//
// The input argument a must be positive and x must be non-negative or Igam
// will panic.
func Igam(a, x float64) float64 {
	// The integral is evaluated by either a power series or continued fraction
	// expansion, depending on the relative values of a and x.
	// Sources:
	// [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov
	// [2] Maddock et. al., "Incomplete Gamma Functions",
	// http://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

	// Check zero integration limit first
	if x == 0 {
		return 0
	}

	if x < 0 || a <= 0 {
		panic(paramOutOfBounds)
	}

	// Asymptotic regime where a ~ x; see [2].
	absxmaA := math.Abs(x-a) / a
	if (igamSmall < a && a < igamLarge && absxmaA < igamSmallRatio) ||
		(igamLarge < a && absxmaA < igamLargeRatio/math.Sqrt(a)) {
		return asymptoticSeries(a, x, igam)
	}

	if x > 1 && x > a {
		return 1 - IgamC(a, x)
	}

	return igamSeries(a, x)
}

// IgamC computes the complemented incomplete Gamma integral.
//
//	IgamC(a,x) = 1 - Igam(a,x)
//	           = (1/ Γ(a)) \int_0^\infty e^{-t} t^{a-1} dt
//
// The input argument a must be positive and x must be non-negative or
// IgamC will panic.
func IgamC(a, x float64) float64 {
	// The integral is evaluated by either a power series or continued fraction
	// expansion, depending on the relative values of a and x.
	// Sources:
	// [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov
	// [2] Maddock et. al., "Incomplete Gamma Functions",
	// http://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

	switch {
	case x < 0, a <= 0:
		panic(paramOutOfBounds)
	case x == 0:
		return 1
	case math.IsInf(x, 0):
		return 0
	}

	// Asymptotic regime where a ~ x; see [2].
	absxmaA := math.Abs(x-a) / a
	if (igamSmall < a && a < igamLarge && absxmaA < igamSmallRatio) ||
		(igamLarge < a && absxmaA < igamLargeRatio/math.Sqrt(a)) {
		return asymptoticSeries(a, x, igamC)
	}

	// Everywhere else; see [2].
	if x > 1.1 {
		if x < a {
			return 1 - igamSeries(a, x)
		}
		return igamCContinuedFraction(a, x)
	} else if x <= 0.5 {
		if -0.4/math.Log(x) < a {
			return 1 - igamSeries(a, x)
		}
		return igamCSeries(a, x)
	}

	if x*1.1 < a {
		return 1 - igamSeries(a, x)
	}
	return igamCSeries(a, x)
}

// igamFac computes
//
//	x^a * e^{-x} / Γ(a)
//
// corrected from (15) and (16) in [2] by replacing
//
//	e^{x - a}
//
// with
//
//	e^{a - x}
func igamFac(a, x float64) float64 {
	if math.Abs(a-x) > 0.4*math.Abs(a) {
		ax := a*math.Log(x) - x - lgam(a)
		return math.Exp(ax)
	}

	fac := a + lanczosG - 0.5
	res := math.Sqrt(fac/math.Exp(1)) / lanczosSumExpgScaled(a)

	if a < 200 && x < 200 {
		res *= math.Exp(a-x) * math.Pow(x/fac, a)
	} else {
		num := x - a - lanczosG + 0.5
		res *= math.Exp(a*log1pmx(num/fac) + x*(0.5-lanczosG)/fac)
	}

	return res
}

// igamCContinuedFraction computes IgamC using DLMF 8.9.2.
func igamCContinuedFraction(a, x float64) float64 {
	ax := igamFac(a, x)
	if ax == 0 {
		return 0
	}

	// Continued fraction
	y := 1 - a
	z := x + y + 1
	c := 0.0
	pkm2 := 1.0
	qkm2 := x
	pkm1 := x + 1.0
	qkm1 := z * x
	ans := pkm1 / qkm1

	for i := 0; i < maxIter; i++ {
		c += 1.0
		y += 1.0
		z += 2.0
		yc := y * c
		pk := pkm1*z - pkm2*yc
		qk := qkm1*z - qkm2*yc
		var t float64
		if qk != 0 {
			r := pk / qk
			t = math.Abs((ans - r) / r)
			ans = r
		} else {
			t = 1.0
		}
		pkm2 = pkm1
		pkm1 = pk
		qkm2 = qkm1
		qkm1 = qk
		if math.Abs(pk) > big {
			pkm2 *= biginv
			pkm1 *= biginv
			qkm2 *= biginv
			qkm1 *= biginv
		}
		if t <= machEp {
			break
		}
	}

	return ans * ax
}

// igamSeries computes Igam using DLMF 8.11.4.
func igamSeries(a, x float64) float64 {
	ax := igamFac(a, x)
	if ax == 0 {
		return 0
	}

	// Power series
	r := a
	c := 1.0
	ans := 1.0

	for i := 0; i < maxIter; i++ {
		r += 1.0
		c *= x / r
		ans += c
		if c <= machEp*ans {
			break
		}
	}

	return ans * ax / a
}

// igamCSeries computes IgamC using DLMF 8.7.3. This is related to the series
// in igamSeries but extra care is taken to avoid cancellation.
func igamCSeries(a, x float64) float64 {
	fac := 1.0
	sum := 0.0

	for n := 1; n < maxIter; n++ {
		fac *= -x / float64(n)
		term := fac / (a + float64(n))
		sum += term
		if math.Abs(term) <= machEp*math.Abs(sum) {
			break
		}
	}

	logx := math.Log(x)
	term := -expm1(a*logx - lgam1p(a))
	return term - math.Exp(a*logx-lgam(a))*sum
}

// asymptoticSeries computes Igam/IgamC using DLMF 8.12.3/8.12.4.
func asymptoticSeries(a, x float64, fun int) float64 {
	maxpow := 0
	lambda := x / a
	sigma := (x - a) / a
	absoldterm := math.MaxFloat64
	etapow := [igamDimN]float64{1}
	sum := 0.0
	afac := 1.0

	var sgn float64
	if fun == igam {
		sgn = -1
	} else {
		sgn = 1
	}

	var eta float64
	if lambda > 1 {
		eta = math.Sqrt(-2 * log1pmx(sigma))
	} else if lambda < 1 {
		eta = -math.Sqrt(-2 * log1pmx(sigma))
	} else {
		eta = 0
	}
	res := 0.5 * math.Erfc(sgn*eta*math.Sqrt(a/2))

	for k := 0; k < igamDimK; k++ {
		ck := igamCoefs[k][0]
		for n := 1; n < igamDimN; n++ {
			if n > maxpow {
				etapow[n] = eta * etapow[n-1]
				maxpow++
			}
			ckterm := igamCoefs[k][n] * etapow[n]
			ck += ckterm
			if math.Abs(ckterm) < machEp*math.Abs(ck) {
				break
			}
		}
		term := ck * afac
		absterm := math.Abs(term)
		if absterm > absoldterm {
			break
		}
		sum += term
		if absterm < machEp*math.Abs(sum) {
			break
		}
		absoldterm = absterm
		afac /= a
	}
	res += sgn * math.Exp(-0.5*a*eta*eta) * sum / math.Sqrt(2*math.Pi*a)

	return res
}