(* Runge-Kutta 9(8) pair *)
(* Ch. Tsitouras, Appl. Numer. Math. 38 (2001) 123-134. *)
(* 8th and 9th order interpolants, accurate at 55 digits *)

(* 9th order weights of the interpolant *)
btilda9={t-19.28431428517097368937448650675238144038622663303086497449077378846282638273977*t^2+151.50966865631955852563787365059035036647682676875622063416475362129592975543201*t^3-632.2742837325336234724385160069622740179153640633805840860253360802437*t^4+1542.8574223810721178204426888024293903430990220982514097365958068897608*t^5-2273.6308766391947999672147161821093163258379363085416616662313029113644*t^6+1991.1939080844981052320447237082429289279635662278489405269229308603976*t^7-953.4870878673746383182057848621251025331174339302738570894374378978395*t^8+192.130472423204038479333053567710230206858938268950733192454211596223*t^9,0,0,0,0,0,0,-0.66684463629785661077095148068171329751989078864774683354148440976669402102196*t^2+16.70472393362983127607986095209445949808337228329555132312171313940123359911126*t^3-139.10888715273347437541225913387879324613131233019647683317278021903899571930769*t^4+564.0944279947974742074591828881617294420582162111847417013805614437581*t^5-1247.9410940334837563860198595111378531024416277885999585104132535448869*t^6+1536.9199723375195560511621041279055687954781235769038112423093086809593*t^7-988.2470939600892328913739918375174249894730712971404368682168416546318*t^8+258.040715069736917216292422785712678981901256118074178426344098707035*t^9,0.74831772101327833743169720155527758675521679995710989167610484098983951703921*t^2-18.74565717971273243472757131634225184778433958541109504537185035626970398910204*t^3+156.10479524098621072631492864482958142013936930990261200157907542514072891304469*t^4-633.0137993414231153396543635420505821652625349235963890881159061675187*t^5+1400.4108072766026245254862732984274182946054912323352405320537292187078*t^6-1724.6962612828023647480052589317110927066518693944582944353983993495524*t^7+1108.9881704017930501467675786396969029124383145716868820247250873696887*t^8-289.5673584504512467409655593010345887338865106301786528393782598755896*t^9,0.41826562711906322971113082093348374900551021653941659014059712840254073787222*t^2-10.47772067379971813863676704850949974423210412704834298535030348761544487182911*t^3+87.25340619937754280845441334662115912287262067660482826905138952651496559449985*t^4-353.8175113614115327171801695747773779063861546238278026375213756933641*t^5+782.7473385726063936529339293566700474668507946004011461924758820116777*t^6-964.0038489781464017411171518133993438450096877506866191899391702892837*t^7+619.8592115828001693839525527089505017276420726290029459564705765874637*t^8-161.8511353860340427214258556807516785480053280020931349544718099512827*t^9,0.73129989641221334052615495668257895959143407531668840389753058257806109713798*t^2-18.31935389040390497890150577575764612818788196431713468997729041205380044806433*t^3+152.55474697913416987332096274912951438979835488690278823029228783926988773506032*t^4-618.6181522723422248052324430066688106334209569360097395212908531128137*t^5+1368.5634450954707676962380167069925209578107029262890498560324310340923*t^6-1685.4741800191374972000926129312504352655425521021496828940895130106113*t^7+1083.7681794292435505840013900064025366898382377726409261711735899856239*t^8-282.9821789499165330733622620359653995926089013498953729221609218496881*t^9,1.29242249920080836219719766495915592018836139780435163042504568520868445350837*t^2-32.3756987453942855636969397537779915191710898168305562801255416718869677617347*t^3+269.6092100149608669893358246162137808156170371658711030691052238909452*t^4-1093.2806394931333344688846381039469740988162018553631706679035280480589*t^5+2418.6550506881990984384840556236553952983431714083637575682047298695345*t^6-2978.7297424296293403544067453328894211274996681215746477173691346382501*t^7+1915.3378605467834480668717230215634550008845129789235787773585749147248*t^8-500.1129314280502609256949538442131837728764497958149149961338343342549*t^9,0.17699205839337255764014279367859467001617742856349556655192749695555696494016*t^2-4.43372160915989723067116079003775296132048144939965296212573042371897456654958*t^3+36.92189595265222451792286125012003416251828139234940965055269965481275122847963*t^4-149.72038238669446309082299571854897335189616859951340697593727737621854639203079*t^5+331.2250724740117927664485536268344497838553208357651321732045284801293*t^6-407.9250468296480761526414547612930383990301860149988074870031014287597*t^7+262.2978094752759075678520432586072720554630634963822308834459569991099*t^8-68.48845266724279112376230601397697852225425821793506929627904108742405850983647*t^9,0.41470011394215677762383985652541889048210675437547266131842274385365988594768*t^2-10.38840314755761168019948514476269926926012343490197060757359018913805048870306*t^3+86.5096129030536894693013151726054776319486191434378916790762349804587884285437*t^4-350.8013873550746417487254236320760096732292233210166139928102707822388*t^5+776.0747942158246459464872073973232703754779030313832067448903466676221*t^6-955.786180101565022426901384660477105008165561533472290329354280649894*t^7+614.5752101171569306322742023452544361944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(* 8th order weights of the interpolant *)
btilda8={t-11.97345176506617656036131101822672784050194395371436147688145910785537679386622*t^2+61.86594333413414574145464196748706812091827311207906644191489875623333605679621*t^3-172.1912849388249664621314568847245829743337613026067919532729045971816*t^4+278.4206016191573084825972867744062453619099182185748035743374671202527*t^5-262.2459610789586336134141585096726928258715694042586133181023273658116*t^6+133.62552314202981516425562987514061211889986695375351056886582500962092715369692*t^7-28.48646129165170814217579603338609643387939119524727756309185187566363985264498*t^8,0,0,0,0,0,0,43.27956037119576633146406873347845166148542486823381588342126313668548926673835*t^2-388.6436150102667676522415538175732297858711832808040677784459050444915*t^3+1375.3613423834438651642549424898352755043742698182696073728064068615602*t^4-2531.7496378716722348176210322352557382845620637060186479789700057371612*t^5+2582.7115024191092842667768582424527415393196971320676514084896117481405*t^6-1389.7113915081824165659549627262147146263663873455510409758367561317121*t^7+308.5481587694519617607381881039358660735753084986763459503538318321166*t^8,-35.94922915751584130869321252248514855413078337973889322545328794856604680250353*t^2+344.660071452673237933047012396128131232718715857603542365716089002504*t^3-1277.8231161852588793034483568957162740405683598830460223150252566569434*t^4+2438.4531969414645043419415426301071211295512372936314532957660477696715*t^5-2556.6772649423957779426859557911790265063330339949570997309165775195886*t^6+1403.5145365712080540414467219761373390303020156735188811592903692001269*t^7-315.9491802941695932889600270996214775311866541867744485408654239135454*t^8,14.50776885556117057739951068486411480875424528037806565555997591807755098982149*t^2-102.87334506907882369949499181653257453701607118049068327459276226160639453117644*t^3+325.7362093496231072258963200065449438570501084930027322039446060049316*t^4-564.4147540188118244433978950045742298770077941533871243795451683419952*t^5+554.7181844974447817963022286053482885119275406184638127780972378065216*t^6-290.7927874648276340793815003237771428065754595774526607569364759011792*t^7+63.24672943260069637936840996386389206560515413837829504493517788768934614808868*t^8,-2.10946955190246465989087827653629821569012002660017056296435168751471619242668*t^2+55.98825195478833795819109523603203326740164807057798858258496510438681258534764*t^3-264.6141321069909665263158653723270381749556596507838474630262693302674*t^4+565.6582238506475888153810338682014718064289397083795205931166013120311*t^5-633.3809162640656841448760442624826540652160056726019717733136563092204*t^6+362.7496970893098976092438472452455433863278091001320330165304647191641*t^7-84.06784870332616761523548776856819862701817422032602976109425959893778415936784*t^8,-100.7937501145973976044095927423400329904707243931647886421427421175049721661033*t^2+912.9912530546392512108894496154954268389648552397263601735252544907565*t^3-3280.683157881327571293130306845950438925830453223783924976560598344694*t^4+6130.8565731590057666459772117731736608014336617789176067479241762584638*t^5-6333.8123210002439365588445068616605143794032463539310286575419956479588*t^6+3440.7580426866649922779331052966785979391847773176378845045626160175759*t^7-768.9211082512041041342098363438324827672091970040226079757988437588728*t^8,-25.44050789487298455050043602806742730135367733016771821475100150607721100636098*t^2+231.5672123873270511552155588874808088143794224185489062489440629726624*t^3-848.8813598243031668696034935741370524572259165490857035380098157578975*t^4+1619.4601461226664780921682769195806044103193388325734171930589173686446*t^5-1701.9954814871161810970517626589562476690776195433527615908376040701176*t^6+936.3762810767528054186629597497882711711398316432965146484973837653934*t^7-211.032123912865932336925419650305349530829630600599323141449627999125*t^8,-53.04452376666895399086534833750822873602099760266029185885658943828392445046406*t^2+472.0673116815203049497687242558929585599936481734446526391132487725422*t^3-1690.9474431179622415653053379847579542550243159891242764923620879700934*t^4+3162.7606215432605215020698402851889613615526673712169957130921951995289*t^5-3273.1311367235474224182518161777664034610762149878383191746780956665675*t^6+1780.7160806537701180827154030095557624863707628959573731950894303532307*t^7-398.2939958738478675232750311974834155792178840699091182028291880588661*t^8,0.06953290488595322428064781586652944645883707692307628577858134518174685518044*t^2-0.40836619698511108536380100612280752425256037002181575435577121592031814696295*t^3+0.64705882352941176470588235294117647058823529411764705882352941176470588235294*t^4+0.13333333333333333333333333333333333333333333333333333333333333333333333333333*t^5-1.24390243902439024390243902439024390243902439024390243902439024390243902439024*t^6+1.12*t^7-0.31818181818181818181818181818181818181818181818181818181818181818181818181818*t^8,-16.273632561649042928632428258068405984592780257411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c={0,0.02173913043478260869565217391304347826086956521739130434782608695652173913043,0.09629581047800066670113001679819925959085658682855574409482938528571542584724,0.14444371571700100005169502519729888938628488024283361600120683094587667512815,0.52205882352941176470588235294117647058823529411764705882352941176470588235294,0.22842443612863469578031459099794265170568729495075968407330185950913875445944,0.54360353589933733219171338103002937626634067707721234383543015413053984186453,0.64335664335664335664335664335664335664335664335664335664335664335664335664336,0.48251748251748251748251748251748251748251748251748251748251748251748251748252,0.06818181818181818181818181818181818181818181818181818181818181818181818181818,0.25060827250608272506082725060827250608272506082725060827250608272506082725061,0.66736715965600568968278165443304378929030003367892917916280609909012198890194,0.85507246376811594202898550724637681159420289855072463768115942028985507246377,0.89795918367346938775510204081632653061224489795918367346938775510204081632653,1.,1.,1.,0.74143720845372055359900703795542553636931134546459043030550372498086335432596,0.85542586221887479770704957375656067573667288069151888238339994185087288882611,0.64180122353767291975285767191286604597746537284710274708385019473061230101582,0.50852519773191487807809775022422859333413052597967341675629882133786038615001,0.04347826086956521739130434782608695652173913043478260869565217391304347826087,0.19047619047619047619047619047619047619047619047619047619047619047619047619048,0.29166666666666666666666666666666666666666666666666666666666666666666666666667,0.64285714285714285714285714285714285714285714285714285714285714285714285714285,0.72727272727272727272727272727272727272727272727272727272727272727272727272728};

(* 9th order weights of the main method *)
b={0.01490902081978461022483617102382552714139242858033627391564914803022648421473,0,0,0,0,0,0,-0.20408044692054151258349120934134791804493401512633635250492851944173090998178,0.22901438600570447264772469337066476035313738023741304162093075399794686088638,0.128005582511473756692082115737292022737723618892437240858502252806873896563,0.22380626846054143649770066956485937727843730877752263380500045862038739178868,0.39553165293700054420552389156421651666967336137950138371378877612251258146312,0.054166467588069811965683645383607437351748871213331552335195878763152313116,0.12691439652445903685643385312168037657766579108701586288175713592830721910645,-0.00052539244262118876455834655383035812936087407347970114202491708574036342437,0.03225806451612903225806451612903225806451612903225806451612903225806451612903,0,0,0,0,0,0,0,0,0,0};

(* 8th order weights of the main method *)
bb={0.0986944409532825112697565371461558184176960142481876980078540548055362,0,0,0,0,0,0,20.9022189036201781133368473213981559419248329198300936096724580106722583,-1.8336644778238788808281184551603890335530985295348754630775272067103917,-0.08273332108398198824168126595166413313192173558539478780446650452208393,0.67131540884240257028965403993665111519129165487313417313361743467742079,-20.9892390399697647058568586436757008589735763090590147452903075985615423,6.13887488578967922129465332603558335891060855328643994801709284618789439,-4.29642051701795891700057824542304185815469652317778501891398050358976501,0.358695652173913043478260869565217391304347826086956521739130434782608696,0.0322580645161290322580645161290322580645161290322580645161290322580645161,0,0,0,0,0,0,0,0,0,0};

(* matrix a *)
a={{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.02173913043478260869565217391304347826086956521739130434782608695652173913043,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{-0.11698050118114486205818241524969622410384223979749965805696164405165374068545,0.21327631165914552875931243204789548369469882662605540205429350381508117932416,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.03611092892925025001292375629932472234657122006070840400030170773646916878204,0,0.10833278678775075003877126889797416703971366018212521200090512320940750634611,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{1.57329743908138605107331820072051125641643664351842805209769690028102798657659,0,-5.98400943754042002888532938159655553518127689802844787698990227392475614910376,4.93277082198844574251789353381722074935307554862766688366377625231796145777087,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.05052046351120380909008334360006234635511181026904002251194423118941950046497,0,0,0.17686653884807108146683657390397612041041458603581793182438474501847191168463,0.00103743376935980522339467349390418494016089864590172962507499344829031727187,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.10543148021953768958529340893598138847228076756096566564586860869775529231537,0,0,-0.16042415162569842979496486916719383758722635620697602559102345943091722514966,0.11643956912829316045688724281285250393566139141568280111760708666061015359453,0.48215663817720491194449759844838932144562487430753990276750905332820882466184,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.07148407148407148407148407148407148407148407148407148407148407148407148407149,0,0,0,0,0.32971116090443908023196389566296464803047147789829324999429860673704318583012,0.24216141096813279233990867620960722454140109397427862246588264600357052639983,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.07162368881118881118881118881118881118881118881118881118881118881118881118881,0,0,0,0,0.32859867301674234161492268975519694162614616072449015216348529457960819195682,0.11622213117906185418927311444060725417804964347131404372371681988432044678195,-0.03392701048951048951048951048951048951048951048951048951048951048951048951049,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0.04861540768024729180628870095388582672433144665590947586795228849109445591019,0,0,0,0,0.03998502200331629058445317782406268415701330226121918489861492002427174285737,0.10715724786209388876739304914053506956742307410964855781939652925101421581225,-0.02177735985419485163815426357369818120744539675289636874965686683769452198667,-0.10579849950964443770179884616296721742314060809206266805403458002612522500602,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{-0.02540141041535143673515871979014924329181062727138312459720832287794796919218,0,0,0,0,0.03333333333333333333333333333333333333333333333333333333333333333333333333334,-0.164048547600691820735035530202387825925934036472683126120563732060474408503,0.0341054889879473778889141456652852683156608503051057486442769496968166589003,0.15836825014108792658008718465091487655106285177705603304446983707186729673991,0.21425115805975734472868683695127609710041268915582174417087194052087239697391,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
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