###################################################################### # # A 'most efficient' Runge--Kutta (13:8(7)) pair # # These are approximate REAL coefficients # computed using MAPLE with 40 digits for # # a TEN-stage conventional pair methods of # orders p=7 and p=8, with dominant # stage order = 4, # # together with approximate coefficients # # for two interpolants of orders 7 and 8 # # which require 4 and 4 extra stages respectively. # (Companion files list only the RATIONAL COEFFICIENTS # of the pair, and 40-digit floating point approximations # otherwise.) # # This procedure is "most efficient" in the sense that # that for a specified maximum coefficient from # b and A, it has a propagating formula which almost # minimizes the 2-norm of the local truncation error as # # T92 ~ .000000282 # # (Formulas with slightly different nodes c_i can # have a slightly smaller error norm perhaps # achieved by having a larger maximum coefficient.) # # Additional stages and interpolating weights allow # for the computation of an approximation at any # point of the domain of solution of order up to p. # These interpolants have continuous derivatives. # # Nodes c[14]=1, c[15] - c[17] were selected in an # attempt to minimize the maximum of the 2-norm of # the local truncation error over the interval [0,1] # for the interpolant of order 7, and this value is # # T_82 ~ .00002010 # # This 2-norm has three local maximum values on [0,1]. # # The remaining four nodes were selected in an attempt # to minimize the maximum of the 2-norm of the local # truncation error on the interval [0,1] for the # interpolant of order 8, and this value is # # T_92 ~ .000004043 # # This 2-norm has two local maximum values on [0,1]. # Attempts to make the order 7 interpolant monotone # increasing on this interval were unsuccessful. # # The formulas scanned for this optimal formula are # those developed in J.H. Verner, SIAM NA 1978, 772-790, # "Explicit Runge--Kutta methods with estimates of the # Local Truncation Error". It is conceivable that the # pairs in J.H. Verner, Annals of Num. Math 1 1994, # 225-244, "Strategies for deriving new explicit Runge-- # Kutta pairs", which require only ten stages, but solve # the order conditions in a different way, or the # 11-stage contemporary or 12-stage FSAL methods # derived by Sharp and Verner, SIAM NA 31, 1994, # 1169--1190, "Completely imbedded Runge--Kutta pairs" # may yield particular pairs of equivalant or more # efficiency. # ###################################################################### # # NODES # ----- c[1] = 0. c[2] = .5e-1 c[3] = .1065625 c[4] = .15984375 c[5] = .39 c[6] = .465 c[7] = .155 c[8] = .943 c[9] = .9018020417358569582597079406783721499560 c[10] = .909 c[11] = .94 c[12] = 1. c[13] = 1. # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- c[1] = 0. # c[2] = .5e-1 a[2,1] = .5e-1 # c[3] = .1065625 a[3,1] = -.69931640625e-2 a[3,2] = .1135556640625 # c[4] = .15984375 a[4,1] = .399609375e-1 a[4,2] = 0. a[4,3] = .1198828125 # c[5] = .39 a[5,1] = .3613975628004575124052940721184028345129 a[5,2] = 0. a[5,3] = -1.341524066700492771819987788202715834917 a[5,4] = 1.370126503900035259414693716084313000404 # c[6] = .465 a[6,1] = .4904720279720279720279720279720279720280e-1 a[6,2] = 0. a[6,3] = 0. a[6,4] = .2350972042214404739862988335493427143122 a[6,5] = .1808555929813567288109039636534544884850 # c[7] = .155 a[7,1] = .6169289044289044289044289044289044289044e-1 a[7,2] = 0. a[7,3] = 0. a[7,4] = .1123656831464027662262557035130015442303 a[7,5] = -.3885046071451366767049048108111244567456e-1 a[7,6] = .1979188712522045855379188712522045855379e-1 # c[8] = .943 a[8,1] = -1.767630240222326875735597119572145586714 a[8,2] = 0. a[8,3] = 0. a[8,4] = -62.5 a[8,5] = -6.061889377376669100821361459659331999758 a[8,6] = 5.650823198222763138561298030600840174201 a[8,7] = 65.62169641937623283799566054863063741227 # c[9] = .9018020417358569582597079406783721499560 a[9,1] = -1.180945066554970799825116282628297957882 a[9,2] = 0. a[9,3] = 0. a[9,4] = -41.50473441114320841606641502701994225874 a[9,5] = -4.434438319103725011225169229846100211776 a[9,6] = 4.260408188586133024812193710744693240761 a[9,7] = 43.75364022446171584987676829438379303004 a[9,8] = .7871425489912310687446475044226307550860e-2 # c[10] = .909 a[10,1] = -1.281405999441488405459510291182054246266 a[10,2] = 0. a[10,3] = 0. a[10,4] = -45.04713996013986630220754257136007322267 a[10,5] = -4.731362069449576477311464265491282810943 a[10,6] = 4.514967016593807841185851584597240996214 a[10,7] = 47.44909557172985134869022392235929015114 a[10,8] = .1059228297111661135687393955516542875228e-1 a[10,9] = -.5746842263844616254432318478286296232021e-2 # c[11] = .94 a[11,1] = -1.724470134262485191756709817484481861731 a[11,2] = 0. a[11,3] = 0. a[11,4] = -60.92349008483054016518434619253765246063 a[11,5] = -5.951518376222392455202832767061854868290 a[11,6] = 5.556523730698456235979791650843592496839 a[11,7] = 63.98301198033305336837536378635995939281 a[11,8] = .1464202825041496159275921391759452676003e-1 a[11,9] = .6460408772358203603621865144977650714892e-1 a[11,10] = -.7930323169008878984024452548693373291447e-1 # c[12] = 1. a[12,1] = -3.301622667747079016353994789790983625569 a[12,2] = 0. a[12,3] = 0. a[12,4] = -118.0112723597525085666923303957898868510 a[12,5] = -10.14142238845611248642783916034510897595 a[12,6] = 9.139311332232057923544012273556827000619 a[12,7] = 123.3759428284042683684847180986501894364 a[12,8] = 4.623244378874580474839807625067630924792 a[12,9] = -3.383277738068201923652550971536811240814 a[12,10] = 4.527592100324618189451265339351129035325 a[12,11] = -5.828495485811622963193088019162985703755 # c[13] = 1. a[13,1] = -3.039515033766309030040102851821200251056 a[13,2] = 0. a[13,3] = 0. a[13,4] = -109.2608680894176254686444192322164623352 a[13,5] = -9.290642497400293449717665542656897549158 a[13,6] = 8.430504981764911142134299253836167803454 a[13,7] = 114.2010010378331313557424041095523427476 a[13,8] = -.9637271342145479358162375658987901652762 a[13,9] = -5.034884088802189791198680336183332323118 a[13,10] = 5.958130824002923177540402165388172072794 a[13,11] = 0. a[13,12] = 0. # # ******************************************************** # High order weights c[ 14] = 1 # ----------------------------------------------------------- # b[1] = .4427989419007951074716746668098518862111e-1 b[2] = 0. b[3] = 0. b[4] = 0. b[5] = 0. b[6] = .3541049391724448744815552028733568354121 b[7] = .2479692154956437828667629415370663023884 b[8] = -15.69420203883808405099207034271191213468 b[9] = 25.08406496555856261343930031237186278518 b[10] = -31.73836778626027646833156112007297739997 b[11] = 22.93828327398878395231483560344797018313 b[12] = -.2361324633071542145259900641263517600737 b[13] = 0. # # ******************************************************** # Low order weights C[extra]:= 1 # -------------------------------------------------- # bh[1] = .4431261522908979212486436510209029764893e-1 bh[2] = 0. bh[3] = 0. bh[4] = 0. bh[5] = 0. bh[6] = .3546095642343226447863179350895055038855 bh[7] = .2478480431366653069619986721504458660016 bh[8] = 4.448134732475784492725128317159648871312 bh[9] = 19.84688636611873369930932399297687935291 bh[10] = -23.58162337746561841969517960870394965085 bh[11] = 0. bh[12] = 0. bh[13] = -.3601679437289775162124536737746202409110 # # ******************************************************* # # Largest coefficient in b or A has magnitude 123.3759 # # ******************************************************* # SUMMARY OF NORMS OF ERRORS: A91, A92, A9inf #---------------------------------------------------- # A_[9, 1] = .2275260758e-5 # A_[9, 2] = .2827866033e-6 # A_[9,oo] = .1060544186e-6 #**************************************************** # # END OF GENERATION OF A PAIR OF RK METHODS # ############################################################## # # START OF GENERATION OF STABILITY INTERVALS # ############################################################## # # Stability Boundaries of High Order Method # ----------------------------------------- # Real Stability Interval is nearly [ -5.864113239, 0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [ -5.698133298, 0] # ############################################################# # # START OF GENERATION OF INTERPOLANT # ############################################################# # # FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7 # c[14] = 1.0 a[14,1] = .4427989419007951074716746668098518862111e-1 a[14,2] = 0. a[14,3] = 0. a[14,4] = 0. a[14,5] = 0. a[14,6] = .3541049391724448744815552028733568354121 a[14,7] = .2479692154956437828667629415370663023884 a[14,8] = -15.69420203883808405099207034271191213468 a[14,9] = 25.08406496555856261343930031237186278518 a[14,10] = -31.73836778626027646833156112007297739997 a[14,11] = 22.93828327398878395231483560344797018313 a[14,12] = -.2361324633071542145259900641263517600737 a[14,13] = 0. # #******************************************************** # # Coupling coefficients for c[15] = .3110177634953863863927417318829099695921 # ---------------------------------------------------- a[15,1] = .4620700646754963101730413150238116432863e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = 0. a[15,5] = 0. a[15,6] = .4503904160842480866828520384400679697151e-1 a[15,7] = .2336816697713424410788701065340221126565 a[15,8] = 37.83901368421067410780338220861855254153 a[15,9] = -15.94911328945424610266139490307397370835 a[15,10] = 23.02836835181610285142510596329590091940 a[15,11] = -44.85578507769412524816130998016948002745 a[15,12] = -.6379858768647444009509067402330140781326e-1 a[15,13] = 0. a[15,14] = -.1259503554386166268241032464519842162533e-1 # #******************************************************** # # Coupling coefficients for c[16] = .1725 # ---------------------------------------------------- a[16,1] = .5037946855482040993065158747220696112586e-1 a[16,2] = 0. a[16,3] = 0. a[16,4] = 0. a[16,5] = 0. a[16,6] = .4109836131046079339916530614028848248545e-1 a[16,7] = .1718054153348195783296309209549424619697 a[16,8] = 4.61410531998151886974342237185977124648 a[16,9] = -1.791667883085396449712744996746836471721 a[16,10] = 2.531658930485041408462243518792913614971 a[16,11] = -5.32497786020573071925718815977276269909 a[16,12] = -.3065532595385634734924449496356513113607e-1 a[16,13] = 0. a[16,14] = -.5254479979429613570549519094377878106127e-2 a[16,15] = -.8399194644224792997538653464258058697156e-1 # #******************************************************** # # Coupling coefficients for c[17] = .7846 # ---------------------------------------------------- a[17,1] = .4082897132997079620207118756242653796386e-1 a[17,2] = 0. a[17,3] = 0. a[17,4] = 0. a[17,5] = 0. a[17,6] = .4244479514247632218892086657732332485609 a[17,7] = .2326091531275234539465100096964845486081 a[17,8] = 2.677982520711806062780528871014035962908 a[17,9] = .7420826657338945216477607044022963622057 a[17,10] = .1460377847941461193920992339731312296021 a[17,11] = -3.579344509890565218033356743825917680543 a[17,12] = .1138844389600173704531638716149985665239 a[17,13] = 0. a[17,14] = .1267790651033190047378693537615687232109e-1 a[17,15] = -.7443436349946674429752785032561552478382e-1 a[17,16] = .4782748079757851554575511473876987663388e-1 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi7 WITH 17 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi7[ 1] bi7[1,1] = 1. u bi7[1,2] = -7.238550783576432811855355839508646327161 u^2 bi7[1,3] = 26.00913483254676138219215542805486438340 u^3 bi7[1,4] = -50.23684777762566731759165474184543812128 u^4 bi7[1,5] = 52.12072084601022449485077581012685809554 u^5 bi7[1,6] = -27.06472451211777193118825764262673140465 u^6 bi7[1,7] = 5.454547288952965694339504452480078562780 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[2] bi7[2,1] = 0. u bi7[2,2] = 0. u^2 bi7[2,3] = 0. u^3 bi7[2,4] = 0. u^4 bi7[2,5] = 0. u^5 bi7[2,6] = 0. u^6 bi7[2,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[3] bi7[3,1] = 0. u bi7[3,2] = 0. u^2 bi7[3,3] = 0. u^3 bi7[3,4] = 0. u^4 bi7[3,5] = 0. u^5 bi7[3,6] = 0. u^6 bi7[3,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[4] bi7[4,1] = 0. u bi7[4,2] = 0. u^2 bi7[4,3] = 0. u^3 bi7[4,4] = 0. u^4 bi7[4,5] = 0. u^5 bi7[4,6] = 0. u^6 bi7[4,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0. u bi7[5,2] = 0. u^2 bi7[5,3] = 0. u^3 bi7[5,4] = 0. u^4 bi7[5,5] = 0. u^5 bi7[5,6] = 0. u^6 bi7[5,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0. u bi7[6,2] = 11.15330887588935170976376962782446833855 u^2 bi7[6,3] = -91.7609656398961659890179437322816238711 u^3 bi7[6,4] = 291.7074241722059450113911477530513089255 u^4 bi7[6,5] = -430.4096692910862817449451677633631387823 u^5 bi7[6,6] = 299.4531188198997479843407054776900024282 u^6 bi7[6,7] = -79.78911199784015209705095616004766020335 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0. u bi7[7,2] = 2.34875229807309355640904629061136935335 u^2 bi7[7,3] = -11.6724894172018429369093778842231443146 u^3 bi7[7,4] = -3.339139076505928386509206543237093540 u^4 bi7[7,5] = 94.885262249720610030798242337479596095 u^5 bi7[7,6] = -143.071126583012024456409244370652716962 u^6 bi7[7,7] = 61.0967097444217359754873031115590556707 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0. u bi7[8,2] = -1027.321675339240679090464776362465090654 u^2 bi7[8,3] = 9198.71432360760879019681406218311101879 u^3 bi7[8,4] = -33189.78048157363822223641020734287802492 u^4 bi7[8,5] = 57750.0831348887181073584126028277545727 u^5 bi7[8,6] = -47698.93315706261990169947144294597707756 u^6 bi7[8,7] = 14951.54365344033382142012769129774268946 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0. u bi7[9,2] = 1568.546608927281956416687915664731868885 u^2 bi7[9,3] = -13995.38852541600542155322174511897930298 u^3 bi7[9,4] = 50256.2124698102445419491620666726469821 u^4 bi7[9,5] = -86974.5128036219909523950692144595063700 u^5 bi7[9,6] = 71494.7977095997701213661747332399327008 u^6 bi7[9,7] = -22324.57139433374168317029445568645401598 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[10] bi7[10,1] = 0. u bi7[10,2] = -2000.882061921041961546811133479107090218 u^2 bi7[10,3] = 17864.36380347691630038038755096765127729 u^3 bi7[10,4] = -64205.1907515562863000297926577113695108 u^4 bi7[10,5] = 111224.8489930378077126420609392735999202 u^5 bi7[10,6] = -91509.3392102130338542605593697286718077 u^6 bi7[10,7] = 28594.46085938937782634638310955782423389 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[11] bi7[11,1] = 0. u bi7[11,2] = 1496.620400693446268810344884971434468267 u^2 bi7[11,3] = -13397.55405171476021512904990709508924800 u^3 bi7[11,4] = 48323.5602199437493999696912750109765015 u^4 bi7[11,5] = -84051.4283423393032636942266780744607468 u^5 bi7[11,6] = 69399.8582111570893316100585838633124312 u^6 bi7[11,7] = -21748.11815446623273761450332307272543593 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0. u bi7[12,2] = -16.41320775560933621675902845723196069900 u^2 bi7[12,3] = 147.6097045407002371315249807692915435608 u^3 bi7[12,4] = -535.719963714732106447158760197417632645 u^4 bi7[12,5] = 938.286247077820650371318861625025573381 u^5 bi7[12,6] = -779.438309639349328345148153897689081893 u^6 bi7[12,7] = 245.4393970278627292916961100938952065362 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0. u bi7[13,2] = 0.u^2 bi7[13,3] = 0. u^3 bi7[13,4] = 0. u^4 bi7[13,5] = 0. u^5 bi7[13,6] = 0. u^6 bi7[13,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0. u bi7[14,2] = -4.29672443178246482824254064733546854251 u^2 bi7[14,3] = 38.6444746111678092366406218271498656093 u^3 bi7[14,4] = -140.3503471762808981414524290552248895548 u^4 bi7[14,5] = 246.3954669697502467443139611011701827640 u^5 bi7[14,6] = -205.8341686964167118696204191085878165880 u^6 bi7[14,7] = 65.44129872356201885836080588282812631205 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0. u bi7[15,2] = -20.41628069294821485579834313809132051248 u^2 bi7[15,3] = 153.5213232524836445391962375168798263930 u^3 bi7[15,4] = -436.5502610211220460266289847121377276100 u^4 bi7[15,5] = 598.214644262650861959065070073603792110 u^5 bi7[15,6] = -398.7823950071290897160364203878571043995 u^6 bi7[15,7] = 104.0129692060648441002024406476025340187 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0. u bi7[16,2] = 16.53007184264271512356106095760699278945 u^2 bi7[16,3] = -96.6861433615782065041742809436987893361 u^3 bi7[16,4] = 268.959934219531723149495873437076657635 u^4 bi7[16,5] = -428.681909788964647271837835032326719249 u^5 bi7[16,6] = 354.578231152433375494079868740183658991 u^6 bi7[16,7] = -114.7001840640649599911246871588418008302 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[17] bi7[17,1] = 0. u bi7[17,2] = -18.63064171313429626683549958846959067803 u^2 bi7[17,3] = 164.1994112280183092456176460821337125030 u^3 bi7[17,4] = -579.272256249540441494196462569641132906 u^4 bi7[17,5] = 980.198255708866731505258442280896479501 u^5 bi7[17,6] = -786.224179015513894176220583239056456901 u^6 bi7[17,7] = 239.7294100413035911863764570341369884827 u^7 # #******************************************************** # # FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 8 # # Coupling coefficients for c[18] = .37 # ---------------------------------------------------- a[18,1] = .5212682393668413629928136927994514676607e-1 a[18,2] = 0. a[18,3] = 0. a[18,4] = 0. a[18,5] = 0. a[18,6] = .5392508396744797718209106862347065628649e-1 a[18,7] = .1660758097434640828541930599928251901718e-1 a[18,8] = -4.454485757926779655418936993298463071587 a[18,9] = 6.835218278632146381711296817968152631469 a[18,10] = -8.711334822181993739847172734848837971169 a[18,11] = 6.491635839232917053651267142703105653517 a[18,12] = -.7072551809844346422069985227700294651922e-1 a[18,13] = 0. a[18,14] = -.1854031491993216429111842937941202966440e-1 a[18,15] = .2350402105435384645116542087045962190647e-1 a[18,16] = .2344795103407822090556377813402774776461 a[18,17] = -.8241072501152898885823089698097768766651e-1 # #******************************************************** # # Coupling coefficients for c[19] = .5 # ---------------------------------------------------- a[19,1] = .5020102870355713598699964419977883461362e-1 a[19,2] = 0. a[19,3] = 0. a[19,4] = 0. a[19,5] = 0. a[19,6] = .1552209034795498114932226104700567642339 a[19,7] = .1264268424089234914713091134864747506300 a[19,8] = -5.14920630353984701704917414605721854951 a[19,9] = 8.46834099903692926607453176331494311551 a[19,10] = -10.66213068108149527544209836207095498430 a[19,11] = 7.54183322495972836290996201569018333903 a[19,12] = -.743696811383214243944066492459357053774e-1 a[19,13] = 0. a[19,14] = -.2055887686618382619339821759221121764364e-1 a[19,15] = .775379526471029807261782993777862395844e-1 a[19,16] = .1046259220352544296313761971333987587377 a[19,17] = -.1179213306451979352145022687063013455111 a[19,18] = 0. # #******************************************************** # # Coupling coefficients for c[20] = .7 # ---------------------------------------------------- a[20,1] = .3737341446457825692757506548800094134977e-1 a[20,2] = 0. a[20,3] = 0. a[20,4] = 0. a[20,5] = 0. a[20,6] = .3504930705338316406767087468339071089224 a[20,7] = .4922652819373025433298989824173484805373 a[20,8] = 8.553695439359312242284304421725315855379 a[20,9] = -10.35317299030591348532574006719207803272 a[20,10] = 13.83320427252914990351082875460544773493 a[20,11] = -12.28092433078461863729523583784519048012 a[20,12] = .1719151595656509762746810113378644307112 a[20,13] = 0. a[20,14] = .3641583114314496380113822384214528216140e-1 a[20,15] = .2961920580288763054890146412520723429115e-1 a[20,16] = -.2651793938627067002647615623738425030047 a[20,17] = .942950396173806655317007970358739475630e-1 a[20,18] = 0. a[20,19] = 0. # #******************************************************** # . # Coupling coefficients for c[21] = .9 # ---------------------------------------------------- a[21,1] = .3939058345528250943410670634923521987132e-1 a[21,2] = 0. a[21,3] = 0. a[21,4] = 0. a[21,5] = 0. a[21,6] = .3558516141234424183136697322755323715063 a[21,7] = .4197382225952610029372225526720065366258 a[21,8] = .872044977807194166293172525204036071060 a[21,9] = .898952083487659486126627160171417043611 a[21,10] = -.630580616105988359023456649527853470403 a[21,11] = -1.121887220595483550736681645425215081433 a[21,12] = .4298219512400197176967511031829197714867e-1 a[21,13] = 0. a[21,14] = .1332557566873915707013495891889190564164e-1 a[21,15] = .1876227053964148034446101291928097773800e-1 a[21,16] = -.1859411132922105570515379368592596513699 a[21,17] = .1773614271924602745226064729836361000042 a[21,18] = 0. a[21,19] = 0. a[21,20] = 0. # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi8 WITH 21 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi8[1] bi8[1,1] = 1. u bi8[1,2] = -10.03915465055451898280745009553727015838 u^2 bi8[1,3] = 53.79210495862331394937504547285261606206 u^3 bi8[1,4] = -165.0579057235472167092186792753028629327 u^4 bi8[1,5] = 298.0264565434610102489744601822776142620 u^5 bi8[1,6] = -311.9125448707900689751032283191627986699 u^6 bi8[1,7] = 174.6059852691171542761046061351126284335 u^7 bi8[1,8] = -40.37066163211959429657758663355894180800 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[2] bi8[2,1] = 0. u bi8[2,2] = 0. u^2 bi8[2,3] = 0. u^3 bi8[2,4] = 0. u^4 bi8[2,5] = 0. u^5 bi8[2,6] = 0. u^6 bi8[2,7] = 0. u^7 bi8[2,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[3] bi8[3,1] = 0. u bi8[3,2] = 0. u^2 bi8[3,3] = 0. u^3 bi8[3,4] = 0. u^4 bi8[3,5] = 0. u^5 bi8[3,6] = 0. u^6 bi8[3,7] = 0. u^7 bi8[3,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[4] bi8[4,1] = 0. u bi8[4,2] = 0. u^2 bi8[4,3] = 0. u^3 bi8[4,4] = 0. u^4 bi8[4,5] = 0. u^5 bi8[4,6] = 0. u^6 bi8[4,7] = 0. u^7 bi8[4,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[5] bi8[5,1] = 0. u bi8[5,2] = 0. u^2 bi8[5,3] = 0. u^3 bi8[5,4] = 0. u^4 bi8[5,5] = 0. u^5 bi8[5,6] = 0. u^6 bi8[5,7] = 0. u^7 bi8[5,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[6] bi8[6,1] = 0. u bi8[6,2] = 158.1976739121776138067531004299642556045 u^2 bi8[6,3] = -1543.961417219490013383329186557376850919 u^3 bi8[6,4] = 6241.398747828780065219699818963300847515 u^4 bi8[6,5] = -13136.51615640610824674042591770724411138 u^5 bi8[6,6] = 15106.94849316959941770760848348143558467 u^6 bi8[6,7] = -8996.489626298230413000758717864256649583 u^7 bi8[6,8] = 2170.776389952444021264933974457050280938 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[7] bi8[7,1] = 0. u bi8[7,2] = 110.7811520079778201620910891542159716196 u^2 bi8[7,3] = -1081.190514535617748557462051373884811281 u^3 bi8[7,4] = 4370.666940459977376891679103587685016930 u^4 bi8[7,5] = -9199.113723922197066947453657458673365167 u^5 bi8[7,6] = 10578.94920962985483690180716390515207397 u^6 bi8[7,7] = -6299.975594978841008450271944308599363057 u^7 bi8[7,8] = 1520.130500554341433782477059435641543286 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[8] bi8[8,1] = 0. u bi8[8,2] = -7011.442038211314089634068023254940106045 u^2 bi8[8,3] = 68429.55220744077890209519664603903716349 u^3 bi8[8,4] = -276623.5714822198169288202316196287008724 u^4 bi8[8,5] = 582220.4545548494658856503006312634684934 u^5 bi8[8,6] = -669551.5244611245601905652331468068626208 u^6 bi8[8,7] = 398731.3087623332757943809792249308827732 u^7 bi8[8,8] = -96210.47174510666745715793578288559674281 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[9] bi8[9,1] = 0. u bi8[9,2] = 11206.39756984814734031374482605836502113 u^2 bi8[9,3] = -109371.0485495066182770525095928736321803 u^3 bi8[9,4] = 442127.8393698154661543505844693555049508 u^4 bi8[9,5] = -930563.7629864562145364082427559715712707 u^5 bi8[9,6] = 1070145.133585590072636708771436125254933 u^6 bi8[9,7] = -637292.8058429046904373075590712408701797 u^7 bi8[9,8] = 153773.3309185793956820086499888593205888 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[10] bi8[10,1] = 0. u bi8[10,2] = -14179.23164045568390825368995504736244876 u^2 bi8[10,3] = 138385.0093196357218693716546019209270760 u^3 bi8[10,4] = -559415.5490240869974273158302752589638112 u^4 bi8[10,5] = 1177423.794699250413603625249340565972051 u^5 bi8[10,6] = -1354033.322790821429356166591306087001182 u^6 bi8[10,7] = 806353.8938825050195016379699232308969498 u^7 bi8[10,8] = -194566.3328138133045593670938904445416121 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[11] bi8[11,1] = 0. u bi8[11,2] = 10247.76176792174468727263230424253072668 u^2 bi8[11,3] = -100015.0532637523107509874155382267979521 u^3 bi8[11,4] = 404306.6240143429367125014776377339233105 u^4 bi8[11,5] = -850959.9711689702682710993795157496434280 u^5 bi8[11,6] = 978601.0462088684697300958464199995189771 u^6 bi8[11,7] = -582776.4729907748855939796622931794117500 u^7 bi8[11,8] = 140619.0037156383022701488158207833280861 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[12] bi8[12,1] = 0. u bi8[12,2] = -105.4930397685096787379931952745881034169 u^2 bi8[12,3] = 1029.580139580310194120073236423148130618 u^3 bi8[12,4] = -4162.034181876452751021493197688100770349 u^4 bi8[12,5] = 8759.996193602336131526447045580160767641 u^5 bi8[12,6] = -10073.96555688604885441046004449728532151 u^6 bi8[12,7] = 5999.247741473950186438936812025268574829 u^7 bi8[12,8] = -1447.567428588892382130036646632729629570 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[13] bi8[13,1] = 0. u bi8[13,2] = 0. u^2 bi8[13,3] = 0. u^3 bi8[13,4] = 0. u^4 bi8[13,5] = 0. u^5 bi8[13,6] = 0. u^6 bi8[13,7] = 0. u^7 bi8[13,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[14] bi8[14,1] = 0. u bi8[14,2] = -14.86361337326743122469601010648237947608 u^2 bi8[14,3] = 145.7635936489486611601020590400812969906 u^3 bi8[14,4] = -587.6557063401913588520708808169444817103 u^4 bi8[14,5] = 1227.372151254555709980234511427063838550 u^5 bi8[14,6] = -1394.493105740553645217117387304216418608 u^6 bi8[14,7] = 816.8562950730668774494805290335070403105 u^7 bi8[14,8] = -192.9796145225588132959328212730088960570 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[15] bi8[15,1] = 0. u bi8[15,2] = 14.34968575290546223276673100484047073648 u^2 bi8[15,3] = -150.2949344481665658851785896351738227010 u^3 bi8[15,4] = 629.4812425700290706612346725243246098946 u^4 bi8[15,5] = -1352.518207309060677914698908083510085133 u^5 bi8[15,6] = 1575.896933708880305858556996706058962503 u^6 bi8[15,7] = -946.7876580472948045886633971120598201035 u^7 bi8[15,8] = 229.8729377727072096359824945955196848017 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[16] bi8[16,1] = 0. u bi8[16,2] = -102.5452470111040085560664290210906322518 u^2 bi8[16,3] = 1074.032661264680594125263250545103109541 u^3 bi8[16,4] = -4498.377917100410634753487685261882069653 u^4 bi8[16,5] = 9665.320624003280508099125255751992581938 u^5 bi8[16,6] = -11261.62224831288113545795903649800929060 u^6 bi8[16,7] = 6765.902468760784366342575368188597359812 u^7 bi8[16,8] = -1642.710341604349689799450723704711058784 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[17] bi8[17,1] = 0. u bi8[17,2] = -38.13206313286473398334122725888547021750 u^2 bi8[17,3] = 399.3854658292328681862496726489289700594 u^3 bi8[17,4] = -1672.748720491971752312231602599596419744 u^4 bi8[17,5] = 3594.107254858566583822606674735752304040 u^5 bi8[17,6] = -4187.701556802926199931725021751236897492 u^6 bi8[17,7] = 2515.941280649063720613355430002270532846 u^7 bi8[17,8] = -610.8516609091004863949139257772330194915 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[18] bi8[18,1] = 0. u bi8[18,2] = -66.38279583069588062871084016403504860018 u^2 bi8[18,3] = 595.8297683881103280237377269355990794854 u^3 bi8[18,4] = -2188.737060092971609278770563269347103559 u^4 bi8[18,5] = 4213.839795282852421559730676511794767863 u^5 bi8[18,6] = -4484.035731929196864370162258757955490985 u^6 bi8[18,7] = 2500.648251425346544829791147364129986790 u^7 bi8[18,8] = -571.1622272434449401356158886201861909946 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[19] bi8[19,1] = 0. u bi8[19,2] = -90.41887573173058787343992868450872085904 u^2 bi8[19,3] = 931.9503884048153706496188381219698380844 u^3 bi8[19,4] = -3962.898377713156165984683269799703910403 u^4 bi8[19,5] = 8733.317420025551238329244389917866097896 u^5 bi8[19,6] = -10445.90818988766053535212385670877957360 u^6 bi8[19,7] = 6426.218942917598693647793004359979629852 u^7 bi8[19,8] = -1592.261308015418013416409177206823360972 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[20] bi8[20,1] = 0. u bi8[20,2] = -59.73884363038871206457816967313835076801 u^2 bi8[20,3] = 544.8870146891724527559861176467523778088 u^3 bi8[20,4] = -2090.430374926312850791322527518588562537 u^4 bi8[20,5] = 4194.418982707226648046953315742901721971 u^5 bi8[20,6] = -4603.369436819628073439413527693451638704 u^6 bi8[20,7] = 2619.201413559297614510795648037620577207 u^7 bi8[20,8] = -604.9687555793670790184208565420961249773 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[21] bi8[21,1] = 0. u bi8[21,2] = -59.20053764683937384859682230934791521325 u^2 bi8[21,3] = 571.7660156218088014286377638724659591261 u^3 bi8[21,4] = -2308.949564445360683785335401047607870804 u^4 bi8[21,5] = 4881.234110686139058221334453291392021952 u^5 bi8[21,6] = -5660.118807771202003386701685793459298252 u^6 bi8[21,7] = 3408.706689037421803199133730396931709513 u^7 bi8[21,8] = -833.4379054819676018284720384103746063216 u^8 # #******************************************************** #Norms of low order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- 0.1000000000 -.8604606225e-5 .1660859515e-4 0.2000000000 -.1035624985e-4 .2010612008e-4 0.3000000000 .2605644067e-5 .6026468628e-5 0.4000000000 .8752903197e-5 .1719724448e-4 0.5000000000 .1061549801e-4 .2009152788e-4 0.6000000000 .5425021990e-5 .9968363589e-5 0.7000000000 -.4239610731e-5 .9493267907e-5 0.8000000000 -.5690887147e-5 .1342251069e-4 0.9000000000 -.2921330306e-5 .6618282044e-5 1.0000000000 -.2073660714e-38 .5085671171e-38 1.1000000000 -.5789313887e-5 .1608720004e-4 1.2000000000 .6258937006e-4 .1211112232e-3 1.3000000000 .3147095520e-3 .5783318358e-3 1.4000000000 .1082463834e-2 .2099310034e-2 1.5000000000 .2980043940e-2 .6214922149e-2 1.6000000000 .7069722660e-2 .1584757337e-1 1.7000000000 .1506110470e-1 .3613616453e-1 1.8000000000 .2956060998e-1 .7557420622e-1 1.9000000000 .5437688342e-1 .1475311521 2.0000000000 .9488788519e-1 .2722341665 # #******************************************************** #Norms of high order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- 0.1000000000 .5668879224e-6 .1758596199e-5 0.2000000000 -.1437119671e-5 .3075049233e-5 0.3000000000 -.2049843371e-5 .4043765109e-5 0.4000000000 -.1611109226e-5 .3637167156e-5 0.5000000000 .1211116351e-5 .2901278908e-5 0.6000000000 .1322396099e-5 .3043272325e-5 0.7000000000 .1084613037e-5 .2817517323e-5 0.8000000000 .5544122070e-6 .1383333910e-5 0.9000000000 .3808958959e-6 .1195384935e-5 1.0000000000 -.1060544186e-6 .2827866033e-6 1.1000000000 .5870394626e-5 .1859240985e-4 1.2000000000 .6067881342e-4 .1920098134e-3 1.3000000000 .3080434709e-3 .9711353584e-3 1.4000000000 .1120440043e-2 .3520408520e-2 1.5000000000 .3324295692e-2 .1041481959e-1 1.6000000000 .8566646664e-2 .2677274604e-1 1.7000000000 .1987921545e-1 .6199572134e-1 1.8000000000 .4250840492e-1 .1323249923 1.9000000000 .8509361698e-1 .2644673158 2.0000000000 .1612914735 .5005889135 #********************************************************#