###################################################################### # # A better `efficient' Runge--Kutta (16:8,9) pair # # These are approximate REAL coefficients computed # using MAPLE with 40 digits for # # a slightly more efficient SIXTEEN-stage conventional # pair of methods of orders p=8 and p=9, with # dominant stage order = 5, # # together with approximate 40-digit coefficients # for two interpolants of orders 8 and 9 that # require 5 and 5 extra stages respectively. # # Exact coefficients of the method are possible using # surds in terms of sqrt(6). Exact coefficients for the # interpolants require that c[18] must be a zero of a # cubic polynomial with coefficients that are surds in # terms of sqrt(6). Hence, exact coefficients for the # interpolants cannot be conveniently represented. # # This procedure is a better efficient pair in the # sense that for an unrestricted maximum coefficient # from b and A, it has a propagating formula with # a minimum 2-norm of the local truncation error as # T_10,2 ~ .00000034399 # This 2-norm is slightly smaller than T10,2 for the # (16:8,9) pair tabulated above. This 2-norm has two # local maximum values on [0,1]. # # Additional stages and interpolating weights allow # for the computation of an approximation of order up # to order p, at any point of the domain of solution. # These interpolants have continuous derivatives. # # Nodes c[17]=1, c[18] - c[21] were selected to # minimize the maximum of the 2-norm of the local # truncation error over the interval [0,1] for the # interpolant of order 8, and this value is # # Ti_9,2 ~ .000003721 # # This 2-norm has three local maximum values on [0,1]. # # The remaining five 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 9, and this value is # # Ti_10,2 ~ .0000007830 # # This 2-norm has two local maximum values on [0,1]. # # The formulas scanned for this optimal formula are # those developed in J.H. Verner, SIAM J 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, Applied Numerical Mathematics 22 # 1996, 345--357, "High-order explicit Runge--Kutta pairs # with low stage order", which also require 16 stages, # but solve the order conditions in a different way # may yield particular pairs of equivalant or more # efficiency. Coefficients for the interpolants are # those developed in J.H. Verner, SIAM J NA 1993, # 1446-1466, "Differentiable Interpolants for high order # Runge--Kutta methods". # # Instructions for using the interpolants are contained # in J.H. Verner, SIAM NA 30, 1993, 1446-1466, "Differentiable # Interpolants for high-order Runge--Kutta methods". # ###################################################################### # NODES # ----- c[1] = 0 c[2] = .3571e-1 c[3] = .9906028091267414072062290548990445721420e-1 c[4] = .1485904213690112110809343582348566858213 c[5] = .6134 c[6] = .2327359473605626756631680289030288192566 c[7] = .5538640526394373243368319710969711807434 c[8] = .6555 c[9] = .491625 c[10] = .6858e-1 c[11] = .253 c[12] = .6620641795412045944786226689660879617864 c[13] = .8309 c[14] = .8998 c[15] = 1 c[16] = 1 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- # for c[1] = 0 # # for c[2] = .3571e-1 a[2,1] = .3571e-1 # # for c[3] = .9906028091267414072062290548990445721420e-1 a[3,1] = -.3833735636677017025757228807792426014327e-1 a[3,2] = .1373976372794443109781951935678287173575 # # for c[4] = .1485904213690112110809343582348566858213 a[4,1] = .3714760534225280277023358955871417145532e-1 a[4,2] = 0 a[4,3] = .1114428160267584083107007686761425143660 # # for c[5] = .6134 a[5,1] = 2.674764429871505119194043347886848033362 a[5,2] = 0 a[5,3] = -9.982382134885293836441318015951354323506 a[5,4] = 7.921017705013788717247274668064506290145 # # for c[6] = .2327359473605626756631680289030288192566 a[6,1] = .5242104050577351069841401193318449756747e-1 a[6,2] = 0 a[6,3] = 0 a[6,4] = .1796911189175953081279466649977364298948 a[6,5] = .6237879371938568368073519721078917943307e-3 # # for c[7] = .5538640526394373243368319710969711807434 a[7,1] = .1592492223647632083060991150129732726703 a[7,2] = 0 a[7,3] = 0 a[7,4] = -.4298429877241087508189185066909803751595 a[7,5] = .6665266542726088051243445942394404211672e-1 a[7,6] = .7578051525715219863372169033510342411158 # # for c[8] = .6555 a[8,1] = .7283333333333333333333333333333333333333e-1 a[8,2] = 0 a[8,3] = 0 a[8,4] = 0 a[8,5] = 0 a[8,6] = .3359344590665103678713422141936031057620 a[8,7] = .2467322076001562987953244524730635609046 # # for c[9] = .491625 a[9,1] = .729755859375e-1 a[9,2] = 0 a[9,3] = 0 a[9,4] = 0 a[9,5] = 0 a[9,6] = .3348009729699333533129043555243262838993 a[9,7] = .1184158239050666466870956444756737161007 a[9,8] = -.345673828125e-1 # # for c[10] = .6858e-1 a[10,1] = .4911213663452096382929799914888692571185e-1 a[10,2] = 0 a[10,3] = 0 a[10,4] = 0 a[10,5] = 0 a[10,6] = .3983857361308652347066808423646954244674e-1 a[10,7] = .1069675288939354812077139476183615284408 a[10,8] = -.2174259165458647598455650055976622663086e-1 a[10,9] = -.1055956474869564925231235304439517699685 # # for c[11] = .253 a[11,1] = -.2707988818641280489175862864774930636734e-1 a[11,2] = 0 a[11,3] = 0 a[11,4] = 0 a[11,5] = 0 a[11,6] = .333e-1 a[11,7] = -.1645526070036057075627040519129748090051 a[11,8] = .3428266306497389946775103799535438271025e-1 a[11,9] = .1585264064439221046855989221358760453749 a[11,10] = .2185234256811225083011127204294936872873 # # for c[12] = .6620641795412045944786226689660879617864 a[12,1] = .5584657769108862727526281006220448941548e-1 a[12,2] = 0 a[12,3] = 0 a[12,4] = 0 a[12,5] = 0 a[12,6] = .9166533166672539087479545790553151183505e-1 a[12,7] = .2392399655523627049569179599100121305027 a[12,8] = .1023834712248414879740316672186561113729e-1 a[12,9] = -.2679331322859542442991625976085644567159e-2 a[12,10] = .4235624181474284646680744174197106205439e-1 a[12,11] = .2253970470166604185504274586005888014087 # # for c[13] = .8309 a[13,1] = -.4802510512725195893103945790000307739284 a[13,2] = 0 a[13,3] = 0 a[13,4] = 0 a[13,5] = 0 a[13,6] = -6.359610162555930097843606206238737368598 a[13,7] = -.2762313898040841385163666588363199062208 a[13,8] = -6.500796633979846747011407331804389650124 a[13,9] = .5734765877040956875980719881692630407295 a[13,10] = 1.347125994868138849716719646646049357523 a[13,11] = 5.936840409706221306933249144503479736029 a[13,12] = 6.590346245333924728433733996560685564589 # # for c[14] = .8998 a[14,1] = .3307533067671401079975082476263855100019 a[14,2] = 0 a[14,3] = 0 a[14,4] = 0 a[14,5] = 0 a[14,6] = 5.956207776829962111958631943398154563177 a[14,7] = -.4868316400481527952968983900652017328476 a[14,8] = 4.462055288206771191616083778225319083232 a[14,9] = .7410258231442071778525406966672954745201 a[14,10] = -.7118192034575913119318346652024763899638 a[14,11] = -5.454619594516665440649148499095762263275 a[14,12] = -4.140803729244709693305868672684077643030 a[14,13] = .2038319723190386517589855611303633981843 # # for c[15] = 1 a[15,1] = -.5847111122998944389651719860007729221315 a[15,2] = 0 a[15,3] = 0 a[15,4] = 0 a[15,5] = 0 a[15,6] = -12.41268417116267068926060969930886386670 a[15,7] = 1.360245445660928146977003413153919520189 a[15,8] = -22.42610531111868229005965344123311160860 a[15,9] = -.8828857055865458252669099251378431470909 a[15,10] = 1.770155128538230466813900572142546329490 a[15,11] = 12.15809651918533877218739262890752294243 a[15,12] = 22.23037520407760703834156132941550000589 a[15,13] = -.6634483760201249006317401316871863808959 a[15,14] = .4509623787258137198642272397482891274306 # # for c[16] = 1 a[16,1] = 1.940575549810648717395482898478643635949 a[16,2] = 0 a[16,3] = 0 a[16,4] = 0 a[16,5] = 0 a[16,6] = 21.97798408114556310164832153356570429956 a[16,7] = .8230747326984728145128172720328075817426 a[16,8] = 68.16441683626354817206843139989369741474 a[16,9] = -3.117097463620266666801525529574905518624 a[16,10] = -4.568841021822439620303730003755793384193 a[16,11] = -18.74190987126264955904919353670543412590 a[16,12] = -66.57711839637831878350036856645874492472 a[16,13] = 1.098915553165441824029764532524025021446 a[16,14] = 0 a[16,15] = 0 # # ******************************************************** # High order weights c[17] = 1 # i.e. This is the propagating stage, and stage 17 as well. # ------------------------------------------------------- # b[1] = .15006690149797247957662887123770409800223878439125635509305e-1 b[2] = 0 b[3] = 0 b[4] = 0 b[5] = 0 b[6] = 0 b[7] = 0 b[8] = -1.05518099274638127859438168508018447470958212998793507706900 b[9] = .238494726378218311263814055777485195395241336180778947138354 b[10] = .128815177428299135462251588714408117237111752488648622524665 b[11] = .227662311104621561461491789666574650857365714720222465635242 b[12] = 1.22953258743751744316032181517357642189586257623541890214594 b[13] = .462497666281038348739730827525612863770345382666305044524257e-1 b[14] = .138619631936629390362830862120184349246824228027665359337989 b[15] = .308001016831943540520356037516240438999181056294446403250787e-1 b[16] = 0 # # ******************************************************** # Low order weights c[extra] = 1 # ------------------------------------------------------- # bh[1] = .1897210532481101330735875918987801428423e-1 bh[2] = 0 bh[3] = 0 bh[4] = 0 bh[5] = 0 bh[6] = 0 bh[7] = 0 bh[8] = 3.408110314549493848404398964228060776034 bh[9] = .1260323883820920906560270507988839661845 bh[10] = .1188375063451149770930378540709095182997 bh[11] = .2491041997838687569190073177760326138188 bh[12] = -3.269966219928978218713853055139116510335 bh[13] = .3023798100228882907409723963501699715077 bh[14] = 0 bh[15] = 0 bh[16] = .4652989552070924159305071272518165020637e-1 # #*******************************************************` # # Largest coefficient in b or A has magnitude `, 68.16442 # #*******************************************************` # SUMMARY OF NORMS OF ERRORS: A101, A102, A10inf` # ----------------------------------------------------` # A_[10, 1] = `, .6298637441e-5 # A_[10, 2] = `, .3439902103e-6 # A_[10,oo] = `, .5849302357e-7 #**************************************************** # # 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 [`, -4.431330766, `,0]` # # Stability Boundaries of Low Order Method` # ----------------------------------------` # Real Stability Interval is nearly [`, -3.828196857, `,0]` # ############################################################ # # START OF GENERATION OF INTERPOLANT # # ******************************************************** # # FIVE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 8 # # Coupling coefficients for j from 1 to 16 # for node c[17] = 1 # ---------------------------------------------------- a[17,1] = .1500669014979724795766288712377040980022e-1 a[17,2] = 0 a[17,3] = 0 a[17,4] = 0 a[17,5] = 0 a[17,6] = 0 a[17,7] = 0 a[17,8] = -1.055180992746381278594381685080184474710 a[17,9] = .2384947263782183112638140557774851953952 a[17,10] = .1288151774282991354622515887144081172371 a[17,11] = .2276623111046215614614917896665746508574 a[17,12] = 1.229532587437517443160321815173576421896 a[17,13] = .4624976662810383487397308275256128637703e-1 a[17,14] = .1386196319366293903628308621201843492468 a[17,15] = .3080010168319435405203560375162404389992e-1 a[17,16] = 0 # # ******************************************************** # # Coupling coefficients for j from 1 to 16 # for node c[17] = 1. # ---------------------------------------------------- c[17] = 1 a[17,1] = .1500669014979724795766288712377040980022e-1 a[17,2] = 0 a[17,3] = 0 a[17,4] = 0 a[17,5] = 0 a[17,6] = 0 a[17,7] = 0 a[17,8] = -1.055180992746381278594381685080184474710 a[17,9] = .2384947263782183112638140557774851953952 a[17,10] = .1288151774282991354622515887144081172371 a[17,11] = .2276623111046215614614917896665746508574 a[17,12] = 1.229532587437517443160321815173576421896 a[17,13] = .4624976662810383487397308275256128637703e-1 a[17,14] = .1386196319366293903628308621201843492468 a[17,15] = .3080010168319435405203560375162404389992e-1 a[17,16] = 0 # ******************************************************** # # Coupling coefficients for j from 1 to 17 # for node c[18] = .7375018139988810429214526568476200466933 # ---------------------------------------------------- a[18,1] = .1571801061417788021580499748292804743973e-1 a[18,2] = 0 a[18,3] = 0 a[18,4] = 0 a[18,5] = 0 a[18,6] = 0 a[18,7] = 0 a[18,8] = .4853403452657363080900487157345670578815 a[18,9] = .2107787568904546684574678306538806242458 a[18,10] = .1269802413053354152906551816812865612400 a[18,11] = .2319687014513919157472314236852033704833 a[18,12] = -.3620214714069096774945761015915844860748 a[18,13] = .5366106712036344076786687672191174826870e-1 a[18,14] = -.2806066613385549629370667652641771053798e-1 a[18,15] = -.2378121372710330174923994809623697280793e-1 a[18,16] = 0 a[18,17] = .2691804261928988988990035710208180655494e-1 # ******************************************************** # # Coupling coefficients for j from 1 to 18 # for node c[19] = .749 # ---------------------------------------------------- a[19,1] = .1569705832522204237038521016179271074328e-1 a[19,2] = 0 a[19,3] = 0 a[19,4] = 0 a[19,5] = 0 a[19,6] = 0 a[19,7] = 0 a[19,8] = .4616075242202111936044636619654233606961 a[19,9] = .2113946516698113234089715994366982070261 a[19,10] = .1270330917167109232354671760560477406684 a[19,11] = .2318540550298708286361100159494550129997 a[19,12] = -.3385266406688372942274816228586594758312 a[19,13] = .5298251972194235542699858758068190007453e-1 a[19,14] = -.2750461365887187780272257875389040879148e-1 a[19,15] = -.2361906185395527177468217904834875318962e-1 a[19,16] = 0 a[19,17] = .2668458089504036186694391521145359328377e-1 a[19,18] = .1139683460285541525554621429934611232038e-1 # ******************************************************** # # Coupling coefficients for j from 1 to 19 # for node c[20] = .65 # ---------------------------------------------------- a[20,1] = .1438964884291216402951708307100553583630e-1 a[20,2] = 0 a[20,3] = 0 a[20,4] = 0 a[20,5] = 0 a[20,6] = 0 a[20,7] = 0 a[20,8] = -1.206901219123788519767229935392578222389 a[20,9] = .2505628554639376294459724790398627595052 a[20,10] = .1303332915702127271150218433265955255131 a[20,11] = .2246717759263522068724895908997148014087 a[20,12] = 1.308419325781946480763221457213122900404 a[20,13] = .2589750180376235922034567811002287469857e-2 a[20,14] = .8070743254562856892787425005539711000810e-2 a[20,15] = -.1267568255392829384757055330207124946149e-1 a[20,16] = 0 a[20,17] = .1129158072373321675992773077865249355325e-1 a[20,18] = .3422056680709749787491027385482635813798e-1 a[20,19] = -.1149726368734142020610819623056729009783 # ******************************************************** # # Coupling coefficients for j from 1 to 20 # for node c[21] = .487 # ---------------------------------------------------- a[21,1] = .1452348029801041972968851600082942318526e-1 a[21,2] = 0 a[21,3] = 0 a[21,4] = 0 a[21,5] = 0 a[21,6] = 0 a[21,7] = 0 a[21,8] = -.5214243102465818231380106106018875132310 a[21,9] = .1866698844204603947548300448071067931901 a[21,10] = .1299316354451273004594203171440036506875 a[21,11] = .2262141085765719298250459612902177827497 a[21,12] = .6104582639466710747030070634860296980845 a[21,13] = .1418715607022412453299982942636029187870e-1 a[21,14] = .1480061054412245756808618945399364705863e-1 a[21,15] = -.3711471609871774475268028540581604884370e-2 a[21,16] = 0 a[21,17] = .1393256979572558890317889514899316376009e-2 a[21,18] = 1.114731020624073270253135239909350910479 a[21,19] = -1.021208555757145859230933049243060207745 a[21,20] = -.2795650792912340738723193626472621878286 # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi8 WITH 21 STAGES for i from 1 to 21, bi8[i] = SUM_j=1^8 bi8[i,j]u^j # ----------------------------------------- # # COEFFICIENTS OF bi8[1] bi8[1,1] = 1 u bi8[1,2] = -11.00917227309954827048799176229874144363 u^2 bi8[1,3] = 53.60376844185445526077321090756616710855 u^3 bi8[1,4] = -143.7818601051773394006248841909965394832 u^4 bi8[1,5] = 227.7802272724059672847516567630398210649 u^5 bi8[1,6] = -212.7667774745397515281313247919614156196 u^6 bi8[1,7] = 108.4765585030943501072304151917413023749 u^7 bi8[1,8] = -23.28773767438833620555341922996682359208 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] = 0 u^2 bi8[6,3] = 0 u^3 bi8[6,4] = 0 u^4 bi8[6,5] = 0 u^5 bi8[6,6] = 0 u^6 bi8[6,7] = 0 u^7 bi8[6,8] = 0 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[7] bi8[7,1] = 0 u bi8[7,2] = 0 u^2 bi8[7,3] = 0 u^3 bi8[7,4] = 0 u^4 bi8[7,5] = 0 u^5 bi8[7,6] = 0 u^6 bi8[7,7] = 0 u^7 bi8[7,8] = 0 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[8] bi8[8,1] = 0 u bi8[8,2] = 179.0879062178872372197637635178056034051 u^2 bi8[8,3] = -1728.403136057631342220019375555018665689 u^3 bi8[8,4] = 6564.460940561142416685202971494559376099 u^4 bi8[8,5] = -12902.91081584121105690497909548869952323 u^5 bi8[8,6] = 13965.17977100265059245172194550171929806 u^6 bi8[8,7] = -7920.424061687375443376559247327959298606 u^7 bi8[8,8] = 1841.954214811791214866274656172513025486 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[9] bi8[9,1] = 0 u bi8[9,2] = -22.88304714767390548521469744522669995060 u^2 bi8[9,3] = 238.8040507685294155366494768970788101399 u^3 bi8[9,4] = -936.8585782296073752493303117453241744494 u^4 bi8[9,5] = 1874.044957762151353867893217234671921697 u^5 bi8[9,6] = -2050.511559353009354817713813307430993559 u^6 bi8[9,7] = 1171.508545192416250747220534524328950392 u^7 bi8[9,8] = -273.8658742664281662882405921023203290746 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[10] bi8[10,1] = 0 u bi8[10,2] = 12.17331549628124801002349940562808097269 u^2 bi8[10,3] = -82.75076476436870272146201057456216825170 u^3 bi8[10,4] = 256.4804438751307755574323340365457074340 u^4 bi8[10,5] = -441.0999429758310962594951067937304764418 u^5 bi8[10,6] = 434.5079831401213158642449792005352310524 u^6 bi8[10,7] = -229.8934605896900265488669050173042421551 u^7 bi8[10,8] = 50.71124099578478523358546133160227550674 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[11] bi8[11,1] = 0 u bi8[11,2] = 2.613203520133622756551032901538192914958 u^2 bi8[11,3] = 16.88002937261184447692017262466190582959 u^3 bi8[11,4] = -134.1727921155237789914884572421173077178 u^4 bi8[11,5] = 340.1204801692637664465059992362156474712 u^5 bi8[11,6] = -420.1258893776879400857436087457229872331 u^6 bi8[11,7] = 258.3234372146557103657079225360621734929 u^7 bi8[11,8] = -63.41080647234860340699156952097105010691 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[12] bi8[12,1] = 0 u bi8[12,2] = -212.3127005913665992194796715806401319940 u^2 bi8[12,3] = 2045.352316395154364712249302798824723141 u^3 bi8[12,4] = -7762.060413644090906790449120787506530500 u^4 bi8[12,5] = 15250.14841459378132543421843792712597249 u^5 bi8[12,6] = -16501.08003544517859797013884904039476397 u^6 bi8[12,7] = 9356.907363957304758100332957460543520093 u^7 bi8[12,8] = -2175.725412678166826823572734962779212842 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[13] bi8[13,1] = 0 u bi8[13,2] = -11.50164708567429764283826551320712022931 u^2 bi8[13,3] = 107.2769704045725235564526349677807440972 u^3 bi8[13,4] = -401.2345332983947627677963563088489537002 u^4 bi8[13,5] = 781.8916989572930300323034641740782180816 u^5 bi8[13,6] = -841.6593371013254430188454534530483533077 u^6 bi8[13,7] = 475.5767391485588457657241425216173583523 u^7 bi8[13,8] = -110.3036412584017920901261933056193320075 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[14] bi8[14,1] = 0 u bi8[14,2] = -38.77239677340365800498371633092550416428 u^2 bi8[14,3] = 358.6391180152970136912934982919377109538 u^3 bi8[14,4] = -1336.215732140226920321570086014013189585 u^4 bi8[14,5] = 2598.193515313187034150047120315374589407 u^5 bi8[14,6] = -2792.878330051016777349083250990468387014 u^6 bi8[14,7] = 1576.586790342810048230642938513691709160 u^7 bi8[14,8] = -365.4143450747101110059836729234767444075 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[15] bi8[15,1] = 0 u bi8[15,2] = -10.00425469998852674211886984302245249578 u^2 bi8[15,3] = 91.67754241560498208460382885503842174321 u^3 bi8[15,4] = -340.0777600652848208705042811235806913159 u^4 bi8[15,5] = 659.6008369976978366087037536255754670281 u^5 bi8[15,6] = -707.8826284112975671515477305189138171574 u^6 bi8[15,7] = 399.1580030260127128211116842683795971037 u^7 bi8[15,8] = -92.44093916106142239619634965972490086213 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[16] bi8[16,1] = 0 u bi8[16,2] = 0 u^2 bi8[16,3] = 0 u^3 bi8[16,4] = 0 u^4 bi8[16,5] = 0 u^5 bi8[16,6] = 0 u^6 bi8[16,7] = 0 u^7 bi8[16,8] = 0 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[17] bi8[17,1] = 0 u bi8[17,2] = 16.21014491337561130525562931390759162785 u^2 bi8[17,3] = -150.4492131048662883122727122117594677554 u^3 bi8[17,4] = 565.1196631722152490306979317160273112948 u^4 bi8[17,5] = -1111.210282651553705321998904060968429058 u^5 bi8[17,6] = 1211.236976000843526549958975517785065570 u^6 bi8[17,7] = -695.3365606918091595268831805414223001360 u^7 bi8[17,8] = 164.4292723617947662752422602664302284566 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[18] bi8[18,1] = 0 u bi8[18,2] = -1872.805019318414830108065147723853146433 u^2 bi8[18,3] = 16400.66633207048634386587617517712424379 u^3 bi8[18,4] = -57270.16260166053287080817377636090729160 u^4 bi8[18,5] = 103330.5912960663516254356794429889210991 u^5 bi8[18,6] = -102051.1888173727465381067712242944167386 u^6 bi8[18,7] = 52424.75260874662694445820923552319700594 u^7 bi8[18,8] = -10961.85379853177067473675470531006517221 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[19] bi8[19,1] = 0 u bi8[19,2] = 1720.562129057968827268151794567408887461 u^2 bi8[19,3] = -15159.62046432254747381864100381352909840 u^3 bi8[19,4] = 53314.59561935523492911472686991966671330 u^4 bi8[19,5] = -96988.37468691576351426414353587220276161 u^5 bi8[19,6] = 96690.39181023687073280170142510495743481 u^6 bi8[19,7] = -50199.31248988246623378558547060878127074 u^7 bi8[19,8] = 10621.75808247070273268378992070248009519 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[20] bi8[20,1] = 0 u bi8[20,2] = 264.2470488396084725751013414957565817480 u^2 bi8[20,3] = -2232.338003905684731509741541080134626580 u^3 bi8[20,4] = 7453.753802419143998655808817000026688315 u^4 bi8[20,5] = -12770.71881400916730561960440987151459948 u^5 bi8[20,6] = 11903.25226767446224225175138830095252030 u^6 bi8[20,7] = -5733.155576507225740169825158561334353021 u^7 bi8[20,8] = 1114.959275488863063816509562716247788712 u^8 # ----------------------------------------- # # COEFFICIENTS OF bi8[21] bi8[21,1] = 0 u bi8[21,2] = -15.60551015563365366165870100287114141984 u^2 bi8[21,3] = 40.66145427098759539731834271499129986994 u^3 bi8[21,4] = 170.1538018759714061560683496064688819112 u^4 bi8[21,5] = -848.0568847386052608898820401778869464852 u^5 bi8[21,6] = 1373.524566531853560108596541516407906615 u^6 bi8[21,7] = -993.1678967729130171884598684827601522494 u^7 bi8[21,8] = 272.4904689883393700780173758256501517585 u^8 # ******************************************************** # # Coupling coefficients for j from 1 to 21 # for node c[22] = .97e-2 # ---------------------------------------------------- a[22,1] = .8711816186418632397551552134391273160887e-2 a[22,2] = 0 a[22,3] = 0 a[22,4] = 0 a[22,5] = 0 a[22,6] = 0 a[22,7] = 0 a[22,8] = .1532993247326560354009297246756198757646e-1 a[22,9] = -.1943250606288015181452523235039064995331e-2 a[22,10] = .1072095950570478409673351209542485874475e-2 a[22,11] = .2601233036074381124605301648955068308034e-3 a[22,12] = -.1817718521410219268803888649170562657549e-1 a[22,13] = -.9877668338996712912794853387601975290298e-3 a[22,14] = -.3332383192417756458522769371199819990865e-2 a[22,15] = -.8605833352714280922890371244764817761631e-3 a[22,16] = 0 a[22,17] = .1392810143886650671629507962655268250424e-2 a[22,18] = -.1617419987897630058856682050471490927435 a[22,19] = .1485156561306044255987175208430691701911 a[22,20] = .2289051095253062003982633508518682297336e-1 a[22,21] = -.1429777169141779172700863258972231247118e-2 # ******************************************************** # # Coupling coefficients for j from 1 to 22 # for node c[23] = .138 # ---------------------------------------------------- a[23,1] = .2710092628714179537584868171073201523422e-1 a[23,2] = 0 a[23,3] = 0 a[23,4] = 0 a[23,5] = 0 a[23,6] = 0 a[23,7] = 0 a[23,8] = .6923104986003885114772983986172611691902 a[23,9] = -.6725243132164494445073150422370697609664e-1 a[23,10] = .8808440659269460280192719417821072306037e-1 a[23,11] = .5982566312199630889836790781586188392095e-1 a[23,12] = -.8251290314815523858109567180972702723517 a[23,13] = -.4886453387508305661509501250020800230341e-1 a[23,14] = -.1682608371566511953625070499451460275789 a[23,15] = -.4443170503743608176862370639456174098455e-1 a[23,16] = 0 a[23,17] = .7037844639432780238885500484343585742713e-1 a[23,18] = -8.818686397504859095347055843377871115177 a[23,19] = 8.028821919603921347277176943010114087042 a[23,20] = 1.306591406499157862229641342220359676579 a[23,21] = -.1624883307224014710941456378572112779619 a[23,22] = 0 # ******************************************************** # # Coupling coefficients for j from 1 to 23 # for node c[24] = .249 # ---------------------------------------------------- a[24,1] = .1466442426857961230207869561545353695312e-1 a[24,2] = 0 a[24,3] = 0 a[24,4] = 0 a[24,5] = 0 a[24,6] = 0 a[24,7] = 0 a[24,8] = .1898399561399124450873282966092465494738 a[24,9] = .3711530651907537053995301753121055816571e-1 a[24,10] = .1316247981199510429999180335475037001729 a[24,11] = .1466622928822685713088228930531811168096 a[24,12] = -.2377331162467838377764310139745632904067 a[24,13] = -.2493071232192949761907468241344412740688e-1 a[24,14] = -.9427777049221307692559326299260474943801e-1 a[24,15] = -.2726667627690941070300686079610559177653e-1 a[24,16] = 0 a[24,17] = .4097659404063362064133671142526181935526e-1 a[24,18] = -5.538725321977494545432278985839620727708 a[24,19] = 4.973442175247713118429917494679987165227 a[24,20] = .8626294449189858956229369969895335684221 a[24,21] = -.2250213948217893084759073334350395278429 a[24,22] = 0 a[24,23] = 0 # ******************************************************** # # Coupling coefficients for j from 1 to 24 # for node c[25] = .439 # ---------------------------------------------------- a[25,1] = .1191252689920919922350166270042464848683e-1 a[25,2] = 0 a[25,3] = 0 a[25,4] = 0 a[25,5] = 0 a[25,6] = 0 a[25,7] = 0 a[25,8] = -.6723137858130876080443132301001699876552 a[25,9] = .1801284266825688933352476317750604591900 a[25,10] = .1365689447449347163335592887842694005196 a[25,11] = .2111038393798906615990851223053694372026 a[25,12] = .7775854726244728792738759030556073621755 a[25,13] = .2362117948505939633346199470372118351743e-1 a[25,14] = .6391325607581230059121991748062989083367e-1 a[25,15] = .1197654289246164466950805226357270574198e-1 a[25,16] = 0 a[25,17] = -.2188622145087031801816344581960035242509e-1 a[25,18] = 3.500218060218659348421859254969160434525 a[25,19] = -3.195765244251373581009030415350620096609 a[25,20] = -.5994401048600148264885439684062323011428 a[25,21] = .1137710737227729377873223163880721563872e-1 a[25,22] = 0 a[25,23] = 0 a[25,24] = 0 # ******************************************************** # # Coupling coefficients for j from 1 to 25 # for node c[26] = .794 # ---------------------------------------------------- a[26,1] = .1184012014074604271397800922251586670558e-1 a[26,2] = 0 a[26,3] = 0 a[26,4] = 0 a[26,5] = 0 a[26,6] = 0 a[26,7] = 0 a[26,8] = -.6641263289768790103963252230448945748496 a[26,9] = .1788914061873194018510863179806909040625 a[26,10] = .1367562642287085016908037148859544845819 a[26,11] = .2107758119730286112356534765669813731420 a[26,12] = .7679184744806335059253231677825531412084 a[26,13] = .2313496869580297355994450248598930620113e-1 a[26,14] = .6230605705999706953753333719939087802061e-1 a[26,15] = .1157099029193866324646273674695267419004e-1 a[26,16] = 0 a[26,17] = -.2129441601042184344575923744216391881547e-1 a[26,18] = 3.163166251075345445930896810245694869610 a[26,19] = -2.819544872276817190650813744596266559171 a[26,20] = -.3978769046309758754403585402077121882020 a[26,21] = .1304821777615737042415746721743137433164 a[26,22] = 0 a[26,23] = 0 a[26,24] = 0 a[26,25] = 0 # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi9 WITH 26 STAGES for i from 1 to 26, bi9[i] = SUM_j=1^9 bi9[i,j]u^j # ----------------------------------------- # # COEFFICIENTS OF bi9[1] bi9[1,1] = 1 u bi9[1,2] = -60.67156499096275825192551363602396761243 u^2 bi9[1,3] = 669.4173339890965298528834018742984105094 u^3 bi9[1,4] = -3377.878946225199351520226690513332256243 u^4 bi9[1,5] = 9286.468967391046692225695247237821541998 u^5 bi9[1,6] = -14780.47713681102054116333940623447596217 u^6 bi9[1,7] = 13604.99386328224720102647507806334531178 u^7 bi9[1,8] = -6724.705443356114485986782323617396379907 u^8 bi9[1,9] = 1381.867933411056511065177869712887072054 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[2] bi9[2,1] = 0 u bi9[2,2] = 0 u^2 bi9[2,3] = 0 u^3 bi9[2,4] = 0 u^4 bi9[2,5] = 0 u^5 bi9[2,6] = 0 u^6 bi9[2,7] = 0 u^7 bi9[2,8] = 0 u^8 bi9[2,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[3] bi9[3,1] = 0 u bi9[3,2] = 0 u^2 bi9[3,3] = 0 u^3 bi9[3,4] = 0 u^4 bi9[3,5] = 0 u^5 bi9[3,6] = 0 u^6 bi9[3,7] = 0 u^7 bi9[3,8] = 0 u^8 bi9[3,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[4] bi9[4,1] = 0 u bi9[4,2] = 0 u^2 bi9[4,3] = 0 u^3 bi9[4,4] = 0 u^4 bi9[4,5] = 0 u^5 bi9[4,6] = 0 u^6 bi9[4,7] = 0 u^7 bi9[4,8] = 0 u^8 bi9[4,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[5] bi9[5,1] = 0 u bi9[5,2] = 0 u^2 bi9[5,3] = 0 u^3 bi9[5,4] = 0 u^4 bi9[5,5] = 0 u^5 bi9[5,6] = 0 u^6 bi9[5,7] = 0 u^7 bi9[5,8] = 0 u^8 bi9[5,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[6] bi9[6,1] = 0 u bi9[6,2] = 0 u^2 bi9[6,3] = 0 u^3 bi9[6,4] = 0 u^4 bi9[6,5] = 0 u^5 bi9[6,6] = 0 u^6 bi9[6,7] = 0 u^7 bi9[6,8] = 0 u^8 bi9[6,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[7] bi9[7,1] = 0 u bi9[7,2] = 0 u^2 bi9[7,3] = 0 u^3 bi9[7,4] = 0 u^4 bi9[7,5] = 0 u^5 bi9[7,6] = 0 u^6 bi9[7,7] = 0 u^7 bi9[7,8] = 0 u^8 bi9[7,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[8] bi9[8,1] = 0 u bi9[8,2] = -.5972167605668447080195299461102185885272 u^2 bi9[8,3] = 47.93604836413167338885438264277611366496 u^3 bi9[8,4] = -565.7170417002332153647406623941231321861 u^4 bi9[8,5] = 2799.622707202858561151696501297251806365 u^5 bi9[8,6] = -7032.161794919678731491945450324980521259 u^6 bi9[8,7] = 9404.154583498766261417380396057323248088 u^7 bi9[8,8] = -6374.661804344304055026346155757703309071 u^8 bi9[8,9] = 1720.369337666279969354526136740485828512 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[9] bi9[9,1] = 0 u bi9[9,2] = .1349844707960069717220566866826298984923 u^2 bi9[9,3] = -10.83462914594452681222359026508445675428 u^3 bi9[9,4] = 127.8648231869934298266652232809372858609 u^4 bi9[9,5] = -632.7779367772187332338007906200698487630 u^5 bi9[9,6] = 1589.427325412255464804263544956685619289 u^6 bi9[9,7] = -2125.551246305557207529871063524735793374 u^7 bi9[9,8] = 1440.817483665754881321490009540733392622 u^8 bi9[9,9] = -388.8423097807010970369815759993713435843 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[10] bi9[10,1] = 0 u bi9[10,2] = .7290747606753265739553888434183906298602e-1 u^2 bi9[10,3] = -5.851972901033216313427390997286151384252 u^3 bi9[10,4] = 69.06202973876299355875761438134035666890 u^4 bi9[10,5] = -341.7744427162097135566927248434513873995 u^5 bi9[10,6] = 858.4775271201373000458419246386670445384 u^6 bi9[10,7] = -1148.047443579864082962792949259101806969 u^7 bi9[10,8] = 778.2107496409214137189987901610599544952 u^8 bi9[10,9] = -210.0205396013539280126185513768554408959 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[11] bi9[11,1] = 0 u bi9[11,2] = .1288534847345785561116203329301814360204 u^2 bi9[11,3] = -10.34252098059179829195550043573389974374 u^3 bi9[11,4] = 122.0572110662541635697477898873947855333 u^4 bi9[11,5] = -604.0372032137003894354892742901494195352 u^5 bi9[11,6] = 1517.235637581122757484801301501208740856 u^6 bi9[11,7] = -2029.008844152904067945761282194453917620 u^7 bi9[11,8] = 1375.375645376320937242441893999796663414 u^8 bi9[11,9] = -371.1811168501315596184350570113265596896 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[12] bi9[12,1] = 0 u bi9[12,2] = .6958971720762387362164791425187492767466 u^2 bi9[12,3] = -55.85670513574830649397484217620113261823 u^3 bi9[12,4] = 659.1926340795728211557235694809221532847 u^4 bi9[12,5] = -3262.215084140846354068888231363556701923 u^5 bi9[12,6] = 8194.112807588290125367213373592291507372 u^6 bi9[12,7] = -10958.03904467296551539422209426971705612 u^7 bi9[12,8] = 7427.971576643475789982515741208313369971 u^8 bi9[12,9] = -2004.632548946417281841423673799397312817 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[13] bi9[13,1] = 0 u bi9[13,2] = .2617668058132623028300450746186815539953e-1 u^2 bi9[13,3] = -2.101090775094604504096600471528167656924 u^3 bi9[13,4] = 24.79601256660032039131082591571668336326 u^4 bi9[13,5] = -122.7106039105787684003939157931166303491 u^5 bi9[13,6] = 308.2275402436813943146648421712827254341 u^6 bi9[13,7] = -412.1946451000669582597347477727378421190 u^7 bi9[13,8] = 279.4085780645544831469608316472195929468 u^8 bi9[13,9] = -75.40571800304908908412026712154566848821 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[14] bi9[14,1] = 0 u bi9[14,2] = .7845665161262074618502665073713581595690e-1 u^2 bi9[14,3] = -6.297381611696195354835186970953877598132 u^3 bi9[14,4] = 74.31851847203782644803150125553425852543 u^4 bi9[14,5] = -367.7877747056500197543410562506219227864 u^5 bi9[14,6] = 923.8184599908639672968037927810710633244 u^6 bi9[14,7] = -1235.428287659738944177317308989226766106 u^7 bi9[14,8] = 837.4423716056154246604113713897017147785 u^8 bi9[14,9] = -226.0057431111080504745753090041214216040 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[15] bi9[15,1] = 0 u bi9[15,2] = .1743239982411997083542666099618776565927e-1 u^2 bi9[15,3] = -1.399224563421080747566112426047905797818 u^3 bi9[15,4] = 16.51294188206731169216889160389405043672 u^4 bi9[15,5] = -81.71931133065192111822155020120302944168 u^5 bi9[15,6] = 205.2645942498128099152915051189419910813 u^6 bi9[15,7] = -274.5016441798805281613128379066882052764 u^7 bi9[15,8] = 186.0725630195001464648099116049710397144 u^8 bi9[15,9] = -50.21655137556766366195319885111250443823 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[16] bi9[16,1] = 0 u bi9[16,2] = 0 u^2 bi9[16,3] = 0 u^3 bi9[16,4] = 0 u^4 bi9[16,5] = 0 u^5 bi9[16,6] = 0 u^6 bi9[16,7] = 0 u^7 bi9[16,8] = 0 u^8 bi9[16,9] = 0 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[17] bi9[17,1] = 0 u bi9[17,2] = -.3120237034739356102149090910458972991542e-1 u^2 bi9[17,3] = 2.505323513858665425821148146925028556689 u^3 bi9[17,4] = -29.63209944182268891094187050958569485080 u^4 bi9[17,5] = 147.1735748991918299269418428659506735081 u^5 bi9[17,6] = -371.6682626297095949736614911358789582154 u^6 bi9[17,7] = 500.8032537625967383990239072447982187459 u^7 bi9[17,8] = -342.9490465144388046578978125156518258548 u^8 bi9[17,9] = 93.79845878067124835173576681254714784020 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[18] bi9[18,1] = 0 u bi9[18,2] = 1.005189184600557566129408870772027118073 u^2 bi9[18,3] = -80.14524762849856177105235169139266835049 u^3 bi9[18,4] = 904.2862871618540482735522275321299612604 u^4 bi9[18,5] = -4159.377900797260916077366144135436208057 u^5 bi9[18,6] = 9382.948440638739069789096069230867579132 u^6 bi9[18,7] = -10865.07436232189535132287509765058944820 u^7 bi9[18,8] = 6171.218731586134383883573674542623005216 u^8 bi9[18,9] = -1354.861137823673230341057786698974248115 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[19] bi9[19,1] = 0 u bi9[19,2] = -1.018790304892149008339208622085912341214 u^2 bi9[19,3] = 81.22968543433086703732306275594490877087 u^3 bi9[19,4] = -916.5220998408498750879913972119183817869 u^4 bi9[19,5] = 4215.658051870923057302442161815205855483 u^5 bi9[19,6] = -9509.908233268860695396133182984225322485 u^6 bi9[19,7] = 11012.08866136426918043466675789153630399 u^7 bi9[19,8] = -6254.720911673139287616309544212696219769 u^8 bi9[19,9] = 1373.193636418218902334341350568238768134 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[20] bi9[20,1] = 0 u bi9[20,2] = .1486756383162591888438322851466844348297e-1 u^2 bi9[20,3] = -1.185413256701197518560901363041446866220 u^3 bi9[20,4] = 13.37512808773948803650703726266929826638 u^4 bi9[20,5] = -61.52057482048160294775557914196592133996 u^5 bi9[20,6] = 138.7814224498311940264420678029949672145 u^6 bi9[20,7] = -160.7032676951998445023342766109082536156 u^7 bi9[20,8] = 91.27733347752088221300420564599480170832 u^8 bi9[20,9] = -20.03949580654054522618693682425811381100 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[21] bi9[21,1] = 0 u bi9[21,2] = -.2611963106691748849299256055037070183077e-2 u^2 bi9[21,3] = .2082557524387797755360510166314825074305 u^3 bi9[21,4] = -2.349769034664475310156364256215305171863 u^4 bi9[21,5] = 10.80805662268303734097853101791700152378 u^5 bi9[21,6] = -24.38139559637047359365680950022456660620 u^6 bi9[21,7] = 28.23266885539013812966436802211627788468 u^7 bi9[21,8] = -16.03578301196446470276182164155397173990 u^8 bi9[21,9] = 3.520578375594150109245344597384118672243 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[22] bi9[22,1] = 0 u bi9[22,2] = 61.75136568508868110286254101938221664703 u^2 bi9[22,3] = -750.1268910552884473493229930939554325311 u^3 bi9[22,4] = 3898.685586244841930988256009725848784326 u^4 bi9[22,5] = -10841.82272864745777154320543067614124209 u^5 bi9[22,6] = 17321.57566462516617366130887449349951271 u^6 bi9[22,7] = -15937.70492491813691151403407581204545506 u^7 bi9[22,8] = 7853.047625862506649651583373646970497362 u^8 bi9[22,9] = -1605.405697796720304997448299303558881367 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[23] bi9[23,1] = 0 u bi9[23,2] = -1.934298042224096808013043975920800289243 u^2 bi9[23,3] = 147.5886431665603647558155476856964112175 u^3 bi9[23,4] = -1181.453462187681932247747922750006635641 u^4 bi9[23,5] = 4124.816268317402451398433861948686547907 u^5 bi9[23,6] = -7708.555106592447263782269817638341609752 u^6 bi9[23,7] = 8052.349419898650085369881466158633077027 u^7 bi9[23,8] = -4443.023055054952034626751291729687203499 u^8 bi9[23,9] = 1010.211590494692425940651200300940213030 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[24] bi9[24,1] = 0 u bi9[24,2] = .9126428307869666507744889095249555815836 u^2 bi9[24,3] = -71.31710609019478576924959388866173624547 u^3 bi9[24,4] = 695.4561446075446391059957659052646164863 u^4 bi9[24,5] = -2620.882193436484363083577561241318740147 u^5 bi9[24,6] = 4957.008141154350233563895859921266117993 u^6 bi9[24,7] = -5038.298085544127560098020485475941711703 u^7 bi9[24,8] = 2633.333935059078624368760947592332675986 u^8 bi9[24,9] = -556.2134785809537547385794217224661779516 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[25] bi9[25,1] = 0 u bi9[25,2] = -.4654439970747991651050782556914121007009 u^2 bi9[25,3] = 37.09099810444154184865658933003385399191 u^3 bi9[25,4] = -417.1382228339854991229014922810663023285 u^4 bi9[25,5] = 1920.758799061188217073321625257641207277 u^5 bi9[25,6] = -4409.960156149581696033720218281819632770 u^6 bi9[25,7] = 5383.383503749768100112081288811655341483 u^7 bi9[25,8] = -3353.518501772742141737984950593448341369 u^8 bi9[25,9] = 839.8490238379862770256522360126952858160 u^9 # ----------------------------------------- # # COEFFICIENTS OF bi9[26] bi9[26,1] = 0 u bi9[26,2] = -.1176451708255218561268102928705214692156 u^2 bi9[26,3] = 9.481894819354298841374880327580666327841 u^3 bi9[26,4] = -114.9156758298319354820100563154045258044 u^4 bi9[26,5] = 591.3193291312467068002224871165564177657 u^5 bi9[26,6] = -1559.765475086581493834896780108830295683 u^6 bi9[26,7] = 2198.545841718649266979102957216738477165 u^7 bi9[26,8] = -1564.562048273728342299716850911579457005 u^8 bi9[26,9] = 440.0137786917170208520501729678092387026 u^9 # #******************************************************** # Norms of low order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, .1227222739e-5, .3348508463e-5 .2000000000, .1615356496e-5, .3721626366e-5 .3000000000, .6667570096e-6, .1125720884e-5 .4000000000, .3385524146e-6, .1371991824e-5 .5000000000, -.4979316877e-6, .1866845810e-5 .6000000000, -.3855275900e-6, .1577140382e-5 .7000000000, -.5333350301e-6, .1597787028e-5 .8000000000, -.8981064587e-6, .1968485379e-5 .9000000000, -.6973183102e-6, .1411646512e-5 1.000000000, .1966258333e-56, .4681281745e-56 1.100000000, -.3763579804e-5, .8746517281e-5 1.200000000, -.2896131271e-4, .7960902000e-4 1.300000000, -.1153029259e-3, .3883837105e-3 1.400000000, .4370475809e-3, .1421009138e-2 1.500000000, .1370581071e-2, .4332655075e-2 1.600000000, .3719164661e-2, .1157394481e-1 1.700000000, .9047924268e-2, .2790864305e-1 1.800000000, .2019562961e-1, .6198056425e-1 1.900000000, .4203146751e-1, .1286244962 2.000000000, .8253193184e-1, .2521568348 # # ******************************************************** # Norms of high order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------------- .1000000000, .5235834963e-7, .1519257373e-6 .2000000000, .2095914343e-6, .5424950965e-6 .3000000000, .3292408958e-6, .7830811505e-6 .4000000000, .3445670411e-6, .7603468569e-6 .5000000000, .3009343118e-6, .5990805940e-6 .6000000000, .2568176444e-6, .4484576869e-6 .7000000000, .2143238993e-6, .3562383778e-6 .8000000000, .1418727695e-6, .2683013543e-6 .9000000000, .5417476349e-7, .2733305329e-6 1.000000000, .5849302357e-7, .3439902103e-6 1.100000000, .8714506430e-6, .3576174326e-5 1.200000000, .1131375469e-4, .4623422258e-4 1.300000000, .6872897170e-4, .2815480596e-3 1.400000000, .2923912562e-3, .1200914903e-2 1.500000000, .9964163854e-3, .4101897910e-2 1.600000000, .2905974139e-2, .1198624121e-1 1.700000000, .7539401031e-2, .3114936931e-1 1.800000000, .1784337214e-1, .7382582027e-1 1.900000000, .3919834562e-1, .1623811325 2.000000000, .8094769658e-1, .3356926326 # ********************************************************