# Coefficients provided in this procedure are used in # explicit Runge--Kutta methods to obtain numerical # approximations of the solution to an initial value # problem in ordinary differential equations: # # y' = f(x,y), y(x_0) = y_0, # x in [x_0,b], y in R^m f: RxR^m -> R^m # # See J.H. Verner, Numerically optimal Runge--Kutta pairs # with interpolants. Numerical Algorithms, 53, (2010) # pp. 383--396 for instructions on using these coefficients. # Coefficients for the interpolants originally posted for # this procedure were in error - the high order interpolant # satisfied some but not all conditions of order 7. To # satisfy all conditions of order 7 using only 15 stages, # node c[12] needed to be a zero of a particular quadratic # polynomial. The coefficients below form one procedure # of a 10 stage pair of Runge-Kutta methods of orders # 6 and 7 together with an interpolant of order 6 using # 13 stages, and an interpolant of order 7 using 16 stages. # In application, stage 10 is used only for local error # estimation, and is not required for either interpolant. # ###################################################################### # # A 'most efficient' Runge--Kutta (10:7(6)) pair # # These are corrected approximate REAL coefficients # computed using MAPLE with 40 digits for # # a TEN-stage conventional pair methods of # orders p=6 and p=7, with dominant # stage order = 3, # # together with approximate coefficients for # two interpolants of orders 6 and 7 that # require 3 and 3 extra stages respectively. # # (Companion files list only the RATIONAL and 40-digit # floating point approximations respectively.) # # 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 # # T_82 ~ .00001675 # # (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. # # Node c[11]=1 was chosen to make the interpolants # differentiable. Node c[12] was chosen as a zero of a # quadratic polynomial to allow the computation of an # interpolant of order 7 in only 15 stages. Node c[13] # was chosen in attempting to minimize the 2-norm of # the interpolant of order 6. # # For these choices, the maximum value of the 2-norm # for the interpolant of order 6 on [0,1] is # # Ti_72 ~ .00002496 # # This 2-norm has two local maximum values on [0,1]. # # The remaining nodes were selected to minimize the # 2-norm of the error in the interpolant of order 7. # This 2-norm had a value of # # Ti_82 ~ .00001675, # # the 2-norm of the endpoint local truncation error. # Moreover, the 2-norm of this interpolant is bounded # by this value on [0,1], and monotone increasing. # # 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. # # 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] = .5e-2 c[3] = .1088888888888888888888888888888888888889 c[4] = .1633333333333333333333333333333333333333 c[5] = .4555 c[6] = .6095094489978381317087004421486024949638 c[7] = .884 c[8] = .9250000000000000000000000000000000000000 c[9] = 1 c[10] = 1 c[11] = 1 c[12] = .6716626365038747067708664674609860468131 c[13] = .125 c[14] = .25 c[15] = .53 c[16] = .79 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- # for c[1] = 0. # # for c[2] = .5e-2 a[2,1] = .5e-2 # # for c[3] = .1088888888888888888888888888888888888889 a[3,1] = -1.076790123456790123456790123456790123457 a[3,2] = 1.185679012345679012345679012345679012346 # # for c[4] = .16333333333333333333333333333333333333333 a[4,1] = .4083333333333333333333333333333333333333e-1 a[4,2] = 0 a[4,3] = .1225 # # for c[5] = .4555 a[5,1] = .6389139236255726780508121615993336109954 a[5,2] = 0 a[5,3] = -2.455672638223656809662640566430653894211 a[5,4] = 2.272258714598084131611828404831320283215 # # for c[6] = .60950944899783813170870044214860249496379 a[6,1] = -2.661577375018757131119259297861818119279 a[6,2] = 0 a[6,3] = 10.80451388645613769565396655365532838482 a[6,4] = -8.353914657396199411968048547819291691541 a[6,5] = .8204875949566569791420417341743839209619 # # for c[7] = .884 a[7,1] = 6.067741434696770992718360183877276714679 a[7,2] = 0 a[7,3] = -24.71127363591108579734203485290746001803 a[7,4] = 20.42751793078889394045773111748346612697 a[7,5] = -1.906157978816647150624096784352757010879 a[7,6] = 1.006172249242068014790040335899474187268 # # for c[8] = .925 a[8,1] = 12.05467007625320299509109452892778311648 a[8,2] = 0 a[8,3] = -49.75478495046898932807257615331444758322 a[8,4] = 41.14288863860467663259698416710157354209 a[8,5] = -4.461760149974004185641911603484815375051 a[8,6] = 2.042334822239174959821717077708608543738 a[8,7] = -.9834843665406107379530801693870224403537e-1 # # for c[9] = 1 a[9,1] = 10.13814652288180787641845141981689030769 a[9,2] = 0 a[9,3] = -42.64113603171750214622846006736635730625 a[9,4] = 35.76384003992257007135021178023160054034 a[9,5] = -4.348022840392907653340370296908245943710 a[9,6] = 2.009862268377035895441943593011827554771 a[9,7] = .3487490460338272405953822853053145879140 a[9,8] = -.2714390051048312842371587140910297407572 # # for c[10] = 1 a[10,1] = -45.03007203429867712435322405073769635151 a[10,2] = 0 a[10,3] = 187.3272437654588840752418206154201997384 a[10,4] = -154.0288236935018690596728621034510402582 a[10,5] = 18.56465306347536233859492332958439136765 a[10,6] = -7.141809679295078854925420496823551192821 a[10,7] = 1.308808578161378625114762706007696696508 a[10,8] = 0 a[10,9] = 0 # # ******************************************************** # High order weights c[ 11] = 1 # i.e. This is the propagating stage, and stage 11, as well. # ----------------------------------------------------------- # b[1] = .4715561848627222170431765108838175679569e-1 b[2] = 0 b[3] = 0 b[4] = .2575056429843415189596436101037687580986 b[5] = .2621665397741262047713863095764527711129 b[6] = .1521609265673855740323133199165117535523 b[7] = .4939969170032484246907175893227876844296 b[8] = -.2943031171403250441557244744092703429139 b[9] = .8131747232495109999734599440136761892478e-1 b[10] = 0 # # ******************************************************** # Low order weights C[extra]:= 1 # -------------------------------------------------- # bh[1] = .4460860660634117628731817597479197781432e-1 bh[2] = 0 bh[3] = 0 bh[4] = .2671640378571372680509102260943837899738 bh[5] = .2201018300177293019979715776650753096323 bh[6] = .2188431703143156830983120833512893824578 bh[7] = .2289871705411202883378173889763552365362 bh[8] = 0 bh[9] = 0 bh[10] = .2029518466335628222767054793810430358554e-1 # # ******************************************************* # # Largest coefficient in b or A has magnitude `, 187.3272 # ********************************************************* # SUMMARY OF NORMS OF ERRORS: A81, A82, A8inf` # ---------------------------------------------------- # A_[8, 1] = `, .1070085665e-3 # A_[8, 2] = `, .1675585009e-4 # A_[8,oo] = `, .8590002884e-5 #*********************************************************** # # 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 [ -4.640024337, 0] # # Stability Boundaries of Low Order Method` # ----------------------------------------` # Real Stability Interval is nearly [ -4.001051008, 0] # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # # Coupling coefficients # for c[11] = 1 # ---------------------------------------------------- a[11,1] = .4715561848627222170431765108838175679569e-1 a[11,2] = 0 a[11,3] = 0 a[11,4] = .2575056429843415189596436101037687580986 a[11,5] = .2621665397741262047713863095764527711129 a[11,6] = .1521609265673855740323133199165117535523 a[11,7] = .4939969170032484246907175893227876844296 a[11,8] = -.2943031171403250441557244744092703429139 a[11,9] = .8131747232495109999734599440136761892478e-1 a[11,10] = 0 # # ******************************************************** # # Coupling coefficients # for c[12] = .6716626365038747067708664674609860468131 # ---------------------------------------------------- a[12,1] = .4608828480030097784573810160682004587659e-1 a[12,2] = 0 a[12,3] = 0 a[12,4] = .2609521155912221411807714016159194601215 a[12,5] = .2593539886572148954104325364777172658208 a[12,6] = .1071284853231358590149526776164009620565 a[12,7] = -.7335737007803501940010147823741528427117e-1 a[12,8] = .9516817597231148848177657505563031988827e-1 a[12,9] = -.5557470923733997764040425874100454845262e-1 a[12,10] = 0 a[12,11] = .3190366547506434187770091206691782577319e-1 # # # ******************************************************** # # Coupling coefficients # for c[13] = .125 # ---------------------------------------------------- a[13,1] = .5837333975598387183322580009499934069289e-1 a[13,2] = 0 a[13,3] = 0 a[13,4] = .8577524236694000499244056599099302286222e-1 a[13,5] = -.3429204581002122987958198322824658635148e-1 a[13,6] = -.2818259731103036151485354567470271399566e-1 a[13,7] = -.2078116148802372182126410976576048400947e-1 a[13,8] = -.3670740144441019619026795845974788863064e-2 a[13,9] = .1140645258436468819102382339952341616568e-1 a[13,10] = 0 a[13,11] = -.2750153913374408098701890663991826389698e-2 a[13,12] = .5912166395960217591673813569316061988858e-1 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 13 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1 bi6[1,2] = -7.311180767629776696958619458278912939491 u^2 bi6[1,3] = 22.60641087875757991848471707711959284217 u^3 bi6[1,4] = -33.12556173257274109690767062478071200357 u^4 bi6[1,5] = 22.95954761030948255642157375784858777935 u^5 bi6[1,6] = -6.082060370378272459335683100820173921669 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[2] bi6[2,1] = 0 bi6[2,2] = 0 u^2 bi6[2,3] = 0 u^3 bi6[2,4] = 0 u^4 bi6[2,5] = 0 u^5 bi6[2,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[3] bi6[3,1] = 0 bi6[3,2] = 0 u^2 bi6[3,3] = 0 u^3 bi6[3,4] = 0 u^4 bi6[3,5] = 0 u^5 bi6[3,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[4] bi6[4,1] = 0 bi6[4,2] = -3.978134876602125774874513076122098772255 u^2 bi6[4,3] = 29.53521721383674756246316784496221836158 u^3 bi6[4,4] = -56.07901433922238873867636581288430847736 u^4 bi6[4,5] = 41.00995040124908700321914205599296950760 u^5 bi6[4,6] = -10.23051275627697853317178740184501186146 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0 bi6[5,2] = -3.621405783448216797833422800997110813199 u^2 bi6[5,3] = 28.45041711946822171033513471060107797760 u^3 bi6[5,4] = -63.35128606151121913627203265781677136093 u^4 bi6[5,5] = 57.40994313705539756150067024527746866855 u^5 bi6[5,6] = -18.62550187179005713295896318748821170090 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0 bi6[6,2] = -1.970687756031132878660315159329504150984 u^2 bi6[6,3] = 16.01712004431506310873418476518440023124 u^3 bi6[6,4] = -38.68332297698044734397617917177627792874 u^4 bi6[6,5] = 38.11100240454455032058494460481644228901 u^5 bi6[6,6] = -13.32195078928064763265032171897854868698 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0 bi6[7,2] = -5.638939259565213264463399815196965599114 u^2 bi6[7,3] = 49.13347122149190058929021203127370650355 u^3 bi6[7,4] = -136.6641495459403814931928844070388199337 u^4 bi6[7,5] = 151.4476239676854048245130375169811088598 u^5 bi6[7,6] = -57.78400946666846223145624773669624214610 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0 bi6[8,2] = 3.291909264109858379301398673578534019466 u^2 bi6[8,3] = -29.01660112374212383008594919395821532205 u^3 bi6[8,4] = 82.40461803966407217175372673786307360001 u^4 bi6[8,5] = -92.69288846738315663538954743462125936920 u^5 bi6[8,6] = 35.71865917021102487026464674272859672887 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0 bi6[9,2] = -.8754345587568475086756683253848824015194 u^2 bi6[9,3] = 7.888496352462982232106613196183765742215 u^3 bi6[9,4] = -23.26703990391624128356349606388132608585 u^4 bi6[9,5] = 26.85823381942063250549390180715909026469 u^5 bi6[9,6] = -10.52293823688557484536400461967527990061 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[10] bi6[10,1] = 0 bi6[10,2] = 0 u^2 bi6[10,3] = 0 u^3 bi6[10,4] = 0 u^4 bi6[10,5] = 0 u^5 bi6[10,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[11] bi6[11,1] = 0 bi6[11,2] = 1.089641947723107946112232140591662983820 u^2 bi6[11,3] = -9.670649677149513754901685091731804680347 u^3 bi6[11,4] = 27.86851681207959711975899768807763771310 u^4 bi6[11,5] = -32.08365238360308475926186866332651332044 u^5 bi6[11,6] = 12.79614330094989344829232392638901730387 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0 bi6[12,2] = 6.126786105894736326474301167193380349952 u^2 bi6[12,3] = -51.20037215254756178931575623288364799491 u^3 bi6[12,4] = 130.7335345131059934451910742013798764814 u^4 bi6[12,5] = -132.3730969921482468283320843728823303778 u^5 bi6[12,6] = 46.71314852569507884598246523719272154142 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[13] bi6[13,1] = 0 bi6[13,2] = 12.88744568430561026957800665394589732332 u^2 bi6[13,3] = -63.74350987689329574711063910675109366105 u^3 bi6[13,4] = 110.1637051952937563558848301108576279957 u^4 bi6[13,5] = -80.64666349713006654874976951724556430152 u^5 bi6[13,6] = 21.33902249442399567039757185919313264356 u^6 # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7 # # Coupling coefficients c[14] = .25 # ---------------------------------------------------- a[14,1] = .3781620167894484389097734217537475940628e-1 a[14,2] = 0. a[14,3] = 0. a[14,4] = .3134678532631074103775642636872622557979e-1 a[14,5] = .2225110077526238187226161613568406825821e-1 a[14,6] = .9958131955421000529301204622879718202274e-2 a[14,7] = .1522309604389992558287426004143093193551e-1 a[14,8] = -.7547047454118317125873194132609456732396e-2 a[14,9] = .1315888206546629705431596397483148558901e-2 a[14,10] = 0. a[14,11] = -.2347519033359896919332911522715968811661e-2 a[14,12] = -.2426983932792809284254784793342448202006e-1 a[14,13] = .1662532018290207842691515078471710556231 # # ******************************************************** # # Coupling coefficients c[15] = .53 # ---------------------------------------------------- a[15,1] = .5345075085525607281977266793633803424059e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = .3430173382138450672760919883610065904944 a[15,5] = .2076742726203398574300787732117594341897 a[15,6] = .7722675619395836334886386873715328439377e-1 a[15,7] = .1327268798561135321764135364638696721663e-3 a[15,8] = .2221995820246240129102290486291650170691e-1 a[15,9] = -.1741022886840232443411414202292540569931e-1 a[15,10] = 0. a[15,11] = .7197593769915030785550602945672682421140e-2 a[15,12] = -.8645504868462665761280182814730150657031e-1 a[15,13] = -.7705411918260392443664124942108348484897e-1 a[15,14] = 0. # # ******************************************************** # # Coupling coefficients c[16] = .79 # ---------------------------------------------------- a[16,1] = .5681087911635884235009187585387593516296e-1 a[16,2] = 0. a[16,3] = 0. a[16,4] = .3685221901618432443784221615000295848158 a[16,5] = .2294060530103075331008808070568391539642 a[16,6] = .8854425014313802531960211711477635403509e-1 a[16,7] = .2937879283098644165855344854841522051205e-1 a[16,8] = .5463456745004647616602723887000652830690e-2 a[16,9] = -.1311750049635594995549114647314453663034e-1 a[16,10] = 0. a[16,11] = .5109093689745296784307624438438730642345e-2 a[16,12] = .1241964089587753256730552835816994961283 a[16,13] = -.1043136241598034069260248955079305914610 a[16,14] = 0. a[16,15] = 0. # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi7 WITH 16 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi7[1] bi7[1,1] = 1 bi7[1,2] = -8.366136708817906707256587496057579251260 u^2 bi7[1,3] = 33.45564674491338463365847175559207954194 u^3 bi7[1,4] = -70.50417466572664954758490309606786411219 u^4 bi7[1,5] = 80.67397907967836612193925652942602271164 u^5 bi7[1,6] = -47.49724826833688304754452975511020270063 u^6 bi7[1,7] = 11.28508943677596076849260971330592556729 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[2] bi7[2,1] = 0 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 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 bi7[4,2] = 13.82288155211713305818998620073363978200 u^2 bi7[4,3] = -112.0415006693310346004392233363617471241 u^3 bi7[4,4] = 378.8018233039261336815452690775193363501 u^4 bi7[4,5] = -626.3590532637976098474387079168772337800 u^5 bi7[4,6] = 497.1667710334456823937660762137016294028 u^6 bi7[4,7] = -151.1334163133759631666637566286118558727 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0 bi7[5,2] = 14.07307810511366058918319369346769382933 u^2 bi7[5,3] = -114.0694712595681529570511115594726966772 u^3 bi7[5,4] = 385.6582019903893875351543858863698253970 u^4 bi7[5,5] = -637.6962607392366345376325547234894566831 u^5 bi7[5,6] = 506.1655757984282987854901337254569241352 u^6 bi7[5,7] = -153.8689573553524332103726607127558372301 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0 bi7[6,2] = 8.167985914503869973087194559385489320966 u^2 bi7[6,3] = -66.20568915797484766768764016216198194931 u^3 bi7[6,4] = 223.8352361965341074040691652677197644734 u^4 bi7[6,5] = -370.1176129731857456919175417204553196825 u^5 bi7[6,6] = 293.7774709021209089951683687288874094120 u^6 bi7[6,7] = -89.30522995543090743868723335345884982106 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0 bi7[7,2] = 26.51771352157190690289662014455758360951 u^2 bi7[7,3] = -214.9395844907078534615508938304133084413 u^3 bi7[7,4] = 726.6906103441299900926874461040345411532 u^4 bi7[7,5] = -1201.602565533698840848055495464915927871 u^5 bi7[7,6] = 953.7610488090023297226041503418530617909 u^6 bi7[7,7] = -289.9332257332942839838911097057931625569 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0 bi7[8,2] = -15.79816691200410691331235500226668535847 u^2 bi7[8,3] = 128.0521953379834910620481085022933698259 u^3 bi7[8,4] = -432.9324828953870620101388161175514606244 u^4 bi7[8,5] = 715.8655619667639243592495914716689368891 u^5 bi7[8,6] = -568.2117438592623676788034649093884368167 u^6 bi7[8,7] = 172.7303332447657961368012115808350057416 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0 bi7[9,2] = 4.365115168111920354195790905604187992273 u^2 bi7[9,3] = -35.38148338938862337683672127675703409068 u^3 bi7[9,4] = 119.6214825670096897492970562896860075250 u^4 bi7[9,5] = -197.7973546092682723492205617236299717431 u^5 bi7[9,6] = 156.9998415407770248868993071180186906449 u^6 bi7[9,7] = -47.72628380491678816433752531852051270954 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[10] bi7[10,1] = 0 bi7[10,2] = 0 u^2 bi7[10,3] = 0 u^3 bi7[10,4] = 0 u^4 bi7[10,5] = 0 u^5 bi7[10,6] = 0 u^6 bi7[10,7] = 0 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[11] bi7[11,1] = 0 bi7[11,2] = -3.274137494621028294117921462492887425262 u^2 bi7[11,3] = 27.15609115402443206034928076549176636157 u^3 bi7[11,4] = -93.85378772108284462706907256963954198093 u^4 bi7[11,5] = 158.5920958593385426059558763830493083967 u^5 bi7[11,6] = -128.8765056984211381015120508066826191705 u^6 bi7[11,7] = 40.25624390076203635639388769027397381848 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0 bi7[12,2] = 5.921404526401594125717004582969923833251 u^2 bi7[12,3] = -27.55299853706615138347543027007787995007 u^3 bi7[12,4] = 37.27203301874088572149686416039966422641 u^4 bi7[12,5] = 3.387133200887933999734182285017083864954 u^5 bi7[12,6] = -37.98539394174189025864225888577125977510 u^6 bi7[12,7] = 18.95782173277762779516963812746246780056 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0 bi7[13,2] = 8.487477767791544489695951710080762138462 u^2 bi7[13,3] = -39.49324209765729749945151001059651498212 u^3 bi7[13,4] = 53.42407366300932909711921671337019664478 u^4 bi7[13,5] = 4.854966015394845744743845722515774464325 u^5 bi7[13,6] = -54.44657346814621123151906009315988962684 u^6 bi7[13,7] = 27.17329811960778939941155595778967136139 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0 bi7[14,2] = -19.18681975136084566194177555055440129993 u^2 bi7[14,3] = 167.1498313610223304029460022072960115639 u^3 bi7[14,4] = -537.8357359160404910114027416626443457624 u^4 bi7[14,5] = 827.7451797910091375839558024726116253803 u^5 bi7[14,6] = -614.6483785211818954357785110337022532294 u^6 bi7[14,7] = 176.7759230365517641222212235669933633475 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0 bi7[15,2] = -15.88375234934266311207464684269517278951 u^2 bi7[15,3] = 109.6403523022280467967594696827166534140 u^3 bi7[15,4] = -328.1504970903914932099922972104443987762 u^4 bi7[15,5] = 514.4013813786596992985172200476181568743 u^5 bi7[15,6] = -403.4939189483437905937221929812938671282 u^6 bi7[15,7] = 123.4864347071902008205124473040986284057 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0 bi7[16,2] = -18.84664333946507880426245544273255438136 u^2 bi7[16,3] = 144.2298527015222759907311975324512825075 u^3 bi7[16,4] = -462.0267827951109828751815728427517245137 u^4 bi7[16,5] = 728.0525498274546535601690866374610011782 u^5 bi7[16,6] = -552.7109453783400684364059676628091869385 u^6 bi7[16,7] = 161.3019689839392005649497117783811821479 u^7 # # ******************************************************** # Norms of low order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, -.3885039665e-5, .9425967885e-5 .2000000000, -.2706418363e-5, .5451335668e-5 .3000000000, -.6323744077e-5, .1503054908e-4 .4000000000, -.1109505358e-4, .2436851197e-4 .5000000000, -.1561429630e-4, .2375422830e-4 .6000000000, -.1793911549e-4, .2496579712e-4 .7000000000, -.1648871388e-4, .2439381179e-4 .8000000000, -.1110040567e-4, .1744281602e-4 .9000000000, .4071262512e-5, .9357746450e-5 1., -.4714285714e-58, .7037675526e-58 1.100000000, .2104979550e-4, .5410208179e-4 1.200000000, .1748922152e-3, .4370083814e-3 1.300000000, .7322270064e-3, .1806204130e-2 1.400000000, .2240502285e-2, .5488435582e-2 1.500000000, .5680564275e-2, .1385707546e-1 1.600000000, .1267062744e-1, .3082358836e-1 1.700000000, .2572356501e-1, .6245739363e-1 1.800000000, .4856351986e-1, .1177485261 1.900000000, .8650783585e-1, .2095275181 2., .1469203095, .3555569340 # ******************************************************** # Norms of high order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, .2479063321e-5, .3669670718e-5 .2000000000, .4125158962e-5, .5614524351e-5 .3000000000, .4217552666e-5, .5714342886e-5 .4000000000, .4547124505e-5, .7060974252e-5 .5000000000, .5187131230e-5, .9351670571e-5 .6000000000, .4855691454e-5, .1019438198e-4 .7000000000, -.6289627505e-5, .1046469463e-4 .8000000000, -.7696968544e-5, .1182772436e-4 .9000000000, -.8447583302e-5, .1347971658e-4 1., -.8590002884e-5, .1675585009e-4 1.100000000, .1578461431e-4, .3262303128e-4 1.200000000, .1561890877e-3, .2990682691e-3 1.300000000, .6797776100e-3, .1304804023e-2 1.400000000, .2166967573e-2, .4180559611e-2 1.500000000, .5752852145e-2, .1116892931e-1 1.600000000, .1347178988e-1, .2633129465e-1 1.700000000, .2873790729e-1, .5654465862e-1 1.800000000, .5699259305e-1, .1128541186 1.900000000, .1065532126, .2122556092 2., .1897004218, .3799906686 # ********************************************************