# 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 one of several quadratic # polynomials. 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 robust' 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 # # which require 2 and 3 extra stages respectively. # (Companion files list only the RATIONAL and 40-digit # floating point approximations respectively.) # # This procedure is "most robust" in the sense that # all weights b_i are non-negative, the maximum # coefficient within b and A is not large, and it # has a propagating formula which almost minimizes # the 2-norm of the local truncation error as # # T_82 ~ .00002701 # # (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 = 7. # # These interpolants have continuous derivatives. # # Nodes c[11]=1, c[12], c[13] were selected to minimize # the maximum of the 2-norm of the local truncation # error on the interval [0,1] for the interpolant # of order 6, and this value is # # Ti_72 ~ .00005084 # # This 2-norm has a single local maximum value on [0,1]. # # The remaining three nodes are 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 7, and this value of # # Ti_82 ~ .00002701 # # reached at the endpoint. This 2-norm is almost monotone # increasing on [0,1]. # # 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] = .5000000000000000000000000000000000000000e-2 c[3] = .1088888888888888888888888888888888888889 c[4] = .1633333333333333333333333333333333333333 c[5] = .4550000000000000000000000000000000000000 c[6] = .6059617471462913245758145021744683294809 c[7] = .8350000000000000000000000000000000000000 c[8] = .9150000000000000000000000000000000000000 c[9] = 1 c[10] = 1 c[11] = 1 c[12] = .3067207177086018949109476427952022863650 c[13] = .7875000000000000000000000000000000000000e-1 c[14] = .2000000000000000000000000000000000000000 c[15] = .5500000000000000000000000000000000000000 c[16] = .8000000000000000000000000000000000000000 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- # for c[1] = 0 # # for c[2] = 1/200 a[2,1] = .5000000000000000000000000000000000000000e-2 # # for c[3] = 49/450 a[3,1] = -1.076790123456790123456790123456790123457 a[3,2] = 1.185679012345679012345679012345679012346 # # for c[4] = 49/300 a[4,1] = .4083333333333333333333333333333333333333e-1 a[4,2] = 0 a[4,3] = .1225000000000000000000000000000000000000 # # for c[5] = 91/200 a[5,1] = .6360714285714285714285714285714285714286 a[5,2] = 0 a[5,3] = -2.444464285714285714285714285714285714286 a[5,4] = 2.263392857142857142857142857142857142857 # # for c[6] = 34704460/57271701 a[6,1] = -2.535121107934924522925638355466021548721 a[6,2] = 0 a[6,3] = 10.29937465444926792043851446075602491361 a[6,4] = -7.951303288599057994949321745826687653648 a[6,5] = .7930114892310059220122601427111526182380 # # for c[7] = 167/200 a[7,1] = 1.001876581252463296196919658309499980821 a[7,2] = 0 a[7,3] = -4.166571282442379833131393800547097145319 a[7,4] = 3.834343292912864241255266521825137866520 a[7,5] = -.5023333356071084754746433022861176561240 a[7,6] = .6676847438841607711538509226985769541026 # # for c[8] = 183/200 a[8,1] = 27.25501835463076713033396381917500571735 a[8,2] = 0 a[8,3] = -42.00461727841063835531864544390929536961 a[8,4] = -10.53571312661948991792108160054652610372 a[8,5] = 80.49553671141193714798365215892682663420 a[8,6] = -67.34388227179051346854907596321297564093 a[8,7] = 13.04865761077793746347118702956696476271 # # for c[9] = 1 a[9,1] = -3.039737805711496514694365865875576322688 a[9,2] = 0 a[9,3] = 10.13816141032980111185794619070970015044 a[9,4] = -6.429305674864721572146282562955529806444 a[9,5] = -1.586437148340827658711531285379861057947 a[9,6] = 1.892178184196842441086430890913135336502 a[9,7] = .1969933540760886906129236016333644283801e-1 a[9,8] = .5441698982793323546510272424795257297790e-2 # # for c[10] = 1 a[10,1] = -1.444951891677773513735100317935571236052 a[10,2] = 0 a[10,3] = 8.031891385995591922411703322301956043504 a[10,4] = -7.583174166340134682079888302367158860498 a[10,5] = 3.581616935319007421124768544245287869686 a[10,6] = -2.436972263219952941118380906569375238373 a[10,7] = .8515899999232617933968976603248614217339 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] = .4742583783370675608356917271757453469893e-1 b[2] = 0 b[3] = 0 b[4] = .2562236165937056265996172745827462344816 b[5] = .2695137683307420661947381725807595288676 b[6] = .1268662240909278284598913836473917324788 b[7] = .2488722594206007162204644942764749276729 b[8] = .3074483740820063133530438847909918476864e-2 b[9] = .4802380998949694330818906334714312332321e-1 b[10] = 0 # # ******************************************************** # Low order weights C[extra]:= 1 # -------------------------------------------------- # bh[1] = .4748524769929963103753127380572796155227e-1 bh[2] = 0 bh[3] = 0 bh[4] = .2559941258869063329715491824590539387050 bh[5] = .2705847808106768872253089109926813573239 bh[6] = .1250561868442599291363882232374691792045 bh[7] = .2520446872374386050718404382019744256218 bh[8] = 0 bh[9] = 0 bh[10] = .4883497152141861455738197130309313759259e-1 # #******************************************************** # # Largest coefficient in b or A has magnitude `, 80.49554 # # ******************************************************* # SUMMARY OF NORMS OF ERRORS: A81, A82, A8inf` # ---------------------------------------------------- # A_[8, 1] = `, .1495076450e-3 # A_[8, 2] = `, .2701546765e-4 # A_[8,oo] = `, .9215639068e-5 # ******************************************************* # # END OF GENERATION OF A PAIR OF RK METHODS # ############################################################# # # START OF GENERATION OF STABILITY REGIONS # ############################################################# # # Stability Boundaries of High Order Method # ----------------------------------------- # Real Stability Interval is nearly [ -4.635489330, 0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [ -3.999541616, 0] # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # # Coupling coefficients # for c[11] = 1 # ---------------------------------------------------- a[11,1] = .4742583783370675608356917271757453469893e-1 a[11,2] = 0 a[11,3] = 0 a[11,4] = .2562236165937056265996172745827462344816 a[11,5] = .2695137683307420661947381725807595288676 a[11,6] = .1268662240909278284598913836473917324788 a[11,7] = .2488722594206007162204644942764749276729 a[11,8] = .3074483740820063133530438847909918476864e-2 a[11,9] = .4802380998949694330818906334714312332321e-1 a[11,10] = 0 # # ******************************************************** # # Coupling coefficients # for c[12] = .3067207177086018949109476427952022863650 # ---------------------------------------------------- a[12,1] = .5084001823323651000647096877290063127003e-1 a[12,2] = 0 a[12,3] = 0 a[12,4] = .2340953776302340927060690222130822879757 a[12,5] = .2957282565868639393723036073846174594572e-1 a[12,6] = -.7891822081774038060777822301275761921904e-2 a[12,7] = -.3124515117284615723495332436165903196571e-3 a[12,8] = .3075900699836012263713563630912227790408e-3 a[12,9] = .1346518474743767274498197636413097501530e-1 a[12,10] = 0 a[12,11] = -.1335600503747387607704868611157222437932e-1 # # # ******************************************************** # # Coupling coefficients # for c[13] = .7875000000000000000000000000000000000000e-1 # ---------------------------------------------------- a[13,1] = .4571638671066279127171353905223268681038e-1 a[13,2] = 0 a[13,3] = 0 a[13,4] = .6135139403136968332208211375799160523016e-1 a[13,5] = .1491507057398255131431717710757995218460e-1 a[13,6] = .7635076575109071067007610291932885992123e-3 a[13,7] = -.2047836064685163003552627495384108018023e-2 a[13,8] = .5715454045443600137343865669766089985382e-5 a[13,9] = .1261351063159691297340795743480869575660e-2 a[13,10] = 0 a[13,11] = -.6786312294583558454434945884425128592581e-3 a[13,12] = -.4253695819658754906329560847232154761272e-1 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 13 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1 u bi6[1,2] = -9.287802493073601957353743786492063345949 u^2 bi6[1,3] = 28.37181737748343338051635305062526378452 u^3 bi6[1,4] = -34.95935239729884889713119812110601563005 u^4 bi6[1,5] = 17.23901766144404601862972727260994049852 u^5 bi6[1,6] = -2.316254310721321788577569242919550772346 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[2] bi6[2,1] = 0 u 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 u 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 u bi6[4,2] = 3.604761077728393845892032481966391933648 u^2 bi6[4,3] = -43.90132051013332790453077632356074711775 u^3 bi6[4,4] = 143.4048773556619116818579457283007520199 u^4 bi6[4,5] = -167.9874957922751812740939887662883530142 u^5 bi6[4,6] = 65.13540148561190927747440415416470241289 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0 u bi6[5,2] = 1.929105425243998668851900157843757873116 u^2 bi6[5,3] = -23.42768366883539572445882853136416615437 u^3 bi6[5,4] = 75.90027184849720003035335216268922252114 u^4 bi6[5,5] = -87.61683178147975516556939032717642089844 u^5 bi6[5,6] = 33.48465194490469425701770471058836618741 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0 u bi6[6,2] = .4542653865647780562449203586421420101885 u^2 bi6[6,3] = -5.484986468997689571085293864412526349045 u^3 bi6[6,4] = 17.46902392549403057979256773227273070628 u^4 bi6[6,5] = -19.53895264570853770054958700399210001131 u^5 bi6[6,6] = 7.227516026738346464057284161137145376363 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0 u bi6[7,2] = -.4595252127874704413283249342362741012014 u^2 bi6[7,3] = 5.737453695169824323343997706759839627335 u^3 bi6[7,4] = -20.07465511716825087735112952294097729875 u^4 bi6[7,5] = 26.26828355650051484730635262820638168633 u^5 bi6[7,6] = -11.22268466229401713575043138351249498605 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0 u bi6[8,2] = -.1150484606476358811080146030071413885522e-1 u^2 bi6[8,3] = .1420638968741292571977227331343062140527 u^3 bi6[8,4] = -.4824858974681316639225996449662084460453 u^4 bi6[8,5] = .6032463910178502874964195648198143162146 u^5 bi6[8,6] = -.2482450606182642295272107538392880268899 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0 u bi6[9,2] = -.2764312406305170345487420982503550738064 u^2 bi6[9,3] = 3.400474146978174062782264777150963447481 u^3 bi6[9,4] = -11.42817994454789068919174383222685588969 u^4 bi6[9,5] = 14.04880527062030898808079610608510047209 u^5 bi6[9,6] = -5.696644422430578383814385889411709832759 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[10] bi6[10,1] = 0 u 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 u bi6[11,2] = .4102535385385108256854385187104749572916 u^2 bi6[11,3] = -5.085955012765407731553513252344879846729 u^3 bi6[11,4] = 17.46979591248517073647128950918854985232 u^4 bi6[11,5] = -22.32274094082816158102379333618435999361 u^5 bi6[11,6] = 9.528646502569887750420578560630215030734 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0 u bi6[12,2] = -5.561170546460708694292346759381826899322 u^2 bi6[12,3] = 64.69697086482037294824481100458495633348 u^3 bi6[12,4] = -187.0019352784693116129945736733753172573 u^4 bi6[12,5] = 202.1576401483203391584241013705230731115 u^5 bi6[12,6] = -74.29150518821069179938199194235088528834 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[13] bi6[13,1] = 0 u bi6[13,2] = 9.198048910941380318959667521498466784890 u^2 bi6[13,3] = -24.44883432059411304045673730057300993898 u^3 bi6[13,4] = -.2973604071858792878839103378358805777295 u^4 bi6[13,5] = 37.14902813238857642129936249139692383284 u^5 bi6[13,6] = -21.60088231554996441191838237448650010102 u^6 # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7# # # Coupling coefficients # for c[14] = .2 # ---------------------------------------------------- a[14,1] = .4895720837021160646063704247238281140243e-2 a[14,2] = 0. a[14,3] = 0. a[14,4] = -.2715965016132070652354391268389482837445e-1 a[14,5] = -.1471118582892732021356233028007905752798e-1 a[14,6] = -.3548741829515423281693629852942855954300e-2 a[14,7] = .3086771782083923184307182280403431753151e-2 a[14,8] = .8149105769962305218267679359657679200786e-4 a[14,9] = .1992488082891027877223378212655417925868e-2 a[14,10] = 0. a[14,11] = -.2859268825507094536557198895337022154448e-2 a[14,12] = .5586163713000076148900856473826299382047e-1 a[14,13] = .1823607377555740483065715654400970625794 # # ******************************************************** # # Coupling coefficients # for c[15] = .55 # ---------------------------------------------------- a[15,1] = .6529904148291268153192118584675719879592e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = .2572420423912244394795082675403296906296 a[15,5] = .1483811462482641701867308875408674270889 a[15,6] = .4007284294116317186672184994560870753991e-1 a[15,7] = -.1000336192875383976289218926700770766316e-1 a[15,8] = -.5059442698140261792783567717843895596137e-3 a[15,9] = -.1424880415164983774939393220447878176376e-1 a[15,10] = 0. a[15,11] = .1681180958703984153237987900953773537091e-1 a[15,12] = .8770029800174049628643329724269360224271e-1 a[15,13] = -.4074907030212709719213088888252348268140e-1 a[15,14] = 0. # # ******************************************************** # # Coupling coefficients # for c[16] = .8 # ---------------------------------------------------- a[16,1] = .1045152970430589502927487092331130990894 a[16,2] = 0. a[16,3] = 0. a[16,4] = .5969208192685441448792898447593605742233 a[16,5] = .3959219441466996682329003000423645782513 a[16,6] = .1298549335245192828418583437596723989380 a[16,7] = .8653312743253928887016075996436582821518e-1 a[16,8] = .3432143531736777732699173398591878436542e-3 a[16,9] = -.6684380434300498891936757211031674244336e-2 a[16,10] = 0. a[16,11] = -.2656538838207228881702404465288483040129e-2 a[16,12] = -.2623495692645475071531194269140313972733 a[16,13] = -.2423988472314797779634692865083841120031 a[16,14] = 0. a[16,15] = 0. # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi7 WITH 16 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi7[1] bi7[1,1] = 1 u bi7[1,2] = -11.10549218310878150294972542535469957571 u^2 bi7[1,3] = 53.71626091747243573007532112571952869213 u^3 bi7[1,4] = -128.5473739979257316562972104983527950703 u^4 bi7[1,5] = 162.0902186531113150098628432997326198709 u^5 bi7[1,6] = -103.5439172019553231638017282714884496779 u^6 bi7[1,7] = 26.43772965023979233919406894246137029559 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] = 4.644609410286821790818608827273831968356 u^2 bi7[4,3] = -55.69548149226319203052268864282320364865 u^3 bi7[4,4] = 247.4659934838965295586705500261724811869 u^4 bi7[4,5] = -479.3835852880289717510165107931536946215 u^5 bi7[4,6] = 417.7216343581429533802164028647928240764 u^6 bi7[4,7] = -134.4969468554404353215667450076794927270 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0 u bi7[5,2] = 4.885522268526039984199515381319689881091 u^2 bi7[5,3] = -58.58436976079942763955284154625277295018 u^3 bi7[5,4] = 260.3018930269609359374580025353201330058 u^4 bi7[5,5] = -504.2488989285943199990512883132144281542 u^5 bi7[5,6] = 439.3885828551885372863055255069464163884 u^6 bi7[5,7] = -141.4732156929510235031641753915382786420 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0 u bi7[6,2] = 2.299725786771036538808491821520966616133 u^2 bi7[6,3] = -27.57698735887394944339902823708961051834 u^3 bi7[6,4] = 122.5299861175378960065024577695198495065 u^4 bi7[6,5] = -237.3613571035572427816676052769237936792 u^5 bi7[6,6] = 206.8301399247779074226007307715733899589 u^6 bi7[6,7] = -66.59464114256471991438515546495341015153 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0 u bi7[7,2] = 4.511350098914579693762784164291257598291 u^2 bi7[7,3] = -54.09751256644427902123991312819451883662 u^3 bi7[7,4] = 240.3659028268276199102501289326196502906 u^4 bi7[7,5] = -465.6295059208482858878798381329519530492 u^5 bi7[7,6] = 405.7367089483621346746982726193020670754 u^6 bi7[7,7] = -130.6380711273911686533709699607900281508 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0 u bi7[8,2] = .5573169368313994584936204877220578262885e-1 u^2 bi7[8,3] = -.6683023780615641949620920893081135755082 u^3 bi7[8,4] = 2.969399087745192949018007907595558266902 u^4 bi7[8,5] = -5.752229471185599429198382650164345796076 u^5 bi7[8,6] = 5.012334109151917503879012762848811609674 u^6 bi7[8,7] = -1.613858557592266711452377540896206369143 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0 u bi7[9,2] = .8705358341293636116640288559710826202416 u^2 bi7[9,3] = -10.43896443277234556265947385790570406634 u^3 bi7[9,4] = 46.38237492673297791283188052518561842363 u^4 bi7[9,5] = -89.85052399936211964597828282043452960223 u^5 bi7[9,6] = 78.29326844889434834893599866050960896097 u^6 bi7[9,7] = -25.20866696763272772148596229997893321296 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[10] bi7[10,1] = 0 u 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 u bi7[11,2] = -1.121142697706800527083012308483258665639 u^2 bi7[11,3] = 13.53715613482686753793893859559118755897 u^3 bi7[11,4] = -60.67323271642655277624479585288063076345 u^4 bi7[11,5] = 119.1225531647168786071241378088451033613 u^5 bi7[11,6] = -105.7683192309275664018545808989967713400 u^6 bi7[11,7] = 34.90298534551717356011931265592436984878 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0 u bi7[12,2] = -2.689843926762734958111868061662373328225 u^2 bi7[12,3] = 18.19249919231253025865171370842278837260 u^3 bi7[12,4] = -47.67419183482623602331838882715584056215 u^4 bi7[12,5] = 60.93531049474041885876501108919761116633 u^5 bi7[12,6] = -38.16882262043857589162237022230698749550 u^6 bi7[12,7] = 9.405048694974597755635902313504801846941 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0 u bi7[13,2] = 14.28744953003485588757315864042649968175 u^2 bi7[13,3] = -96.63178277714716633518302982563517208997 u^3 bi7[13,4] = 253.2275582788345611682109830360906534503 u^4 bi7[13,5] = -323.6656835842511583761765553542772427904 u^5 bi7[13,6] = 202.7385757904130191505864877006907151810 u^6 bi7[13,7] = -49.95611723788411149501104419729545343269 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0 u bi7[14,2] = -7.739397697899142305765936845963446864682 u^2 bi7[14,3] = 109.4264747285538449776642447474549631075 u^3 bi7[14,4] = -443.5472686552890108735394957236035556608 u^4 bi7[14,5] = 788.5368317313452219414265070471096598338 u^5 bi7[14,6] = -645.4407679215430796440618216834112707918 u^6 bi7[14,7] = 198.7641278148321659042765024584136503760 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0 u bi7[15,2] = -3.698276994746987304944919677286868631635 u^2 bi7[15,3] = 47.05688801191757696865859836946877899521 u^3 bi7[15,4] = -221.5772626255367244048612897506928728139 u^4 bi7[15,5] = 458.8717731648818424448053323962660771148 u^5 bi7[15,6] = -422.7479255270888830249365906318943086105 u^6 bi7[15,7] = 142.0948039705731753212788692941391939461 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0 u bi7[16,2] = -5.200771122121390853820487420824887082602 u^2 bi7[16,3] = 61.76412178127866875453025078055184895922 u^3 bi7[16,4] = -271.2237779185314577086808300798182492601 u^4 bi7[16,5] = 516.3350970870320210089846316999689163456 u^5 bi7[16,6] = -440.0514919329773896409453391785660453349 u^6 bi7[16,7] = 138.3768221053195484399317741986884163727 u^7 # # ******************************************************** # Norms of low order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, -.8045336405e-6, .1811756534e-5 .2000000000, -.3555642339e-5, .7459758922e-5 .3000000000, -.8923420109e-5, .1642370371e-4 .4000000000, -.1565720887e-4, .1947065459e-4 .5000000000, -.2059246145e-4, .2867590170e-4 .6000000000, -.2058891871e-4, .4227582738e-4 .7000000000, .2509644606e-4, .5084467104e-4 .8000000000, .2841198317e-4, .4943678762e-4 .9000000000, .1554791789e-4, .2709556104e-4 1.000000000, .8455357143e-59,.1518029595e-58 1.100000000, .5517307687e-4, .1076016763e-3 1.200000000, .3742566172e-3, .7699692244e-3 1.300000000, .1356494570e-2, .2920290730e-2 1.400000000, .3724349557e-2, .8328796023e-2 1.500000000, .8676061405e-2, .2003295890e-1 1.600000000, .1807810001e-1, .4288845296e-1 1.700000000, .3470201252e-1, .8425528548e-1 1.800000000, .6251016491e-1, .1548335342 1.900000000, .1069948778, .2696631154 2.000000000, .1755754567, .4493020141 # ******************************************************** # Norms of high order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, .7146457327e-6, .1148631867e-5 .2000000000, .6593063031e-6, .9509186303e-6 .3000000000, .1530931854e-5, .2931198400e-5 .4000000000, .4261739287e-5, .8099792011e-5 .5000000000, .6880380330e-5, .1296214819e-4 .6000000000, .6873899781e-5, .1418943399e-4 .7000000000, -.7823707017e-5, .1417616235e-4 .8000000000, -.8893985096e-5, .1754004272e-4 .9000000000, -.9208947413e-5, .2099989620e-4 1.000000000, .9215639068e-5, .2701546765e-4 1.100000000, .2679044351e-4, .5018917630e-4 1.200000000, .2680344082e-3, .4466103490e-3 1.300000000, .1117856989e-2, .1869178476e-2 1.400000000, .3421180995e-2, .5768150919e-2 1.500000000, .8763430974e-2, .1492165887e-1 1.600000000, .1989444416e-1, .3422892997e-1 1.700000000, .4130742624e-1, .7181517682e-1 1.800000000, .8000604352e-1, .1405148188 1.900000000, .1464958931, .2598097239 2.000000000, .2560397429, .4583063307 # 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