###################################################################### # # A 'most robust' Runge--Kutta (10:(7)6) pair # # These are 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 # # T82 ~ .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. # 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 # # T_72 ~ .000144 # # This 2-norm has three local maximum values 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 is # # T_82 ~ .000146 # # This 2-norm has three local maximum values on [0,1]. # Attempts to make the order 7 interpolant monotone # increasing on this interval were unsuccessful. # # The formulas scanned for this optimal formula are # those developed in J.H. Verner, SIAM NA 1978, 772-790, # "Explicit Runge--Kutta methods with estimates of the # Local Truncation Error". It is conceivable that the # pairs in J.H. Verner, Annals of Num. Math 1 1994, # 225-244, "Strategies for deriving new explicit Runge-- # Kutta pairs", which require only ten stages, but solve # the order conditions in a different way, or the # 11-stage contemporary or 12-stage FSAL methods # derived by Sharp and Verner, SIAM NA 31, 1994, # 1169--1190, "Completely imbedded Runge--Kutta pairs" # may yield particular pairs of equivalant or more # efficiency. # ###################################################################### # # NODES # ----- c[1] = 0. c[2] = .5e-2 c[3] = .1088888888888888888888888888888888888889 c[4] = .1633333333333333333333333333333333333333 c[5] = .455 c[6] = .6059617471462913245758145021744683294809 c[7] = .835 c[8] = .915 c[9] = 1. c[10] = 1. # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- for c[1] = 0 # for c[2] = 1/200 a[2,1] = .5e-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] = .1225 # 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 THE PAIR OF RK METHODS # ############################################################# # # Stability Boundaries of High Order Method # ----------------------------------------- # Real Stability Interval is nearly [ -4.635489312, 0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [ -3.999541582, 0] # ############################################################# # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # # Coupling coefficients for c[11] = 1.0 # ---------------------------------------------------- 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] = 0.31 # ---------------------------------------------------- a[12,1] = .5066241952051105127788297308453867174270e-1 a[12,2] = 0. a[12,3] = 0. a[12,4] = .2353613267913092058590156113311702525494 a[12,5] = .3382676132945503842494133507440782016457e-1 a[12,6] = -.1074936533903229219678329288306058529581e-1 a[12,7] = .1488702140174965282241649383558610867045e-2 a[12,8] = -.9053636405072999107368243588707874406946e-3 a[12,9] = .1309326349808933126343854836825601741284e-1 a[12,10] = 0. a[12,11] = -.127777443e-1 # # ******************************************************** # Coupling coefficients for c[13] = 0.07875 # ---------------------------------------------------- a[13,1] = .4595034513235430055355771235567511908718e-1 a[13,2] = 0. a[13,3] = 0. a[13,4] = .5950766032868800357330455854854096115023e-1 a[13,5] = .1123110285422930714110889559809514841189e-1 a[13,6] = .3568247494079217477812290559247208500749e-2 a[13,7] = -.3785335544994492974095302988946805485750e-2 a[13,8] = .1124499874300357574804800614166179956714e-2 a[13,9] = .1467321291908930936613307632813763748509e-2 a[13,10] = 0. a[13,11] = -.1057056823387449353878340780915210332955e-2 a[13,12] = -.3925678460717817492922792153867636503657e-1 # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 13 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1. u bi6[1,2] = -9.222458461776465463100824040911752089450 u^2 bi6[1,3] = 27.61162474291097029482246094954360245409 u^3 bi6[1,4] = -32.76207033062347726830830019785289820593 u^4 bi6[1,5] = 14.86365530662214604105392874702744461557 u^5 bi6[1,6] = -1.443325419299466848383696285088822239587 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] = 3.768150194295932160252420337046168502776 u^2 bi6[4,3] = -45.80213991962371291589803020582945936796 u^3 bi6[4,4] = 148.8990589755754723308812692360106572669 u^4 bi6[4,5] = -173.9269572699013007954804255552211330341 u^5 bi6[4,6] = 67.31811163624731484684438346257651286686 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0. u bi6[5,2] = 2.018896751524469760916499967262265967827 u^2 bi6[5,3] = -24.47228871587506611749344547617658484138 u^3 bi6[5,4] = 78.91962751209125308358716761155071165033 u^4 bi6[5,5] = -90.88089327267073446119156962813617547461 u^5 bi6[5,6] = 34.68417149326081980037608569808054222671 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0. u bi6[6,2] = .4765361642210359223025444562903568646128 u^2 bi6[6,3] = -5.744077939242350184694137450273159969185 u^3 bi6[6,4] = 18.21790916234892922024949020276795823756 u^4 bi6[6,5] = -20.34853181930938460486739757799351363115 u^5 bi6[6,6] = 7.525030656072697475469391752855750230638 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0. u bi6[7,2] = -.4753503227130887393197381886797786010772 u^2 bi6[7,3] = 5.921558222667530049692519098543014752195 u^3 bi6[7,4] = -20.60679599602846437124395219769283901821 u^4 bi6[7,5] = 26.84355217143029784801208682013459775019 u^5 bi6[7,6] = -11.43409181593567407092045103802851995542 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0. u bi6[8,2] = -.1195714294707985012627456807778736988000e-1 u^2 bi6[8,3] = .1473257815750197262643078339825482551275 u^3 bi6[8,4] = -.4976949962733083027675626953763837898280 u^4 bi6[8,5] = .6196881220547972060484827942037318046548 u^5 bi6[8,6] = -.2542872806686087162854229258841989815974 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0. u bi6[9,2] = -.2877581036380653644605787499185131972526 u^2 bi6[9,3] = 3.532247434610136728559192325104808696757 u^3 bi6[9,4] = -11.80906114309226738673685562037219434001 u^4 bi6[9,5] = 14.46055525684336770548758364518687411870 u^5 bi6[9,6] = -5.847959634733674739541152536653832154869 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] = .4258709641264577536971692043885314747955 u^2 bi6[11,3] = -5.267643403690579182548896984019644299966 u^3 bi6[11,4] = 17.99495312257509439758667164618835137692 u^4 bi6[11,5] = -22.89045989058428226231532915787189575312 u^5 bi6[11,6] = 9.737279207573309293580385291314657201371 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0. u bi6[12,2] = -5.680390506643039793340485777683717079573 u^2 bi6[12,3] = 66.08393970995500145695244835138885737531 u^3 bi6[12,4] = -191.0108688459008575301668235408710443617 u^4 bi6[12,5] = 206.4914805885088698628382451383103849157 u^5 bi6[12,6] = -75.88416094591997399628338417114448084979 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[13] bi6[13,1] = 0. u bi6[13,2] = 8.988460463549843613179267360284225527221 u^2 bi6[13,3] = -22.01054591328694985565641844226398305499 u^3 bi6[13,4] = -7.345057460672374173081104444352318815998 u^4 bi6[13,5] = 44.76791080700622346041439477435968468808 u^5 bi6[13,6] = -24.40076789659674304485613924802760834431 u^6 # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # Coupling coefficients for c[14] = 0.2 # ---------------------------------------------------- a[14,1] = .4239343814515501060094146280117196578909e-2 a[14,2] = 0. a[14,3] = 0. a[14,4] = -.2880088440588024929203309302323774895311e-1 a[14,5] = -.1561313451836167612990532973676732282680e-1 a[14,6] = -.3772450505529428268179663178917428336072e-2 a[14,7] = .3245734097808960001585260640853677548242e-2 a[14,8] = .8603435377441466872862706303528262476251e-4 a[14,9] = .2106265767977773913696812615616385200009e-2 a[14,10] = 0. a[14,11] = -.3016144964048450805517493734589401784947e-2 a[14,12] = .5705919474826100789407933989567458123953e-1 a[14,13] = .1844660416114821469574513931782147787095 # # ******************************************************** # Coupling coefficients for c[15] = 0.55 # ---------------------------------------------------- a[15,1] = .6426855860713734548879590122226790728583e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = .2546653772640717047142159333872138729205 a[15,5] = .1469651267237810497371860857486384530552 a[15,6] = .3972163022059171542176946411211235355657e-1 a[15,7] = -.9753798128328710721988795220527122768430e-2 a[15,8] = -.4988114960512004205518135180980765880723e-3 a[15,9] = -.1407017822157130628860770362533290157622e-1 a[15,10] = 0. a[15,11] = .1656552099246425529226193691931890027173e-1 a[15,12] = .8958041046158950043216150022903491960228e-1 a[15,13] = -.3744383642368435365524250925462830575941e-1 a[15,14] = 0. # # ******************************************************** # Coupling coefficients for c[16] = 0.8 # ---------------------------------------------------- a[16,1] = .1075979175672079826953485518098848485555 a[16,2] = 0. a[16,3] = 0. a[16,4] = .6046287404689268948988736418736263782581 a[16,5] = .4001578717408221353737548224265739545530 a[16,6] = .1309055628617231640109206578434294250405 a[16,7] = .8578657403377824100564440983273504216235e-1 a[16,8] = .3218771380561001089571419248053695059297e-3 a[16,9] = -.7218727943754251027984473312384460361185e-2 a[16,10] = 0. a[16,11] = -.1919782998444934106496795375722316713643e-2 a[16,12] = -.2679737998002351527614635573862155840129 a[16,13] = -.2522862330680801801975543996367326569876 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] = -11.37572105906188521906019076059155652702 u^2 bi7[1,3] = 55.54392777895945305477116068327137121945 u^3 bi7[1,4] = -133.3368501245211258414858426113445009661 u^4 bi7[1,5] = 168.2119420352797030402114618556752638832 u^5 bi7[1,6] = -107.3784586526884673871641005984217053680 u^6 bi7[1,7] = 27.38258585986602910881108060412870229317 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] = 3.764611432634984039903132308223507696439 u^2 bi7[4,3] = -49.74370029842448052870158113693855125945 u^3 bi7[4,4] = 231.8691062506197269034588350783889366613 u^4 bi7[4,5] = -459.4482464866084934706620619577444255385 u^5 bi7[4,6] = 405.2344835680367475324356026090379312901 u^6 bi7[4,7] = -131.4200308496647788498343096263846526154 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0. u bi7[5,2] = 3.959879370211703741288805094387629727735 u^2 bi7[5,3] = -52.32387356159419078919054341680435212277 u^3 bi7[5,4] = 243.8960054341099904293657282539963116164 u^4 bi7[5,5] = -483.2795271167810874192663514284842962186 u^5 bi7[5,6] = 426.2537317048656424641168334983242341424 u^6 bi7[5,7] = -138.2367020624813163601197338288387676163 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0. u bi7[6,2] = 1.864004747014688262367427230872878615173 u^2 bi7[6,3] = -24.63003025665240043618749231689047920183 u^3 bi7[6,4] = 114.8073639129052756803494633289773387482 u^4 bi7[6,5] = -227.4905997029230452884450266657036462728 u^5 bi7[6,6] = 200.6472685273029187679737361432045421597 u^6 bi7[6,7] = -65.07114100355650915759821633681324231601 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0. u bi7[7,2] = 3.656600299129136064799156532103501278242 u^2 bi7[7,3] = -48.31649498118212356800456213800157020580 u^3 bi7[7,4] = 225.2165086481128865004017776474457454433 u^4 bi7[7,5] = -446.2661354564780436778961198499293194912 u^5 bi7[7,6] = 393.6078292136444468161526241089455015782 u^6 bi7[7,7] = -127.6494354638057014192324118062873836751 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0. u bi7[8,2] = .4517240367617976558505711070424158297399e-1 u^2 bi7[8,3] = -.5968856415692682438271718613262997636234 u^3 bi7[8,4] = 2.782248594579874229578281575195171805187 u^4 bi7[8,5] = -5.513020939874078422441628476240079888510 u^5 bi7[8,6] = 4.862498030090448745466527190614004930422 u^6 bi7[8,7] = -1.576937963162336011227535100099128747972 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0. u bi7[9,2] = .7055984398652487275717229483991248679920 u^2 bi7[9,3] = -9.323426322149200256846360592590936191858 u^3 bi7[9,4] = 43.45906145986305362100006707944347210841 u^4 bi7[9,5] = -86.11405764468659400250258360659287129165 u^5 bi7[9,6] = 75.95281066898106313468911704665344854880 u^6 bi7[9,7] = -24.63196279188407428060377381196509491838 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] = -.9130305083551434822983181343648307928497 u^2 bi7[11,3] = 12.12960947421187239704665387250349881095 u^3 bi7[11,4] = -56.98469884291763823074264631229912416495 u^4 bi7[11,5] = 114.4080116444043401695016428643930257816 u^5 bi7[11,6] = -102.8152121151275378234703716100785203479 u^6 bi7[11,7] = 34.17532034778410696996303931984595071315 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0. u bi7[12,2] = -1.089361877675816715652166099697730979938 u^2 bi7[12,3] = 7.367793678499725409008764424878709646361 u^3 bi7[12,4] = -19.30760614663886339734555929848880537519 u^4 bi7[12,5] = 24.67823638273292482765868587392167489166 u^5 bi7[12,6] = -15.45801965116907752105492105340311734352 u^6 bi7[12,7] = 3.808957614251107397385196152789269160621 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0. u bi7[13,2] = 15.34643468290882235994017578456731974122 u^2 bi7[13,3] = -103.7941264159905549588960840068113711636 u^3 bi7[13,4] = 271.9967741526741977011774512306704327562 u^4 bi7[13,5] = -347.6557703166652450001832129657750510608 u^5 bi7[13,6] = 217.7655504247260049327175293439476898012 u^6 bi7[13,7] = -53.65886252765322503475585938659902007421 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0. u bi7[14,2] = -8.636518599210444916732749188482957747725 u^2 bi7[14,3] = 115.4940651601499860632591305558940326346 u^3 bi7[14,4] = -459.4476387558029721846960403537832782697 u^4 bi7[14,5] = 808.8600706110512695514054481911093486843 u^5 bi7[14,6] = -658.1708925992413422180955516020302043595 u^6 bi7[14,7] = 201.9009141830535037048597623972930590580 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0. u bi7[15,2] = -3.141201937089953234322068229426412874354 u^2 bi7[15,3] = 43.28916474899943161688190029029136086060 u^3 bi7[15,4] = -211.7037889637762149503604716066206691997 u^4 bi7[15,5] = 446.2518805125744167680030438274272909210 u^5 bi7[15,6] = -414.8430434443681489808419328490259533135 u^6 bi7[15,7] = 140.1469890836604687806395285673543836059 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi[16] bi7[16,1] = 0. u bi7[16,2] = -4.186467394047519393389984596694714587897 u^2 bi7[16,3] = 54.90397663674175024068618564252458673687 u^3 bi7[16,4] = -253.2464856192081904607010440115810311633 u^4 bi7[16,5] = 493.3572164779739329246167023379430856002 u^5 bi7[16,6] = -425.6585456750526984629250922277678517183 u^6 bi7[16,7] = 134.8303055735927251517132328555759251325 u^7 # ******************************************************** Norms of low order INTERPOLANT coefficients on [0,2] u Max norm 2-norm ------------------------------------------------- 0.1000000000 .6877499670e-4 .7560702416e-4 0.2000000000 .1115556142e-3 .1225347122e-3 0.3000000000 .5378050527e-4 .6079192797e-4 0.4000000000 -.5299626255e-4 .6199337652e-4 0.5000000000 -.1285966983e-3 .1441743737e-3 0.6000000000 -.1261057244e-3 .1436942322e-3 0.7000000000 -.5785335813e-4 .7780106254e-4 0.8000000000 .2839910212e-4 .4757430745e-4 0.9000000000 .2822949586e-4 .3995061594e-4 1.0000000000 -.1785714286e-40 .2867855660e-40 1.1000000000 .1495801478e-3 .1938120877e-3 1.2000000000 .1056344629e-2 .1376361813e-2 1.3000000000 .3855327694e-2 .5087887862e-2 1.4000000000 .1047097356e-1 .1405008638e-1 1.5000000000 .2388919144e-1 .3266877310e-1 1.6000000000 .4846801531e-1 .6765641912e-1 1.7000000000 .9028716820e-1 .1287887519 1.8000000000 .1575368313 .2298098607 1.9000000000 .2609459175 .3895003528 2.0000000000 .4142501498 .6329231477 ******************************************************** Norms of high order INTERPOLANT coefficients on [0,2] u Max norm 2-norm ------------------------------------------------- 0.1000000000 .6548842207e-4 .8984067666e-4 0.2000000000 .1074302983e-3 .1462683236e-3 0.3000000000 .7505949244e-4 .1016947857e-3 0.4000000000 .1949540598e-4 .3161431815e-4 0.5000000000 -.1629417853e-4 .3174926596e-4 0.6000000000 -.1755186366e-4 .2914477751e-4 0.7000000000 -.1034477430e-4 .1895974955e-4 0.8000000000 -.1298553454e-4 .2120568349e-4 0.9000000000 -.9894483078e-5 .2191195191e-4 1.0000000000 .9215639068e-5 .2701546765e-4 1.1000000000 -.3124309630e-4 .6719012739e-4 1.2000000000 -.3086102140e-3 .6356438758e-3 1.3000000000 -.1463914592e-2 .2847908104e-2 1.4000000000 -.4899753445e-2 .9215694921e-2 1.5000000000 -.1329157797e-1 .2454009656e-1 1.6000000000 -.3125672593e-1 .5716651072e-1 1.7000000000 -.6621372830e-1 .1206503665 1.8000000000 -.1294542389 .2359027885 1.9000000000 -.2374497558 .4338939756 2.0000000000 -.4134151238 .7589985323 ********************************************************