# 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 ~ .000003389 # # (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 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. # # For these choices, the maximum value of the 2-norm # for the interpolant of order 6 on [0,1] is # # Ti_72 ~ .0000165 # # This 2-norm has four local maximum values on [0,1]. # # The choice of all interpolating nodes to minimize # 2-norm of the error in the interpolant order 7 gave # this value as # # Ti_82 ~ .000003389, # # 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 nearly 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] = .69e-1 c[3] = .118 c[4] = .177 c[5] = .501 c[6] = .7737799115305331003715765296862487670813 c[7] = .994 c[8] = .998 c[9] = 1 c[10] = 1 c[11] = 1 c[12] = .3206919742864321488153177349612198724445 c[13] = .107 c[14] = .3 c[15] = .549 c[16] = .757 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- # for c[1] = 0. # # for c[2] = .69e-1 a[2,1] = .69e-1 # # for c[3] = .118 a[3,1] = .1710144927536231884057971014492753623188e-1 a[3,2] = .1008985507246376811594202898550724637681 # # for c[4] = .177 a[4,1] = .4425e-1 a[4,2] = 0 a[4,3] = .13275 # # for c[5] = .501 a[5,1] = .7353445130709566216604424016087331226659 a[5,2] = 0 a[5,3] = -2.830160657856937661591496696351623096811 a[5,4] = 2.595816144785981039931054294742889974145 # # for c[6] = .7737799115305331003715765296862487670812 a[6,1] = -12.21580485360407974005910916471598682362 a[6,2] = 0 a[6,3] = 48.82665485823736062335980699373053427134 a[6,4] = -38.55615592319928364666616600329792491404 a[6,5] = 2.719085830096535863737044703969626233400 # # for c[7] = .994 a[7,1] = 108.8614188704176574066699618897203578466 a[7,2] = 0 a[7,3] = -432.4521181775777896358931629332707752654 a[7,4] = 343.9115281800118289547200158889409233641 a[7,5] = -20.55041135925273709189369488701721016265 a[7,6] = 1.223582486401040366396880041626704217305 # # for c[8] = .998 a[8,1] = 113.4755131883738522204615568160304033854 a[8,2] = 0 a[8,3] = -450.8122021555997002820400438087344405365 a[8,4] = 358.5132765190089889943579090008312808216 a[8,5] = -21.45046667648445540174055882443151176550 a[8,6] = 1.274053318605952891766776667539031508649 a[8,7] = -.2174193904638422805639851234763413667602e-2 # # for c[9] = 1 a[9,1] = 115.6996223324232534824963925993127275021 a[9,2] = 0 a[9,3] = -459.6635446100248030478961869239726305957 a[9,4] = 365.5534717131745930309149378867953890507 a[9,5] = -21.88511586349784824146225495848432937529 a[9,6] = 1.298718109698721459187976480852777474315 a[9,7] = -.5318700918481883515898878747322241917739e-4 a[9,8] = -.3098494764731864405706095716460833640254e-2 # # for c[10] = 1 a[10,1] = 124.1543935612464600014576130437603883332 a[10,2] = 0 a[10,3] = -493.2318713314597046194663569971348299332 a[10,4] = 392.2086219315800762927575562172365337929 a[10,5] = -23.48641564290853341361596821616234280392 a[10,6] = 1.362322948908907509911149920532561575254 a[10,7] = -.7051467367205771043993968232310964220061e-2 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] = .5163520172057869163393251056217968836723e-1 b[2] = 0 b[3] = 0 b[4] = .2767172535461648728769641534539952501983 b[5] = .3374175285287150670818592701488271741753 b[6] = .1884488267810967803491085059046161195540 b[7] = 24.54134121634868026791753618430192161716 b[8] = -68.81190284469011946382716084194838780382 b[9] = 44.41634281776488378396776021757684795437 b[10] = 0 # # ******************************************************** # Low order weights C[extra]:= 1 # -------------------------------------------------- # bh[1] = .5089676583692947576073561095512200263213e-1 bh[2] = 0 bh[3] = 0 bh[4] = .2793777374763233901369432426263934138476 bh[5] = .3281330142746535239936396881369403928344 bh[6] = .2241721218186151033581794837350130009230 bh[7] = .7874574778015076584344903106189416715189 bh[8] = 0 bh[9] = 0 bh[10] = -.6700371172080291516839883360724104817561 # #*******************************************************` # # Largest coefficient in b or A has magnitude `, 493.2319 # #******************************************************** # SUMMARY OF NORMS OF ERRORS: A81, A82, A8inf` # ----------------------------------------------------` # A_[8, 1] = `, .2719852586e-4 # A_[8, 2] = `, .3389335684e-5 # A_[8,oo] = `, .8639228411e-6 #***************************************************** # # END OF GENERATION OF A PAIR OF RK METHODS # # ############################################################# # # START OF GENERATION OF STABILITY INTERVALS # # ############################################################# # # Stability Boundaries of High Order Method` # -----------------------------------------` # Real Stability Interval is nearly [ -4.910807773, 0] # # Stability Boundaries of Low Order Method` # ----------------------------------------` # Real Stability Interval is nearly [ -3.999588993, 0] # # START OF GENERATION OF INTERPOLANT # #******************************************************* # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # # Coupling coefficients # for c[11] = 1 # ---------------------------------------------------- a[11,1] = .5163520172057869163393251056217968836723e-1 a[11,2] = 0 a[11,3] = 0 a[11,4] = .2767172535461648728769641534539952501983 a[11,5] = .3374175285287150670818592701488271741753 a[11,6] = .1884488267810967803491085059046161195540 a[11,7] = 24.54134121634868026791753618430192161716 a[11,8] = -68.81190284469011946382716084194838780382 a[11,9] = 44.41634281776488378396776021757684795437 a[11,10] = 0 # # ******************************************************** # # Coupling coefficients # for c[12] = .3206919742864321488153177349612198724445 # ---------------------------------------------------- a[12,1] = .5595947882055415742583956081637901452552e-1 a[12,2] = 0 a[12,3] = 0 a[12,4] = .2481262830322509327139022005325437452965 a[12,5] = .2103052990591148013587987543073879445482e-1 a[12,6] = -.1086652415454008449407261022406674546105e-1 a[12,7] = 5.710311335454033086222886871347348898943 a[12,8] = -16.59749151785386284126327966975768025349 a[12,9] = 10.90440686603898964031684903021468235733 a[12,10] = 0 a[12,11] = -.1078447695690422224268752339872593914865e-1 # # ******************************************************** # # Coupling coefficients # for c[13] = .1070000000000000000000000000000000000000 # ---------------------------------------------------- a[13,1] = .5365176199729764566758655497507692759771e-1 a[13,2] = 0 a[13,3] = 0 a[13,4] = .9828513377552059696782507697799478414148e-1 a[13,5] = .2236190536978546688920574502539925812364e-1 a[13,6] = -.3196168594283279436559562148268690316884e-2 a[13,7] = .5451003257307239985591851329642383758416 a[13,8] = -1.638732002009432676897859879241483267007 a[13,9] = 1.093896392979466849054156189707873746269 a[13,10] = 0 a[13,11] = -.4657483105197485920611507484730881742515e-4 a[13,12] = -.6432077441802662594433314318598382583251e-1 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 13 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1 bi6[1,2] = -7.582446684249578716313054661512721933240 u^2 bi6[1,3] = 21.65339697937461679177171822544604176620 u^3 bi6[1,4] = -26.50313170111562393346845843813196066333 u^4 bi6[1,5] = 13.68567041142918450667757590934976189120 u^5 bi6[1,6] = -2.201853803718019957033848524588941372473 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] = 5.889034278272535871631390961135086381406 u^2 bi6[4,3] = -57.96960154935775491821473273509196795388 u^3 bi6[4,4] = 174.1066668861731135303452708623511580529 u^4 bi6[4,5] = -196.2003627160861165553101224432427862687 u^5 bi6[4,6] = 74.45098035454438694442515750830250503840 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0 bi6[5,2] = 3.196839349796654162957352828525358112148 u^2 bi6[5,3] = -31.32397572700528310044209947396113560366 u^3 bi6[5,4] = 93.05298102970384392438258309708366195791 u^4 bi6[5,5] = -102.8968871064061647967771234654923865084 u^5 bi6[5,6] = 38.30845998243966487696114628399332921615 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0 bi6[6,2] = -.8787886596190119861699566896289502028596e-1 u^2 bi6[6,3] = 1.013797983077119427760544302767092045959 u^3 bi6[6,4] = -4.100173002382458810769471031390844351476 u^4 bi6[6,5] = 6.641160480067744814819942865759689363544 u^5 bi6[6,6] = -3.278457768019407452844911962268425918186 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0 bi6[7,2] = -208.3972099204096382331570396018818793788 u^2 bi6[7,3] = 2077.909119755834934676745809403064182251 u^3 bi6[7,4] = -6428.966291404445362174964426975619327549 u^4 bi6[7,5] = 7605.042110521116554859824801255385155563 u^5 bi6[7,6] = -3021.046387735747808860531607896646209269 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0 bi6[8,2] = 594.3593292228751204027648218620251141015 u^2 bi6[8,3] = -5925.388971699224459341618710912833599773 u^3 bi6[8,4] = 18326.52193738665917733083306878451733081 u^4 bi6[8,5] = -21667.18569363528617503083225733032464553 u^5 bi6[8,6] = 8602.881495880286217175025916754667412586 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0 bi6[9,2] = -386.8811915634934821635422144581019035900 u^2 bi6[9,3] = 3856.672838888456624774287551141874741487 u^3 bi6[9,4] = -11926.20151472972403478718897872831668331 u^4 bi6[9,5] = 14096.40733595464142660949072316887784424 u^5 bi6[9,6] = -5595.581125732115650649079320906757150875 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] = .6111099397023196267050121119527364301482 u^2 bi6[11,3] = -6.151956371109289894189681549817017604566 u^3 bi6[11,4] = 19.45873699463664083616118469228645171564 u^4 bi6[11,5] = -23.90604463475469049657337318293279633818 u^5 bi6[11,6] = 9.988154071525019927896857928510625796953 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0 bi6[12,2] = -8.007634773786193306062361780345514409318 u^2 bi6[12,3] = 74.60031392403404792779391526165329391946 u^3 bi6[12,4] = -199.6748294772657466686035644277447081048 u^4 bi6[12,5] = 207.5792562775741227780748301919115920885 u^5 bi6[12,6] = -74.49710595055623073120281924547466349383 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[13] bi6[13,1] = 0 bi6[13,2] = 6.900049017254163553633088407166619306472 u^2 bi6[13,3] = -11.01496218408055634389431366310163053450 u^3 bi6[13,4] = -27.69438198223954924672720783503507856034 u^4 bi6[13,5] = 60.83345444770411331060500303070857149829 u^5 bi6[13,6] = -29.02415929863817127361656993973848170993 u^6 # # ******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7 # # Coupling coefficients c[14] = .3 # ---------------------------------------------------- a[14,1] = .1919717775847849685111579308732385545808e-1 a[14,2] = 0. a[14,3] = 0. a[14,4] = -.4739427173175533989609356107380842465934e-1 a[14,5] = -.2641522514821109801913974252340613156326e-1 a[14,6] = .7054089167217366089095708745682724510887e-7 a[14,7] = 1.551079892101621437320838156104778064997 a[14,8] = -4.428095538025376982599451861908192890036 a[14,9] = 2.883718325674217149748617484261428760772 a[14,10] = 0. a[14,11] = -.4297481934497428346001203166435386609045e-2 a[14,12] = .1262644300588589746368092621052046424260 a[14,13] = .2259426207057731181296447821560200523865 # # ******************************************************** # # Coupling coefficients c[15] = .549 # ---------------------------------------------------- a[15,1] = .6125242269008585247666695476683627174961e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = .2525310401834353816786786047482953692850 a[15,5] = .1507054877413180556949606462372746086457 a[15,6] = .1024270639019054271328153617559339591462e-1 a[15,7] = -6.438444503804215943508588627336983785574 a[15,8] = 18.44870904095403871125278855846631959613 a[15,9] = -12.03592743273174801641739335902460845726 a[15,10] = 0. a[15,11] = .1513193748819522199500460159517896993473e-1 a[15,12] = .1043447705299915390244100081194588662149 a[15,13] = -.1954546944129134490980892374736483504656e-1 a[15,14] = 0. # # ******************************************************** # # Coupling coefficients c[16] = .757 # ---------------------------------------------------- a[16,1] = .8959466056806525077749567658988579412839e-1 a[16,2] = 0. a[16,3] = 0. a[16,4] = .6389214946562193863310204803419201536875 a[16,5] = .4309120296822662010167723442150653884245 a[16,6] = .7695592670277012062550066233926247449166e-1 a[16,7] = -7.189894742007330283923979037259652075040 a[16,8] = 21.03645589658795446951415976277135782026 a[16,9] = -13.86140603343527732092225205611764539171 a[16,10] = 0. a[16,11] = .8282681779726606982694318905099680481702e-2 a[16,12] = -.2148456743069278028823832921628374810433 a[16,13] = -.2579762402274666275190288596224563636723 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] = -8.503890289208650086322705078936310611299 u^2 bi7[1,3] = 31.18234339085710879255131238050971776003 u^3 bi7[1,4] = -56.83047976918539469582020937384021018043 u^4 bi7[1,5] = 52.79739831471022178930385951784095486118 u^5 bi7[1,6] = -22.95183302720539338830128746758333934610 u^6 bi7[1,7] = 3.358096581752686280222962532571367204997 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] = 6.267709807177576432406478077069849818602 u^2 bi7[4,3] = -68.59145123448141827278305837000589905312 u^3 bi7[4,4] = 270.1006378954180870537780616882949819305 u^4 bi7[4,5] = -486.2388094162716631231341727978491179719 u^5 bi7[4,6] = 407.1399818231500101241727526996656040234 u^6 bi7[4,7] = -128.4013516214464273415630971437214234972 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0 bi7[5,2] = 7.642585077624110415092972164105836819717 u^2 bi7[5,3] = -83.63756743442629097888399107640792828691 u^3 bi7[5,4] = 329.3495021534571296433374383575409388467 u^4 bi7[5,5] = -592.8994135546816065453818037938193613121 u^5 bi7[5,6] = 496.4495876982774414703954108911602253524 u^6 bi7[5,7] = -156.5672764117220689374781672724308842456 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0 bi7[6,2] = 4.268409521381499328229656167910411979775 u^2 bi7[6,3] = -46.71186327091233753264841481404164445942 u^3 bi7[6,4] = 183.9428068612486847316162527859995659975 u^4 bi7[6,5] = -331.1363205426106275865463338454994779573 u^5 bi7[6,6] = 277.2687677656847319301330472909470886979 u^6 bi7[6,7] = -87.44335150801085409043509907941132813885 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0 bi7[7,2] = 555.8670558189042010639619738936099948255 u^2 bi7[7,3] = -6083.199322405529992849514286236321590545 u^3 bi7[7,4] = 23954.53059900739546845895162864872763414 u^4 bi7[7,5] = -43123.26890207833867031432260116626186577 u^5 bi7[7,6] = 36108.19740617653066320546034515369066848 u^6 bi7[7,7] = -11387.58549530261298929661952410914291952 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0 bi7[8,2] = -1558.605518026587403407644982678592484625 u^2 bi7[8,3] = 17056.79070544840796438857322691726272923 u^3 bi7[8,4] = -67166.53412450698972069889474017010525280 u^4 bi7[8,5] = 120914.1001657429967184958527822159285576 u^5 bi7[8,6] = -101244.4165095385499516578794640912685653 u^6 bi7[8,7] = 31929.85337803603227341616601696482662805 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0 bi7[9,2] = 1006.040439872400448623697106213084759679 u^2 bi7[9,3] = -11009.72698072282062279582145142481879227 u^3 bi7[9,4] = 43354.29892541348545701322502088633140394 u^4 bi7[9,5] = -78046.99336079503441879814959011469619894 u^5 bi7[9,6] = 65350.71006860324690110919871372728749245 u^6 bi7[9,7] = -20609.91274955351288136818203906961181690 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] = -1.612970387190759595570543259931261611538 u^2 bi7[11,3] = 17.72909772735324772041234368381732499671 u^3 bi7[11,4] = -70.34937384312194980443642443527448780299 u^4 bi7[11,5] = 128.1350088912801760638876471983594315219 u^5 bi7[11,6] = -109.0734352266536956182626795265083915641 u^6 bi7[11,7] = 35.17167283833298123396965633953738445994 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0 bi7[12,2] = -2.626599644929271923086786435983631061624 u^2 bi7[12,3] = 9.920024380612826013243113812430759529521 u^3 bi7[12,4] = -7.644130389085101218500146944756885977802 u^4 bi7[12,5] = -14.86430552714213266618722882949033593695 u^5 bi7[12,6] = 26.11390292373462455033637542344644699734 u^6 bi7[12,7] = -10.89889174319094475580532702564635355049 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0 bi7[13,2] = 9.561675242924823427014852060688684933761 u^2 bi7[13,3] = -36.11210856303548088470642279130799743208 u^3 bi7[13,4] = 27.82711572968801973495325529709252051216 u^4 bi7[13,5] = 54.11089673926027907217912866089741112920 u^5 bi7[13,6] = -95.06308263006681094546659235128381873539 u^6 bi7[13,7] = 39.67550348122916959602577912391319959234 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0 bi7[14,2] = -9.791432054831754863321754176936907826271 u^2 bi7[14,3] = 116.6733547596190133707339789196320683056 u^3 bi7[14,4] = -427.0178661529826169224279390827040458460 u^4 bi7[14,5] = 710.0693817808117189359000209005264369724 u^5 bi7[14,6] = -556.8214238669928662708433693467844704976 u^6 bi7[14,7] = 166.8879855343765057499590627862669188919 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0 bi7[15,2] = .7372370767192203283333752534265429181322e-1 u^2 bi7[15,3] = 2.611540012355412116044967235469929822777 u^3 bi7[15,4] = -30.67846770866969010553103505851633192303 u^4 bi7[15,5] = 100.4285900318910859666056494362334084911 u^5 bi7[15,6] = -119.6365555255543602449647502655108119632 u^6 bi7[15,7] = 47.20116948230563023501183112698115128060 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0 bi7[16,2] = -8.581188645336741447289604471431596612033 u^2 bi7[16,3] = 93.07222791200057091279868176378132240393 u^3 bi7[16,4] = -360.9951446906583731902511625987898308351 u^4 bi7[16,5] = 635.7596704131289187099926426178301572747 u^5 bi7[16,6] = -517.9168751756012942639785021372581285996 u^6 bi7[16,7] = 158.6613101864669192787279448258680763681 u^7 # # ************************************************************* # Norms of low order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, -.1966985250e-5, .4576909238e-5 .2000000000, -.3738182523e-6, .9758011213e-6 .3000000000, .3609599513e-5, .8302962692e-5 .4000000000, .1829059747e-5, .4823538039e-5 .5000000000, -.4045610375e-5, .9286345914e-5 .6000000000, -.7843618135e-5, .1655653441e-4 .7000000000, -.4901688495e-5, .1015566985e-4 .8000000000, .4305845243e-5, .1192303763e-4 .9000000000, .4163562013e-5, .1261211912e-4 1.000000000, -.2504160714e-55,.4157641108e-55 1.100000000, .2882470479e-4, .7577780068e-4 1.200000000, .2281538039e-3, .5872216924e-3 1.300000000, .9224621928e-3, .2343378272e-2 1.400000000, .2748749637e-2, .6921168266e-2 1.500000000, .6825008317e-2, .1707532013e-1 1.600000000, .1496759182e-1, .3726652126e-1 1.700000000, .2996348558e-1, .7432221220e-1 1.800000000, .5590347874e-1, .1382444552 1.900000000, .9858223750e-1, .2431793093 2.000000000, .1659712799, .4085621466 # # ************************************************************* # Norms of high order INTERPOLANT error coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- .1000000000, .9335296894e-6, .2045237747e-5 .2000000000, .2319416590e-6, .7364213512e-6 .3000000000, -.4834758334e-6, .1044854089e-5 .4000000000, .3242862601e-6, .8995141237e-6 .5000000000, .1149154526e-5, .2624471402e-5 .6000000000, .1156636214e-5, .2814595135e-5 .7000000000, -.1109603298e-5, .2816499596e-5 .8000000000, -.1250356017e-5, .3015944386e-5 .9000000000, -.8611121620e-6, .3055778488e-5 1.000000000, .8639228411e-6, .3389335684e-5 1.100000000, .7654654611e-5, .1761517021e-4 1.200000000, .7487737684e-4, .1733064158e-3 1.300000000, .3616057743e-3, .8324583165e-3 1.400000000, .1254616518e-2, .2875780179e-2 1.500000000, .3557286353e-2, .8128353883e-2 1.600000000, .8774170609e-2, .2000163490e-1 1.700000000, .1951674389e-1, .4440807726e-1 1.800000000, .4005980054e-1, .9101431927e-1 1.900000000, .7708116255e-1, .1749060806 2.000000000, .1406204924, .3187449716 # 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