###################################################################### # # A 'most robust' Runge--Kutta (13:8(7)) pair # # These are approximate REAL coefficients # computed using MAPLE with 40 digits for # # a TEN-stage conventional pair methods of # orders p=7 and p=8, with dominant # stage order = 4, # # together with approximate coefficients # # for two interpolants of orders 7 and 8 # # which require 4 and 4 extra stages respectively. # (Companion files list only the RATIONAL COEFFICIENTS # of the pair, and 40-digit floating point approximations # otherwise.) # # 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 # # T92 ~ .000007546 # # (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[14]=1, c[15] - c[17] were selected in an # attempt to minimize the maximum of the 2-norm of # the local truncation error over the interval [0,1] # for the interpolant of order 7, and this value is # # T_82 ~ .0002008 # # This 2-norm has three maximum values on [0,1]. # # The remaining four nodes were selected in an attempt # to minimize the maximum of the 2-norm of the local # truncation error on the interval [0,1] for the # interpolant of order 8, and this value is # # T_92 ~ .00003905 # # This 2-norm has a single maximum value 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. # ###################################################################### # # NODES # ----- c[1] = 0. c[2] = .25 c[3] = .1128884514435695538057742782152230971129 c[4] = .1693326771653543307086614173228346456693 c[5] = .424 c[6] = .509 c[7] = .867 c[8] = .15 c[9] = .7090680365138684008060140010282474786750 c[10] = .32 c[11] = .45 c[12] = 1. c[13] = 1. # #********************************************************* # COUPLING COEFFICIENTS # --------------------- c[1] = 0. # c[2] = .25 a[2,1] = .25 # c[3] = .1128884514435695538057742782152230971129 a[3,1] = .8740084650491523205268632759487741197705e-1 a[3,2] = .2548760493865432175308795062034568513581e-1 # c[4] = .1693326771653543307086614173228346456693 a[4,1] = .4233316929133858267716535433070866141732e-1 a[4,2] = 0. a[4,3] = .1269995078740157480314960629921259842520 # c[5] = .424 a[5,1] = .4260950588874226149488144523757227409094 a[5,2] = 0. a[5,3] = -1.598795284659152326542773323065718111709 a[5,4] = 1.596700225771729711593958870689995370799 # c[6] = .509 a[6,1] = .5071933729671392951509061813851363923933e-1 a[6,2] = 0. a[6,3] = 0. a[6,4] = .2543337726460040758275471440887777803137 a[6,5] = .2039468900572819946573622377727085804470 # c[7] = .867 a[7,1] = -.2900037471752311097038837928542589612409 a[7,2] = 0. a[7,3] = 0. a[7,4] = 1.344187391026078988943868110941433700318 a[7,5] = -2.864777943361442730961110382703656282947 a[7,6] = 2.677594299510594851721126064616481543870 # c[8] = .15 a[8,1] = .9853501133799354646974040298072701428476e-1 a[8,2] = 0. a[8,3] = 0. a[8,4] = 0. a[8,5] = .2219268063075138484202403649819738790358 a[8,6] = -.1814062291180699431269033828807395245747 a[8,7] = .1094441147256254823692261491803863125415e-1 # c[9] = .7090680365138684008060140010282474786750 a[9,1] = .3871105254573114467944461816516637340565 a[9,2] = 0. a[9,3] = 0. a[9,4] = -1.442445497485527757125674555307792776717 a[9,5] = 2.905398189069950931769134644923384844174 a[9,6] = -1.853771069630105929084333267581197802518 a[9,7] = .1400364809872815426949732510977124147922 a[9,8] = .5727394081149581657574677462444770648875 # c[10] = .32 a[10,1] = -.1612440344443930810063001619791348059544 a[10,2] = 0. a[10,3] = 0. a[10,4] = -.1733960295735898408357840447396256789490 a[10,5] = -1.301289281406514740601681274517249252974 a[10,6] = 1.137950375173861730855879213143100347212 a[10,7] = -.3174764966396688010692352113804302469898e-1 a[10,8] = .9335129382493366643981106448605688485659 a[10,9] = -.8378631833473385270330085562961643320150e-1 # c[11] = .45 a[11,1] = -.1919944488158953328151080465148357607314e-1 a[11,2] = 0. a[11,3] = 0. a[11,4] = .2733085726526428490794232625401612427562 a[11,5] = -.6753497320694437291969161121094238085624 a[11,6] = .3415184981384601607173848997472838271198 a[11,7] = -.6795006480337577247892051619852462939191e-1 a[11,8] = .9659175224762387888426558649121637650975e-1 a[11,9] = .1325308251118210118072103846654538995123 a[11,10] = .3685495936038611344690632995153166681295 # c[12] = 1. a[12,1] = .6091877403645289867688841211158881778458 a[12,2] = 0. a[12,3] = 0. a[12,4] = -2.272569085898001676899980093141308839972 a[12,5] = 4.757898342694029006815525588191478549755 a[12,6] = -5.516106706692758482429468966784424824484 a[12,7] = .2900596369680119270909581856594617437818 a[12,8] = .5691423963359036822910985845480184914563 a[12,9] = .7926795760332167027133991620589332757995 a[12,10] = .1547372045328882289412619077184989823205 a[12,11] = 1.614970895662181624708321510633454443497 # c[13] = 1. a[13,1] = .8873576220853471966321169405198102270488 a[13,2] = 0. a[13,3] = 0. a[13,4] = -2.975459782108536755851363280470930158198 a[13,5] = 5.600717009488163059799039254835009892383 a[13,6] = -5.915607450536674468001493018994165735184 a[13,7] = .2202968915613492701687914254080763833124 a[13,8] = .1015509782446221666614327134090299699755 a[13,9] = 1.151434564738605590978039775212585055356 a[13,10] = 1.929710166527123939613436190080584365307 a[13,11] = 0. a[13,12] = 0. # #********************************************************* # High order weights c[ 14] = 1 # ----------------------------------------------------------- # b[1] = .4472956466669571420301584042904938246647e-1 b[2] = 0. b[3] = 0. b[4] = 0. b[5] = 0. b[6] = .1569103352770819981336869801072664540918 b[7] = .1846097340815163774070245187352627789204 b[8] = .2251638060208699104247941940035072197092 b[9] = .1479461565197023468700517988544914175374 b[10] = .7605554244495582526979836191033649101273e-1 b[11] = .1227729023501861961082434631592143738854 b[12] = .4181195863899163158338484280087188237679e-1 b[13] = 0. # #********************************************************* # Low order weights C[extra]:= 1 # -------------------------------------------------- # bh[1] = .4584711140049592587866473012201028209588e-1 bh[2] = 0. bh[3] = 0. bh[4] = 0. bh[5] = 0. bh[6] = .2623189140415238743744335658484580339239 bh[7] = .1916937233785261190448573863568842900803 bh[8] = .2170917232790261833097840742290644856820 bh[9] = .1273818962483370679680316945065673786790 bh[10] = .1151053038536532625824051575004319214889 bh[11] = 0. bh[12] = 0. bh[13] = .4056132779843756684182339143658360805005e-1 # #******************************************************** # # Largest coefficient in b or A has magnitude 5.915607 # #***************************************************** # SUMMARY OF NORMS OF ERRORS: A91, A92, A9inf # ---------------------------------------------------- # A_[9, 1] = .9530161092e-4 # A_[9, 2] = .7546770279e-5 # A_[9,oo] = .1502264202e-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 [ -4.819754354, 0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [ -5.005465977, 0] # ############################################################# # # START OF GENERATION OF INTERPOLANT # #******************************************************* # # FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7 # c[14] = 1.0 a[14,1] = .4472956466669571420301584042904938246647e-1 a[14,2] = 0. a[14,3] = 0. a[14,4] = 0. a[14,5] = 0. a[14,6] = .1569103352770819981336869801072664540918 a[14,7] = .1846097340815163774070245187352627789204 a[14,8] = .2251638060208699104247941940035072197092 a[14,9] = .1479461565197023468700517988544914175374 a[14,10] = .7605554244495582526979836191033649101273e-1 a[14,11] = .1227729023501861961082434631592143738854 a[14,12] = .4181195863899163158338484280087188237679e-1 a[14,13] = 0. # #******************************************************** # # Coupling coefficients for c[15] = .3110177634953863863927417318829099695921 # ---------------------------------------------------- a[15,1] = .4584909962591746145269856107990713535990e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = 0. a[15,5] = 0. a[15,6] = .781870674437554054886342907851634313370e-2 a[15,7] = .6155870622618810523423270651592335069895e-2 a[15,8] = .2195571449623681030222448202857462664973 a[15,9] = -.1222581732697346808194498111979519895319e-1 a[15,10] = .3261300492958830346952381858768844254700e-1 a[15,11] = .1254797805520437910933978712641756586385e-1 a[15,12] = .1129681142614891903100335083803550169907e-1 a[15,13] = 0. a[15,14] = -.1259503554386166268241032464519842162533e-1 # #******************************************************** # # Coupling coefficients for c[16] = .536 # ---------------------------------------------------- a[16,1] = .4167847284138612914768499355650240017377e-1 a[16,2] = 0. a[16,3] = 0. a[16,4] = 0. a[16,5] = 0. a[16,6] = -.4073849103699890037905839225012252772752 a[16,7] = -.2302256719981637874461745430790814930259e-1 a[16,8] = .2482392242653016164318596667033368515282 a[16,9] = .7400026817560596139676184623176512480681e-1 a[16,10] = -.1930141806259700048703973978096116667950 a[16,11] = .6690280284382524116505423177469790757302 a[16,12] = -.1099229564510122581232500661965260824893e-1 a[16,13] = 0. a[16,14] = .1480389368968139520045852342144953350362e-1 a[16,15] = .1226640664306490993906164335783647158793 # #******************************************************** # # Coupling coefficients for c[17] = .132 # ---------------------------------------------------- a[17,1] = .5646609456796632256456589489425447704092e-1 a[17,2] = 0. a[17,3] = 0. a[17,4] = 0. a[17,5] = 0. a[17,6] = -.3522910440496784686626294493945401506422e-1 a[17,7] = .6568161065081292073153281369860458994523e-1 a[17,8] = .7383545165330150178251163230257161341400e-1 a[17,9] = -.1925857315058937001203418153699886798201 a[17,10] = .3765563615652640794139928459441174753686e-1 a[17,11] = -.5186009042220349899736698512682662022351 a[17,12] = .9190279569743070828256110335276636243207e-1 a[17,13] = 0. a[17,14] = -.1036134627781298571789386786962991910815 a[17,15] = .1051965910775021413022577292180480274178 a[17,16] = .5512910231074863915343848322133512704141 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi7 WITH 17 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi7[1] bi7[1,1] = 1. u bi7[1,2] = -10.95147587229614758318365889116594477641 u^2 bi7[1,3] = 59.39797899385550603482439108081375286705 u^3 bi7[1,4] = -169.0781428890998540020990176377966746661 u^4 bi7[1,5] = 253.7649460499865317473901174861233707133 u^5 bi7[1,6] = -188.8168930939479177124413410432786593372 u^6 bi7[1,7] = 54.72831637616857722971252484573320458188 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] = 0. u^2 bi7[4,3] = 0. u^3 bi7[4,4] = 0. u^4 bi7[4,5] = 0. u^5 bi7[4,6] = 0. u^6 bi7[4,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0. u bi7[5,2] = 0. u^2 bi7[5,3] = 0. u^3 bi7[5,4] = 0. u^4 bi7[5,5] = 0. u^5 bi7[5,6] = 0. u^6 bi7[5,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0. u bi7[6,2] = -46.41241316458094841876068921071099715321 u^2 bi7[6,3] = 459.3688444435924971928954242395515651436 u^3 bi7[6,4] = -1703.186404409985164695630781250364423728 u^4 bi7[6,5] = 2931.752704829931582483862497010150786386 u^5 bi7[6,6] = -2358.261136034433343571675092313108711217 u^6 bi7[6,7] = 716.8953146707524590074423285045890470229 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0. u bi7[7,2] = 5.227285917918245735074668338738074166332 u^2 bi7[7,3] = -52.19138815536680122729086617107330011960 u^3 bi7[7,4] = 196.4757401286306792465022894549961307879 u^4 bi7[7,5] = -345.7645783562404348575962872133481782689 u^5 bi7[7,6] = 286.0233274970354228513250006834576332734 u^6 bi7[7,7] = -89.58577729789559537060778057403509706021 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0. u bi7[8,2] = -139.7815653180968236698505509266862629309 u^2 bi7[8,3] = 1384.049425472940970524118718645766012801 u^3 bi7[8,4] = -5135.222952667993413775003372854969246585 u^4 bi7[8,5] = 8848.685618357147715240773947755273958860 u^5 bi7[8,6] = -7127.416107369448863530786337537248363980 u^6 bi7[8,7] = 2169.910745331471285121172389111867409053 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0. u bi7[9,2] = -16.96405472831162058414220808777770911590 u^2 bi7[9,3] = 167.6990823447546947011045824626866813115 u^3 bi7[9,4] = -620.4432204586682643279439958288987997443 u^4 bi7[9,5] = 1064.602309196311009392134268605482807150 u^5 bi7[9,6] = -852.8153896584399852560534765441459548119 u^6 bi7[9,7] = 258.0692194608738684217708811915074666277 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[10] bi7[10,1] = 0. u bi7[10,2] = -35.50994970740373879938889812282954732412 u^2 bi7[10,3] = 351.5600123280243130483990434691207951469 u^3 bi7[10,4] = -1304.112648012321285003482011531506371013 u^4 bi7[10,5] = 2246.458668426436193382212046412329433917 u^5 bi7[10,6] = -1808.737304793872399173741152959102843326 u^6 bi7[10,7] = 550.4172773015818723712707710938988690896 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[11] bi7[11,1] = 0. u bi7[11,2] = -42.87270298332566600048011069895021463798 u^2 bi7[11,3] = 424.3841818542901776042952235614934093368 u^3 bi7[11,4] = -1573.800189653220093465588727670908499903 u^4 bi7[11,5] = 2709.864860007943442330920092084058612822 u^5 bi7[11,6] = -2180.642953240307681307096637664499789475 u^6 bi7[11,7] = 663.1895769169700070340584038519656962316 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0. u bi7[12,2] = 6.218398007581899699884002463701971016620 u^2 bi7[12,3] = -61.68791833935996795617147670399618109064 u^3 bi7[12,4] = 229.6400112156388281316309946615500701789 u^4 bi7[12,5] = -397.6368377365357121925105031172906622204 u^5 bi7[12,6] = 322.3060088561582547364776106470120863604 u^6 bi7[12,7] = -98.79785004484431078772724310817641236242 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0. u bi7[13,2] = 0. u^2 bi7[13,3] = 0. u^3 bi7[13,4] = 0. u^4 bi7[13,5] = 0. u^5 bi7[13,6] = 0. u^6 bi7[13,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0. u bi7[14,2] = -7.110162725994358177205600320842605947063 u^2 bi7[14,3] = 70.66890227222425341394718826882626305471 u^3 bi7[14,4] = -263.9692979293415718655045106287526669683 u^4 bi7[14,5] = 459.4726066154985415344035502609253525348 u^5 bi7[14,6] = -375.1621149018975902420543201066847266481 u^6 bi7[14,7] = 116.1000666695107253364136925265283839740 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0. u bi7[15,2] = 70.40432711482233995064655388213490544057 u^2 bi7[15,3] = -711.0254431217091600248329870566067339554 u^3 bi7[15,4] = 2707.824092614873928508406676691788293167 u^4 bi7[15,5] = -4757.024287614610501419698230223675358615 u^5 bi7[15,6] = 3882.656434297324157660275609187738246346 u^6 bi7[15,7] = -1192.835123290700764674797622481379352384 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0. u bi7[16,2] = 82.90873370326421893341477780944311342039 u^2 bi7[16,3] = -817.3868372557512888355927128098679221775 u^3 bi7[16,4] = 3011.227972172782346412032450058909021963 u^4 bi7[16,5] = -5142.113838585539226231277961112123447877 u^5 bi7[16,6] = 4105.547441159415473901755534239775951473 u^6 bi7[16,7] = -1240.183471194171524180332088186136716802 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[17] bi7[17,1] = 0. u bi7[17,2] = 134.8435797564225989139917137649452178416 u^2 bi7[17,3] = -1274.836840837495194475696528986714342318 u^3 bi7[17,4] = 4624.645039888703864836680006535953166510 u^4 bi7[17,5] = -7872.062171190329141410613537947906675403 u^5 bi7[17,6] = 6295.318687282414471644014603410085131340 u^6 bi7[17,7] = -1907.908294899716599508376256776362497972 u^7 # #******************************************************** # # FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 8 # # Coupling coefficients for c[18] = .334 # ---------------------------------------------------- a[18,1] = .3934466647027139074996525141009606759203e-1 a[18,2] = 0. a[18,3] = 0. a[18,4] = 0. a[18,5] = 0. a[18,6] = -.1291667299515080090913774283193174596360e-1 a[18,7] = .1945891642767310407515289952358063928465e-2 a[18,8] = -.3950219424001926036159581502570781977097e-1 a[18,9] = -.4501175554198611234137687445953085443754e-2 a[18,10] = -.998931335985195283267955477426531672585e-2 a[18,11] = -.1198539244102761951947005140289460975181e-1 a[18,12] = .1883152441766023211187579107594459282194e-2 a[18,13] = 0. a[18,14] = -.2288623796054366081491763462894660549547e-2 a[18,15] = .1239520837697704951665852820403595312797 a[18,16] = .1852238959793020024702401439703066043180e-1 a[18,17] = .2295351884637971911562351980362084556913 # #******************************************************** # # Coupling coefficients for c[19] = .35 # ---------------------------------------------------- a[19,1] = .3884096081168368537927354267981728648614e-1 a[19,2] = 0. a[19,3] = 0. a[19,4] = 0. a[19,5] = 0. a[19,6] = -.2427245527708823693410501445101278168941e-1 a[19,7] = .3128428375929178417630603820449548457213e-2 a[19,8] = -.7358478051063769623956621836398499737217e-1 a[19,9] = -.8694978890870701525550785891871065419813e-2 a[19,10] = -.1865660321520561173012739484485694591980e-1 a[19,11] = -.2246454061884046921515692264772804996290e-1 a[19,12] = .3374661498794678376921336894109603882993e-2 a[19,13] = 0. a[19,14] = -.3968398066426385004345096186031114335021e-2 a[19,15] = .1587081125691119989004468338247673365291 a[19,16] = .3991436969512407771883001041914220455046e-1 a[19,17] = .2576752236284254818557491047471989747932 a[19,18] = 0. # #******************************************************** # # Coupling coefficients for c[20] = .755 # ---------------------------------------------------- a[20,1] = .7198494848014507493654612298114036727341e-1 a[20,2] = 0. a[20,3] = 0. a[20,4] = 0. a[20,5] = 0. a[20,6] = .5076533962269455568306708785536594617654 a[20,7] = -.1572363214423407410943664004339326120598e-1 a[20,8] = 1.478215037037996550905747792359144808839 a[20,9] = .2041154554617372630941975148106197833715 a[20,10] = .3793878839474901527036840682829095687643 a[20,11] = .4643932887538339042346694891381803095294 a[20,12] = -.5513989032591925973677454467001044028380e-1 a[20,13] = 0. a[20,14] = .5665552314958980596114050893400471586229e-1 a[20,15] = -.6894099744391873790684724028801677847224 a[20,16] = -.5600773486771496063800966221681643192010 a[20,17] = -1.087054687471247989371876165297923209992 a[20,18] = 0. a[20,19] = 0. # #******************************************************** # # Coupling coefficients for c[21] = .95 # ---------------------------------------------------- a[21,1] = .5696655426838587836617455691105627859732e-1 a[21,2] = 0. a[21,3] = 0. a[21,4] = 0. a[21,5] = 0. a[21,6] = .3386438327349868685266036706413028229066 a[21,7] = .1470519205552199854380086863315794888932 a[21,8] = .7933978111737925583755765349543256757167 a[21,9] = .2067171591185578546324388929571733817983 a[21,10] = .2188160879075981870381792015062558329913 a[21,11] = .2925192926438936110272980692420782663877 a[21,12] = .1215446220886579630956353870439258394935e-1 a[21,13] = 0. a[21,14] = -.229214343240334151600210819062643523147e-2 a[21,15] = -.3225954701993023482175429441563279152604 a[21,16] = -.2993120292318899024302980359438239792811 a[21,17] = -.4920674777477051475500000629573860014675 a[21,18] = 0. a[21,19] = 0. a[21,20] = 0. # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi8 WITH 21 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi8[1] bi8[1,1] = 1. u bi8[1,2] = -11.55704394784499918973648798632772511167 u^2 bi8[1,3] = 66.18403990020051743236903334037514157008 u^3 bi8[1,4] = -212.8883506941218729828317511272778745000 u^4 bi8[1,5] = 399.4004901483858070711526889252294996443 u^5 bi8[1,6] = -429.9081253043036662426013097689484863324 u^6 bi8[1,7] = 244.9480836433383768932694452108430096114 u^7 bi8[1,8] = -57.13436418098746726741860275346451549916 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[2] bi8[2,1] = 0. u bi8[2,2] = 0. u^2 bi8[2,3] = 0. u^3 bi8[2,4] = 0. u^4 bi8[2,5] = 0. u^5 bi8[2,6] = 0. u^6 bi8[2,7] = 0. u^7 bi8[2,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[3] bi8[3,1] = 0. u bi8[3,2] = 0. u^2 bi8[3,3] = 0. u^3 bi8[3,4] = 0. u^4 bi8[3,5] = 0. u^5 bi8[3,6] = 0. u^6 bi8[3,7] = 0. u^7 bi8[3,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[4] bi8[4,1] = 0. u bi8[4,2] = 0. u^2 bi8[4,3] = 0. u^3 bi8[4,4] = 0. u^4 bi8[4,5] = 0. u^5 bi8[4,6] = 0. u^6 bi8[4,7] = 0. u^7 bi8[4,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[5] bi8[5,1] = 0. u bi8[5,2] = 0. u^2 bi8[5,3] = 0. u^3 bi8[5,4] = 0. u^4 bi8[5,5] = 0. u^5 bi8[5,6] = 0. u^6 bi8[5,7] = 0. u^7 bi8[5,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[6] bi8[6,1] = 0. u bi8[6,2] = -42.88922570505273788365519949305668899811 u^2 bi8[6,3] = 425.5475961105596329132496562303262822062 u^3 bi8[6,4] = -1807.123332957443252495058649433756787057 u^4 bi8[6,5] = 4010.773671047197687381499367304969039148 u^5 bi8[6,6] = -4809.951565585724745424257451572811185168 u^6 bi8[6,7] = 2946.928106219364357405003810613309255701 u^7 bi8[6,8] = -723.1283387936238598986478466688726493811 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[7] bi8[7,1] = 0. u bi8[7,2] = -50.46046545238869136081894062532947585815 u^2 bi8[7,3] = 500.6695602190411786594352333204214003658 u^3 bi8[7,4] = -2126.135014374059182964888708235690192332 u^4 bi8[7,5] = 4718.795989860559440653550274879729262542 u^5 bi8[7,6] = -5659.052846326103098152323613917830337143 u^6 bi8[7,7] = 3467.149650058542992090544909438985611543 u^7 bi8[7,8] = -850.7822642515111225480921303415510063376 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[8] bi8[8,1] = 0. u bi8[8,2] = -61.54534868582553215219632681074345272951 u^2 bi8[8,3] = 610.6539522338304019033626076716055757386 u^3 bi8[8,4] = -2593.192901406338786915611945700737213570 u^4 bi8[8,5] = 5755.395674010682538975119157804159619650 u^5 bi8[8,6] = -6902.203091790272687084115416681080706158 u^6 bi8[8,7] = 4228.794408567821047585084418811092303084 u^7 bi8[8,8] = -1037.677529123876112401217700900292618796 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[9] bi8[9,1] = 0. u bi8[9,2] = -40.43899395131401863982670046792893150134 u^2 bi8[9,3] = 401.2363567357150926587491099667139404762 u^3 bi8[9,4] = -1703.883628799929574818846276854906886788 u^4 bi8[9,5] = 3781.640949616386017429269840047398446571 u^5 bi8[9,6] = -4535.162364656994274251736104261557535873 u^6 bi8[9,7] = 2778.572144946015062811222863165387097153 u^7 bi8[9,8] = -681.8165177333586028419626797962516386195 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[10] bi8[10,1] = 0. u bi8[10,2] = -20.78870917127129874079845808963581439385 u^2 bi8[10,3] = 206.2659110452030826664217713788909901528 u^3 bi8[10,4] = -875.9253819087933783828895850153439169417 u^4 bi8[10,5] = 1944.050190427419106780488125885993567656 u^5 bi8[10,6] = -2331.417333399951240477805177887212533230 u^6 bi8[10,7] = 1428.396767293990699860544596716462658787 u^7 bi8[10,8] = -350.5053887441520158806914746272446155388 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[11] bi8[11,1] = 0. u bi8[11,2] = -33.55824018897895572837516004790427388334 u^2 bi8[11,3] = 332.9654058184216671783675245910608579966 u^3 bi8[11,4] = -1413.965345878223082936041069997120760387 u^4 bi8[11,5] = 3138.189230139873654299800789780778645865 u^5 bi8[11,6] = -3763.497878122456923849159234919970142220 u^6 bi8[11,7] = 2305.794044598752104590359665523482456704 u^7 bi8[11,8] = -565.8044434650382773588442714671675697003 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[12] bi8[12,1] = 0. u bi8[12,2] = -11.42870881048945757140602864400156748303 u^2 bi8[12,3] = 113.3958349912202465410122334052048357541 u^3 bi8[12,4] = -481.5448639488654903809985997613465881910 u^4 bi8[12,5] = 1068.752434618471738029840258830539430923 u^5 bi8[12,6] = -1281.709681906451142915388457964819055743 u^6 bi8[12,7] = 785.2690893288964790412926220637983726667 u^7 bi8[12,8] = -192.6922923141433811127686430865745560445 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[13] bi8[13,1] = 0. u bi8[13,2] = 0. u^2 bi8[13,3] = 0. u^3 bi8[13,4] = 0. u^4 bi8[13,5] = 0. u^5 bi8[13,6] = 0. u^6 bi8[13,7] = 0. u^7 bi8[13,8] = 0. u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[14] bi8[14,1] = 0. u bi8[14,2] = .280089315230128997622478115178850386938 u^2 bi8[14,3] = -5.75259286164653381090533534683541280573 u^3 bi8[14,4] = 36.3347704124721225090548424148590963218 u^4 bi8[14,5] = -109.4571937962587692612406962628393165216 u^5 bi8[14,6] = 172.1129037001349520787244027502513430785 u^6 bi8[14,7] = -135.1108792445301913411542783283171601724 u^7 bi8[14,8] = 41.59290247459829082789858665770259971247 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[15] bi8[15,1] = 0. u bi8[15,2] = -378.7724238112221117869223214531251760019 u^2 bi8[15,3] = 3494.225776154715851391625819397967337338 u^3 bi8[15,4] = -14132.52357273095598444453267608181100334 u^4 bi8[15,5] = 30339.16554647382622556866309310821888312 u^5 bi8[15,6] = -35569.49413669981403762620160352927123363 u^6 bi8[15,7] = 21453.09158699572675008794946379004420055 u^7 bi8[15,8] = -5205.692776382276693190581775232023008049 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[16] bi8[16,1] = 0. u bi8[16,2] = 40.50201956693941882963694376959127659632 u^2 bi8[16,3] = -373.6364947931285715555150537385335867510 u^3 bi8[16,4] = 1511.186428287240743583533813015489631327 u^4 bi8[16,5] = -3244.157703572217854474133486062797737913 u^5 bi8[16,6] = 3803.435141911907645680964061921597226771 u^6 bi8[16,7] = -2293.972529692118357473909311641639490934 u^7 bi8[16,8] = 556.6431382913769754094230327362926809033 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[17] bi8[17,1] = 0. u bi8[17,2] = 82.87874270864547488242848049989674796094 u^2 bi8[17,3] = -764.5673783585078165656167350318583716114 u^3 bi8[17,4] = 3092.320642624178405145663924749925351635 u^4 bi8[17,5] = -6638.476661052820557878445016324923017517 u^5 bi8[17,6] = 7782.918627416922583360469786834023795786 u^6 bi8[17,7] = -4694.125406631430880134747431533068575276 u^7 bi8[17,8] = 1139.051433293012791190246990806004069021 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[18] bi8[18,1] = 0. u bi8[18,2] = 669.6598644458484065321915951171526447991 u^2 bi8[18,3] = -5850.820901650828884055491790472630555044 u^3 bi8[18,4] = 23333.04084833567997469401260543798026834 u^4 bi8[18,5] = -50385.82830346012115338587125105399979569 u^5 bi8[18,6] = 59783.07749724851222493390930711927469434 u^6 bi8[18,7] = -36504.68815588032690740194590116823416855 u^7 bi8[18,8] = 8955.559150961236338683195435020456911796 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[19] bi8[19,1] = 0. u bi8[19,2] = -244.0795068338021105727923077508888853942 u^2 bi8[19,3] = 1861.191773381140645800960136422632989480 u^3 bi8[19,4] = -6957.119234219611767500376947472980748193 u^4 bi8[19,5] = 14850.45916369077493827029151043862042156 u^5 bi8[19,6] = -17815.70975006014296656139467211914493777 u^6 bi8[19,7] = 11067.03712002351762274537576720651996859 u^7 bi8[19,8] = -2761.779565981876362182063486724758808276 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[20] bi8[20,1] = 0. u bi8[20,2] = 68.11738686016824405368089974720930254050 u^2 bi8[20,3] = -685.1013161863014613169244675973321265324 u^3 bi8[20,4] = 2941.971632577083079535931825823755311343 u^4 bi8[20,5] = -6597.927348517829753096039008764528017500 u^5 bi8[20,6] = 7995.221810955869422544639292159703889221 u^6 bi8[20,7] = -4947.745846896084061682351921817440153897 u^7 bi8[20,8] = 1225.463681207094529961063380448631794824 u^8 # -------------------------------------------------------- # # COEFFICIENTS OF bi8[21] bi8[21,1] = 0. u bi8[21,2] = 34.08056366135824033096753411991316907144 u^2 bi8[21,3] = -332.4575227396350498410997435380092983347 u^3 bi8[21,4] = 1389.447304681688048353879198238962312338 u^4 bi8[21,5] = -3030.776129634329066363945648536548931551 u^5 bi8[21,6] = 3561.340792618867953986276191837795204058 u^6 bi8[21,7] = -2130.338183331475095076538718051225385564 u^7 bi8[21,8] = 508.7031747435249686104611859291129299849 u^8 # #********************************************************* # Norms of low order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- 0.1000000000 .2074696978e-4 .5872001591e-4 0.2000000000 .1812883350e-4 .4922010621e-4 0.3000000000 .9767555002e-5 .1857505917e-4 0.4000000000 .1867470939e-4 .4767680527e-4 0.5000000000 .4160079529e-4 .1154596507e-3 0.6000000000 .3980010252e-4 .1046538885e-3 0.7000000000 -.1277916379e-4 .4098399288e-4 0.8000000000 -.7157117712e-4 .2008587738e-3 0.9000000000 -.6063780086e-4 .1669497593e-3 1.0000000000 .3427083333e-39 .8591494797e-39 1.1000000000 -.3109713578e-3 .8454429619e-3 1.2000000000 -.2326846155e-2 .6306072934e-2 1.3000000000 -.9065160156e-2 .2450995249e-1 1.4000000000 -.2630930204e-1 .7100116467e-1 1.5000000000 -.6404813969e-1 .1725817295 1.6000000000 -.1383221756 .3722329703 1.7000000000 -.2735245957 .7352447754 1.8000000000 -.5052054251 1.356681569 1.9000000000 -.8834268569 2.370314399 2.0000000000 -1.476717665 3.959149340 # #********************************************************* # Norms of high order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- 0.1000000000 -.1665824802e-5 .6031979566e-5 0.2000000000 .3059560620e-5 .1176533042e-4 0.3000000000 .4162638850e-5 .1391787347e-4 0.4000000000 .5421689981e-5 .1602396643e-4 0.5000000000 .8193303739e-5 .2321234401e-4 0.6000000000 .1175823472e-4 .3463314495e-4 0.7000000000 .1336349885e-4 .3905569962e-4 0.8000000000 .9051330855e-5 .2680668071e-4 0.9000000000 .3098184004e-5 .8683840998e-5 1.0000000000 -.1502264202e-5 .7546770279e-5 1.1000000000 -.9910405350e-5 .3009955922e-4 1.2000000000 -.1183146747e-3 .3588019824e-3 1.3000000000 -.6296684661e-3 .1900188141e-2 1.4000000000 -.2276921991e-2 .6866059786e-2 1.5000000000 -.6568021535e-2 .1984236280e-1 1.6000000000 -.1629517566e-1 .4939747499e-1 1.7000000000 -.3625107922e-1 .1103910261 1.8000000000 -.7419999639e-1 .2271753485 1.9000000000 -.1421564181 .4379155451 2.0000000000 -.2580288079 .8002914335 ********************************************************