###################################################################### # # A 'most efficient' 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 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 # # T82 ~ .00001676 # # (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 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 6, and this value is # # T_72 ~ .00003665 # # This 2-norm has two local maximum values on [0,1]. # # The remaining three 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 7, and this value is # # T_82 ~ .00001676, # # the 2-norm the endpoint local truncation error. # Moreover, this 2-norm is 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. # ###################################################################### # # NODES # ----- c[1] = 0. c[2] = .5e-2 c[3] = .1088888888888888888888888888888888888889 c[4] = .1633333333333333333333333333333333333333 c[5] = .4555 c[6] = .6095094489978381317087004421486024949638 c[7] = .884 c[8] = .925 c[9] = 1. c[10] = 1. # #********************************************************* # COUPLING COEFFICIENTS # --------------------- for c[1]= 0 # for c[2]= 0.005 a[2,1] = .5e-2 # for c[3]= 0.1088888888888888888888888888888888888889 a[3,1] = -1.076790123456790123456790123456790123457 a[3,2] = 1.185679012345679012345679012345679012346 # for c[4]= 0.1633333333333333333333333333333333333333 a[4,1] = .4083333333333333333333333333333333333333e-1 a[4,2] = 0. a[4,3] = .1225 # for c[5]= 0.4555 a[5,1] = .6389139236255726780508121615993336109954 a[5,2] = 0. a[5,3] = -2.455672638223656809662640566430653894211 a[5,4] = 2.272258714598084131611828404831320283215 # for c[6]= 0.6095094489978381317087004421486024949638 a[6,1] = -2.661577375018757131119259297861818119279 a[6,2] = 0. a[6,3] = 10.80451388645613769565396655365532838482 a[6,4] = -8.353914657396199411968048547819291691541 a[6,5] = .8204875949566569791420417341743839209619 # for c[7]= 0.884 a[7,1] = 6.067741434696770992718360183877276714679 a[7,2] = 0. a[7,3] = -24.71127363591108579734203485290746001803 a[7,4] = 20.42751793078889394045773111748346612697 a[7,5] = -1.906157978816647150624096784352757010879 a[7,6] = 1.006172249242068014790040335899474187268 # for c[8]= 0.925 a[8,1] = 12.05467007625320299509109452892778311648 a[8,2] = 0. a[8,3] = -49.75478495046898932807257615331444758322 a[8,4] = 41.14288863860467663259698416710157354209 a[8,5] = -4.461760149974004185641911603484815375051 a[8,6] = 2.042334822239174959821717077708608543738 a[8,7] = -0.9834843665406107379530801693870224403537e-1 # for c[9]= 1.0 a[9,1] = 10.13814652288180787641845141981689030769 a[9,2] = 0. a[9,3] = -42.64113603171750214622846006736635730625 a[9,4] = 35.76384003992257007135021178023160054034 a[9,5] = -4.348022840392907653340370296908245943710 a[9,6] = 2.009862268377035895441943593011827554771 a[9,7] = .3487490460338272405953822853053145879140 a[9,8] = -.2714390051048312842371587140910297407572 # for c[10]= 1.0 a[10,1] = -45.03007203429867712435322405073769635151 a[10,2] = 0. a[10,3] = 187.3272437654588840752418206154201997384 a[10,4] = -154.0288236935018690596728621034510402582 a[10,5] = 18.56465306347536233859492332958439136765 a[10,6] = -7.141809679295078854925420496823551192821 a[10,7] = 1.308808578161378625114762706007696696508 a[10,8] = 0. a[10,9] = 0. # #********************************************************* # High order weights c[11] = 1 # ----------------------------------------------------------- # b[1] = .4715561848627222170431765108838175679569e-1 b[2] = 0. b[3] = 0. b[4] = .2575056429843415189596436101037687580986 b[5] = .2621665397741262047713863095764527711129 b[6] = .1521609265673855740323133199165117535523 b[7] = .4939969170032484246907175893227876844296 b[8] = -.2943031171403250441557244744092703429139 b[9] = .8131747232495109999734599440136761892478e-1 b[10] = 0. # #********************************************************* # Low order weights C[extra]:= , 1 # -------------------------------------------------- # bh[1] = .4460860660634117628731817597479197781432e-1 bh[2] = 0. bh[3] = 0. bh[4] = .2671640378571372680509102260943837899738 bh[5] = .2201018300177293019979715776650753096323 bh[6] = .2188431703143156830983120833512893824578 bh[7] = .2289871705411202883378173889763552365362 bh[8] = 0. bh[9] = 0. bh[10] = .2029518466335628222767054793810430358554e-1 # #******************************************************** # # Largest coefficient in b or A has magnitude , 187.3272 # #******************************************************** # SUMMARY OF NORMS OF ERRORS: A81, A82, A8inf # ---------------------------------------------------- # A_[8, 1] = , .1070085665e-3 # A_[8, 2] = , .1675585009e-4 # A_[8,oo] = , .8590002884e-5 #***************************************************** # # END OF GENERATION OF A PAIR OF RK METHODS # ############################################################# # # START OF GENERATION OF STABILITY INTERVALS # ############################################################# # # Stability Boundaries of High Order Method # ----------------------------------------- # Real Stability Interval is nearly [, -4.640024335, ,0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [, -4.001051009, ,0] # ############################################################# # # START OF GENERATION OF INTERPOLANT # #******************************************************* # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 8 # c[11] = 1.0 a[11,1] = .4715561848627222170431765108838175679569e-1 a[11,2] = 0. a[11,3] = 0. a[11,4] = .2575056429843415189596436101037687580986 a[11,5] = .2621665397741262047713863095764527711129 a[11,6] = .1521609265673855740323133199165117535523 a[11,7] = .4939969170032484246907175893227876844296 a[11,8] = -.2943031171403250441557244744092703429139 a[11,9] = .8131747232495109999734599440136761892478e-1 a[11,10] = 0. # # ******************************************************** # # Coupling coefficients for c[12] = 0.29 # ---------------------------------------------------- a[12,1] = .5232227691599689815470932256735029887614e-1 a[12,2] = 0. a[12,3] = 0. a[12,4] = .2249586182670571550244187743667190903405 a[12,5] = .1744370924877637539031751304611402542578e-1 a[12,6] = -.7669379876829393188009028209348812321417e-2 a[12,7] = .3435896044073284645684381456417912794447e-1 a[12,8] = -.4102097230093949839125144540100346681769e-1 a[12,9] = .2565113300520561655297104906598973655221e-1 a[12,10] = 0. a[12,11] = -.160443457e-1 # #******************************************************** # # Coupling coefficients for c[13] = 0.125 # ---------------------------------------------------- a[13,1] = .5305334125785908638834747243817578898946e-1 a[13,2] = 0. a[13,3] = 0. a[13,4] = .1219530101140188607092225622195251463666 a[13,5] = .1774684073760249704011573985936092552347e-1 a[13,6] = -.5928372667681494328907467430302313286925e-3 a[13,7] = .8381833970853750873624781948796072714855e-2 a[13,8] = -.1293369259698611956700998079778496462996e-1 a[13,9] = .9412056815253860804791356641605087829772e-2 a[13,10] = 0. a[13,11] = -.5353253107275676032399320754008272222345e-2 a[13,12] = -.6666729992455811078380186481263955324311e-1 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 13 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1. u bi6[1,2] = -7.579486522562013856370387489358761917136 u^2 bi6[1,3] = 24.84859042701758998305254381572114767883 u^3 bi6[1,4] = -38.85067748922540724491934058373805286459 u^4 bi6[1,5] = 28.75646349856373329638850558427800090199 u^5 bi6[1,6] = -8.127734295307629956447003675813952042307 u^6 bi6[1,7] = 0. u^7 # -------------------------------------------------------- # # 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 bi6[2,7] = 0. u^7 # -------------------------------------------------------- # # 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 bi6[3,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[4] bi6[4,1] = 0. u bi6[4,2] = 4.551232240400547830289289496617659221741 u^2 bi6[4,3] = -41.74306197989451542904654379577378740985 u^3 bi6[4,4] = 125.9208614672431114278470271124781332255 u^4 bi6[4,5] = -143.2724320984988187759537191634829285599 u^5 bi6[4,6] = 54.80090601373401646582358996026469228056 u^6 bi6[4,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0. u bi6[5,2] = 2.347388766837311148481149416205748182220 u^2 bi6[5,3] = -21.42965001437327943727021057869980940196 u^3 bi6[5,4] = 64.01102018753913070410229000503296635821 u^4 bi6[5,5] = -71.54964616066291046169022808133078061275 u^5 bi6[5,6] = 26.88305376043387425114838554836832824539 u^6 bi6[5,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0. u bi6[6,2] = .6629628602943922170984684021172226061984 u^2 bi6[6,3] = -5.991791209485099663334522931412659054138 u^3 bi6[6,4] = 17.51358806151491209369363907599542681959 u^4 bi6[6,5] = -18.79068837634778062158370304672269638025 u^5 bi6[6,6] = 6.758089590590961548158431819939217762152 u^6 bi6[6,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0. u bi6[7,2] = -1.894931483197030289667134961741273285056 u^2 bi6[7,3] = 17.84551891193619310038245879582238106213 u^3 bi6[7,4] = -56.77440614878830815775145830865327266897 u^4 bi6[7,5] = 70.55596299657564872116838561274122139834 u^5 bi6[7,6] = -29.23814735952325494944153354884626882202 u^6 bi6[7,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0. u bi6[8,2] = 1.489077233496668566079506992903531187388 u^2 bi6[8,3] = -13.95068086791358924071990604089950620891 u^3 bi6[8,4] = 43.93573738643896559384998172064294274652 u^4 bi6[8,5] = -53.74155980596578799479262013665711367333 u^5 bi6[8,6] = 21.97312293680341803142731298960087560542 u^6 bi6[8,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0. u bi6[9,2] = -.5934749977615343231780548830714052210176 u^2 bi6[9,3] = 5.532214575131257019966932786240153623467 u^3 bi6[9,4] = -17.25057884887540359301582894972755244557 u^4 bi6[9,5] = 20.76631879735288001882715503942847061836 u^5 bi6[9,6] = -8.373162053522248022602857998468298956311 u^6 bi6[9,7] = 0. u^7 # -------------------------------------------------------- # # 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 bi6[10,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[11] bi6[11,1] = 0. u bi6[11,2] = .5229705513661462700547131555553062786922 u^2 bi6[11,3] = -4.935085895387111016017254136967884797119 u^3 bi6[11,4] = 15.77686659404020360951757145807650520227 u^4 bi6[11,5] = -19.84035770738365925120223312747058112796 u^5 bi6[11,6] = 8.475606457364420387647202650806654444113 u^6 bi6[11,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0. u bi6[12,2] = -8.546914399459162196852655523910775929846 u^2 bi6[12,3] = 71.42491845557421929764191001202432467195 u^3 bi6[12,4] = -182.3743002138117003976306073059834582214 u^4 bi6[12,5] = 184.6615026587373916897461066715370461463 u^5 bi6[12,6] = -65.16520650104074839290475385366713666699 u^6 bi6[12,7] = 0. u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[13] bi6[13,1] = 0. u bi6[13,2] = 9.041175750584674634065105394682748876817 u^2 bi6[13,3] = -31.60097240260566461465540792605436016441 u^3 bi6[13,4] = 28.09188900392449596430672577587636184837 u^4 bi6[13,5] = 2.454436197629303379092350647679361289224 u^5 bi6[13,6] = -7.986528549532809362808773892184111850000 u^6 bi6[13,7] = 0. u^7 # #******************************************************** # # THREE ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 7 # # Coupling coefficients for c[14] = 0.25 # ---------------------------------------------------- a[14,1] = .3887903257436303686399931060834951327899e-1 a[14,2] = 0. a[14,3] = 0. a[14,4] = -.2440320330830131517910045090190069290791e-2 a[14,5] = -.1392891721467262281273220992320214734208e-2 a[14,6] = -.4744629155868013465038358934145339168472e-3 a[14,7] = .3920793241315951369383517310870803393356e-3 a[14,8] = -.4055473328512800136385880031750264996936e-3 a[14,9] = .1989709314771672628794304728258886009267e-3 a[14,10] = 0. a[14,11] = -.1027819879317916884712606136811051029682e-3 a[14,12] = .3385661513870266715302548402957613704604e-1 a[14,13] = .1814893063199928004309543737509423302792 # #******************************************************** # # Coupling coefficients for c[15] = 0.53 # ---------------------------------------------------- a[15,1] = .5723681204690012909606837582140921695189e-1 a[15,2] = 0. a[15,3] = 0. a[15,4] = .2226594806676118099285816235023183680020 a[15,5] = .1234486420018689904911221497830317287757 a[15,6] = .4006332526666490875113688731927762275433e-1 a[15,7] = -.5269894848581452066926326838943832327366e-1 a[15,8] = .4765971214244522856887315416093212596338e-1 a[15,9] = -.2138895885042213036387863538386958914368e-1 a[15,10] = 0. a[15,11] = .1519389106403640165459624646184297766866e-1 a[15,12] = .1206054671628965554251364472502413614358 a[15,13] = -.2277942301618737288237298052574548913451e-1 a[15,14] = 0. # # ******************************************************** # # Coupling coefficients for c[16] = 0.79 # ---------------------------------------------------- a[16,1] = .5137203880275681426595607279552927584506e-1 a[16,2] = 0. a[16,3] = 0. a[16,4] = .5414214473439405582401399378307410450482 a[16,5] = .3503998066921840081154745647747846804810 a[16,6] = .1419311226969218216861835872156617148040 a[16,7] = .1052737747842942254816302629823570359198 a[16,8] = -.3108184780587401700842726199589213259835e-1 a[16,9] = -.7401883149519145061791854716430279714483e-2 a[16,10] = 0. a[16,11] = -.6377932504865363437569726480040013149706e-2 a[16,12] = -.1732549590836186403386348310205265959935 a[16,13] = -.1822815677762202619429607513861847306420 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] = -8.413387198332767469319987751201351965810 u^2 bi7[1,3] = 33.67550888449089654479469983556967202215 u^3 bi7[1,4] = -70.80159089484886164618905961010838757357 u^4 bi7[1,5] = 80.64695108301297872968868805293298389704 u^5 bi7[1,6] = -47.19413969837521580145883430419406103536 u^6 bi7[1,7] = 11.13381344253924186418881142808952641234 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] = 8.754921980674397160629587282876763437696 u^2 bi7[4,3] = -88.45968286997709426134300934922618655402 u^3 bi7[4,4] = 346.9017638429916309499891288356321692825 u^4 bi7[4,5] = -629.2580030059837046812187141184986252218 u^5 bi7[4,6] = 529.6773755604192983874116479833480529304 u^6 bi7[4,7] = -167.3588698651401860365089970240284051167 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[5] bi7[5,1] = 0. u bi7[5,2] = 8.913387586637921662996190126913331844214 u^2 bi7[5,3] = -90.06081846893217794712014609702916991513 u^3 bi7[5,4] = 353.1807459217057824951538014683541349020 u^4 bi7[5,5] = -640.6476819744374433668701027882567716886 u^5 bi7[5,6] = 539.2646279047155261551781390920363285084 u^6 bi7[5,7] = -170.3880944299154827945664954924414008798 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[6] bi7[6,1] = 0. u bi7[6,2] = 5.173312029847800338889849068990984974299 u^2 bi7[6,3] = -52.27111590005538823385270070373176751689 u^3 bi7[6,4] = 204.9853867374073094711024260808085419491 u^4 bi7[6,5] = -371.8306118563602890875634623992262437796 u^5 bi7[6,6] = 312.9880934374529000210073972654145891826 u^6 bi7[6,7] = -98.89290352172494693555119599233959305606 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[7] bi7[7,1] = 0. u bi7[7,2] = 16.79537744079695986364946329034055578253 u^2 bi7[7,3] = -169.7004000005972744435739149730966805754 u^3 bi7[7,4] = 665.4937727009246303131700313781960584913 u^4 bi7[7,5] = -1207.163889233600728395392916633015853882 u^5 bi7[7,6] = 1016.129151581854603280159105697386989470 u^6 bi7[7,7] = -321.0600155723749421933210511704882816019 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[8] bi7[8,1] = 0. u bi7[8,2] = -10.00599753609866476866352971232058330270 u^2 bi7[8,3] = 101.1005433052275068199636113246449312792 u^3 bi7[8,4] = -396.4739151237843754958939772727577263768 u^4 bi7[8,5] = 719.1787707014182914108130834128646525498 u^5 bi7[8,6] = -605.3681033918824350795711030652978269725 u^6 bi7[8,7] = 191.2743989279793520691961908384572824802 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[9] bi7[9,1] = 0. u bi7[9,2] = 2.764708833638599139713222853969606774131 u^2 bi7[9,3] = -27.93460263739046178114640484830267988046 u^3 bi7[9,4] = 109.5477918613789217803046856340175757800 u^4 bi7[9,5] = -198.7128113064482116421691972646370773711 u^5 bi7[9,6] = 167.2663357164031670694252647113936863857 u^6 bi7[9,7] = -52.85010499525706346613022509203974406942 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] = -2.169632028016350481156919876642428429100 u^2 bi7[11,3] = 22.01669603756987625585768587320929912766 u^3 bi7[11,4] = -86.90152427798948350846176288615482496306 u^4 bi7[11,5] = 159.2238897386147443720253338471077193471 u^5 bi7[11,6] = -135.9618306534587908363115231453760181702 u^6 bi7[11,7] = 43.79240118328000419804718618785625308759 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[12] bi7[12,1] = 0. u bi7[12,2] = -4.890070188793803933769786966428026149549 u^2 bi7[12,3] = 22.75407737425176120799532459991506803585 u^3 bi7[12,4] = -30.78034218537730965082079824005797506535 u^4 bi7[12,5] = -2.797194317207249021142015125037024035537 u^5 bi7[12,6] = 31.36945663750840183161406140272783187147 u^6 bi7[12,7] = -15.65592732038180043387678567111987465689, u^, 7 , + # -------------------------------------------------------- # # COEFFICIENTS OF bi7[13] bi7[13,1] = 0. u bi7[13,2] = 10.86217092955196715517224349929627754387 u^2 bi7[13,3] = -50.54297141782710697188187875653305700081 u^3 bi7[13,4] = 68.37148040407511827604242008548181691494 u^4 bi7[13,5] = 6.213326521632409162585500428935637861213 u^5 bi7[13,6] = -69.68006323194158104163196358466588618336 u^6 bi7[13,7] = 34.77605679450919341971367832748521086414 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[14] bi7[14,1] = 0. u bi7[14,2] = -11.37286691922922915922346687401389055763 u^2 bi7[14,3] = 130.7905807824671644130452602841032046030 u^3 bi7[14,4] = -488.6511367778560207543260583489312609826 u^4 bi7[14,5] = 832.2148793276440873476229585070779183432 u^5 bi7[14,6] = -664.7743368554426242883314487337054193624 u^6 bi7[14,7] = 201.7928804424166224412127551654694479565 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[15] bi7[15,1] = 0. u bi7[15,2] = -5.919778732715006698693070786679427540601 u^2 bi7[15,3] = 63.27679965889218829298274978013773800731 u^3 bi7[15,4] = -265.4326820887379575820873554556433306580 u^4 bi7[15,5] = 520.1009254140610824835871087519714692468 u^5 bi7[15,6] = -467.4121095339020118993777963241667608460 u^6 bi7[15,7] = 155.3868452824017054035883640343803117904 u^7 # -------------------------------------------------------- # # COEFFICIENTS OF bi7[16] bi7[16,1] = 0. u bi7[16,2] = -10.49214619796182281022379415510181241136 u^2 bi7[16,3] = 105.3553852518801101042787230303396283676 u^3 bi7[16,4] = -409.4397501198893846479834816688367917005 u^4 bi7[16,5] = 732.8314489076540326880337353277812147333 u^5 bi7[16,6] = -606.3044574733512377981129469949015057785 u^6 bi7[16,7] = 188.0495196316683024640077644607192667895 u^7 # # ******************************************************** # Norms of low order INTERPOLANT coefficients on [0,2] # Max norm 2-norm # ------------------------------------------------- 0.1000000000 -0.1975126393e-5 0.4791674308e-5 0.2000000000 -0.2915769943e-5 0.4517785291e-5 0.3000000000 -0.6101166967e-5 0.7581780719e-5 0.4000000000 -0.1042167756e-4 0.1359773445e-4 0.5000000000 -0.1487976057e-4 0.2168385956e-4 0.6000000000 -0.1763768620e-4 0.2508686013e-4 0.7000000000 -0.1683568020e-4 0.2896021896e-4 0.8000000000 0.1631270190e-4 0.3665552763e-4 0.9000000000 0.1007291116e-4 0.2391375359e-4 1.0000000000 0.6250000000e-41 0.1321056123e-40 1.1000000000 0.4198697828e-4 0.1060965823e-3 1.2000000000 0.3090235953e-3 0.7709653344e-3 1.3000000000 0.1189836039e-2 0.2945382802e-2 1.4000000000 0.3422823047e-2 0.8429176384e-2 1.5000000000 0.8275555133e-2 0.2030505023e-1 1.6000000000 0.1777453226e-1 0.4349357287e-1 1.7000000000 0.3499119994e-1 0.8544374244e-1 1.8000000000 0.6439022379e-1 0.1569735568 1.9000000000 0.1122460225 0.2732750197 2.0000000000 0.1871335593 0.4550980156 # ******************************************************** # Norms of high order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm # ------------------------------------------------- 0.1000000000 0.1174808084e-5 0.2464855744e-5 0.2000000000 0.1362182154e-5 0.2610887526e-5 0.3000000000 0.1431670571e-5 0.2822434271e-5 0.4000000000 0.2764918097e-5 0.5414972078e-5 0.5000000000 0.4222836385e-5 0.8238427478e-5 0.6000000000 -0.4367622017e-5 0.8990852297e-5 0.7000000000 -0.6103392844e-5 0.9292849651e-5 0.8000000000 -0.7555168043e-5 0.1101288265e-4 0.9000000000 -0.8393871387e-5 0.1246534058e-4 1.0000000000 -0.8590002884e-5 0.1675585009e-4 1.1000000000 0.1735855901e-4 0.3631023520e-4 1.2000000000 0.1669088305e-3 0.3309924501e-3 1.3000000000 0.7185629143e-3 0.1415258191e-2 1.4000000000 0.2272910223e-2 0.4467185634e-2 1.5000000000 0.5997914014e-2 0.1180181455e-1 1.6000000000 0.1397828872e-1 0.2758716633e-1 1.7000000000 0.2970179417e-1 0.5885152362e-1 1.8000000000 0.5871361543e-1 0.1168482232 1.9000000000 0.1094731473 0.2188527671 2.0000000000 0.1944506521 0.3904741651 # ********************************************************